SlideShare a Scribd company logo

Geo1004 lecture 1_topology&topological_datamodels_final

•
•
P
Pirouz Nourian

My lecture notes for the course Geo1004: 3D modelling of built environment, MSc Geomatics, 2018

Engineering
1 of 71
Download to read offline
11 Topology On Topology and Topological Data Models in Geometric Modeling of Space Dr.ir. Pirouz Nourian Assistant Professor of Design Informatics Department of Architectural Engineering & Technology Faculty of Architecture and Built Environment Lecture notes of the course Geo1004: 3D Modelling of Built Environment, MSc Geomatics, TU Delft
22 • Geographical Modelling, replicating space using geometric objects with global coordinates • Geometrical Modelling, replicating space using geometric objects with local coordinates • Topological Modelling, replicating space using topological constructs such as simplexes • Graphical Modelling, replicating space using space objects and proximity links • Spectral Modelling, replicating space using Eigen space of differential operators Spectral Modelling for Spatial Network Analysis Abstractions of Space in Modelling Built Environments Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
33 • Geographical Modelling, replicating space using geometric objects with global coordinates • Geometrical Modelling, replicating space using geometric objects with local coordinates • Topological Modelling, replicating space using topological constructs such as simplexes • Graphical Modelling, replicating space using space objects and proximity links • Spectral Modelling, replicating space using Eigen space of differential operators Spectral Modelling for Spatial Network Analysis Abstractions of Space in Modelling Built Environments Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
44 0D: Point (the space therein everywhere resembles a 0-dimensional Euclidean Space) 1D: Curve (the space therein everywhere resembles a 1-dimensional Euclidean Space) 2D: Surface (the space therein everywhere resembles a 2-dimensional Euclidean Space) 3D: Solid (the space therein everywhere resembles a 3-dimensional Euclidean Space) Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
55 Discrete Models of Manifold Spaces Mesh Graph/Network Grid Discrete Representations of Continuous Space Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
66 Topology could be described as the study of topological invariants/properties of spatial objects, such as connectedness, continuity, closeness and dimension which remain invariant while applying topological two-directional continuous one-to-one or homoeomorphic transformations, such as those applied to a rubber-sheet or a piece of dough to change their shapes without tearing them apart, making cuts, making holes, gluing pieces. See also a glossary here: http://www.solitaryroad.com/c775.html http://www.gmanetwork.com/news/scitech/science/583886/when-is-a-coffee-mug-a-donut-topology-explains-it/story/ Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
77 a riddle: Can you wear a piece of dough as a T-Shirt? What do we do with the dough to wear it? Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
88 a riddle: Can you wear a piece of dough as a T-Shirt? What do we do with the dough to wear it? Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
99 a riddle: Can you wear a piece of dough as a T-Shirt? What do we do with the dough to wear it? Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
1010 a riddle: Can you wear a piece of dough as a T-Shirt? What do we do with the dough to wear it? Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
1111 Geometry without Dimensions / Rubber-Sheet Geometry Image from Encyclopaedia Britannica Why is topology important in spatial modelling?Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
1212 Geometry without Dimensions / Rubber-Sheet Geometry Escher’s Balcony, 1945 Escher’s Print Gallery, 1956Image from Encyclopaedia Britannica Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
1313 Here is a formal definition of what qualifies as a topological model: Think of it as an abstraction of Euclidean space in which neighbourhoods are open balls [open sets, the space inside the balls without the ball surfaces] around points. Concretely, a topology [can be loosely considered as a topological model] on a point set 𝕏 is a collection 𝒰 [that is e.g. a geometric model whose inner space is abstracted as a collection of such neighbourhoods] of subsets of 𝕏, called open sets, such that: (i) 𝕏 is open and the empty set ∅ is open; (ii) if 𝑈1 and 𝑈2 are open, then 𝑈1 ∩ 𝑈2 is open; //[i.e. the intersection of two open sets is open] (iii) if 𝑈𝑖 is open for all 𝑖 in some possibly infinite, possible uncountable, index set, then the union of all 𝑈𝑖 is open. //[i.e. the unison of open sets is open] From Computational Topology, An Introduction, Edelsbrunner, Herbert and Harer, John, 2010, p.15, annotations in brackets from Pirouz Nourian. Here is an example of what this definition could mean e.g. in case of modelling with point clouds: A User’s Guide to Topological Data Analysis Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
1414 Topology is a very broad subject area, in which at least two fields are relevant to us: Point-Set Topology (a.k.a. General Topology) Algebraic Topology (f.k.a. Combinatorial Topology) Point-Set Topology: Point-Set topology, also called set-theoretic topology or general topology, is the study of the general abstract nature of continuity or "closeness" on spaces. Basic point-set topological notions are ones like continuity, dimension, and connectedness. In the context of Point Set Topology we discuss: Nodes, Arcs, Disks, Balls (topological models use to describe the space inside Point Neighbourhoods, Curves, Surfaces, and Solids) Loosely speaking, this sense of topology is evident in the global structure of a geometric model. Algebraic Topology: Algebraic Topology, which includes Combinatorial Topology, is the study of “topological invariants” or ”topological properties” of combinatorial maps or decompositions of spaces such as [but not limited to] simplicial complexes. In the context of Algebraic Topology we discuss: Vertices, Edges, Faces, Bodies, (topological models used to tessellate/approximate/decompose Points, Curves, Surfaces, and Solids) Loosely speaking, this sense of topology is evident in the local structure of a geometric model. We shall see how both of these senses come together in one equation describing topological invariants. Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
1515 https://www.nature.com/articles/nphoton.2015.244 Rubber-Sheet Geometry? Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
1616 “If it looks like a duck, swims like a duck, and quacks like a duck, then it probably is a duck.” Image: DUCK: GETTY Images; ILLUSTRATION: MARTIN O'NEILL, from http://www.nature.com/nature/journal/v484/n7395/full/484451a.html?message-global=remove how do we model a curve in 3D space? Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
1717 1D 2D 3D Parametric Modelling: modelling objects explicitly using their homeomorphic topological space Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
1818 • 1D: Curves (t parameter) • 2D: Surfaces (u & v parameters) • (3D?): B-Reps (each face is a surface) NURBS Objects Non Uniform Rational Basis Splines are used for accurately modeling free-form geometries Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
1919 NURBS Representation Non Uniform Rational Basis Splines (NURBS) are used for modeling free-form geometries accurately Image courtesy of http://www.boatdesign.netImage courtesy of Wikimedia • An elegant mathematical description of a physical drafting aid as a (set of) parametric equation(s). Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
2020 Parametric Curves in General How do numeric weights correspond to physical weights? Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
2121 Splines in Computer Graphics All types of curves can be modeled as splines • Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
2222 Bezier interpolationModelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
2323 NURBS equations All from a summary by Markus Altmann: http://web.cs.wpi.edu/~matt/courses/cs563/talks/nurbs.html Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
2424 NURBS Representation Non Uniform Rational Basis Splines (NURBS) are used for modeling free-form geometries accurately • An elegant mathematical description of a physical drafting aid as a (set of) parametric equation(s). • offer one common mathematical form for both, standard analytical shapes (e.g. conics) and free form shapes; • provide the flexibility to design a large variety of shapes; • can be evaluated reasonably fast by numerically stable and accurate algorithms; • are invariant under affine as well as perspective transformations; • are generalizations of non-rational B-splines and non-rational and rational Bezier curves and surfaces. Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
2525 NURBS Surfaces Rectangular Patches Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
2626 Parametric Space Images courtesy of David Rutten, from Rhinoscript 101 Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
2727 Parametric Locations:Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
2828 Bilbao Guggenheim Museum Bus stop near Sebastiaansbrug Delft Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
2929 Curvature 𝐾 = 1 𝑅 One over the radius of the osculation (kissing) circle Why did people [for such a long time] believe that the Earth was flat? Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
3030 2D Curvature: Gaussian Images courtesy of Raja Issa, Essential Mathematics for Computational Design 𝐾 𝐺 = 𝑲 𝑚𝑖𝑛. 𝑲 𝑚𝑎𝑥 Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
3131 Gaussian Curvature 𝐾 𝐺 = 𝐾 𝑚𝑖𝑛 𝐾 𝑚𝑎𝑥 Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
3232 Continuity Images courtesy of Raja Issa, Essential Mathematics for Computational Design Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
3333 2D Curvature: Mean 𝐻 = 𝑲 𝑚𝑖𝑛 +𝑲 𝑚𝑎𝑥 2 Anish Kapoor, Marsyas, 2002, installation view, Tate Modern https://en.wikipedia.org/wiki/Mean_curvature Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
3434 Boundary RepresentationsModelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
3535 Boundary Representations • 0D: Point • 1D: Curve (represented by boundary Points) • 2D: Surface (represented by boundary Curves) • 3D: Solid (represented by boundary Surfaces) Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
3636 Imagine the Boundary of Manifolds • 0D: Point (the space therein everywhere resembles a 0-dimensional Euclidean Space) • 1D: Curve (the space therein everywhere resembles a 1-dimensional Euclidean Space) • 2D: Surface (the space therein everywhere resembles a 2-dimensional Euclidean Space) • 3D: Solid (the space therein everywhere resembles a 3-dimensional Euclidean Space) Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
3737 Boundary Representations • 0D: Point • 1D: Curve (represented by boundary Points) • 2D: Surface (represented by boundary Curves) • 3D: Solid (represented by boundary Surfaces) • Free-Form Parametric surfaces • Piecewise Linear: Polygon Meshes Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
3838 A simplex cell [actually the space inside it] is defined as the locus of points defined by linear interpolations of the corner vertices with barycentric coordinates denoted by 𝛼𝑖, as follows: 𝑣0, 𝑣1, … , 𝑣 𝑘 ∈ ℝ 𝑛 𝐶 = {𝛼0 𝑣0 + 𝛼1 𝑣1 + ⋯ . +𝛼 𝑘 𝑣 𝑘|𝛼𝑖 ≥ 0, 0 ≤ 𝑖 ≤ 𝑘, ෍ 𝑖 𝛼𝑖 = 1} • 0-simplex: Point • 1-simplex: Line • 2-simplex: Triangle • 3-simplex: Tetrahedron Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
3939 A simplex cell [actually the space inside it] is defined as the locus of points defined by linear interpolations of the corner vertices with barycentric coordinates denoted by 𝛼𝑖, as follows: 𝑣0, 𝑣1, … , 𝑣 𝑘 ∈ ℝ 𝑛 𝐶 = {𝛼0 𝑣0 + 𝛼1 𝑣1 + ⋯ . +𝛼 𝑘 𝑣 𝑘|𝛼𝑖 ≥ 0, 0 ≤ 𝑖 ≤ 𝑘, ෍ 𝑖 𝛼𝑖 = 1} http://beachpackagingdesign.com/boxvox/the-tetra-pak-t Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
4040 A simplex cell is defined as the locus of points defined by linear interpolations of the corner vertices with barycentric coordinates denoted by 𝛼𝑖, as follows: 𝑣0, 𝑣1, … , 𝑣 𝑘 ∈ ℝ 𝑛 𝐶 = {𝛼0 𝑣0 + 𝛼1 𝑣1 + ⋯ . +𝛼 𝑘 𝑣 𝑘|𝛼𝑖 ≥ 0, 0 ≤ 𝑖 ≤ 𝑘, ෍ 𝑖 𝛼𝑖 = 1} http://beachpackagingdesign.com/boxvox/the-tetra-pak-t A simplicial complex is a collection of simplices (simplexes) 𝐾 in ℝ 𝑛 such that: 1. Every face of a simplex of 𝐾 is in 𝐾, and 2. The intersection of any two simplexes of 𝐾 is a [generalized] face of each of them [with dimension 𝑘 − 1] Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
4141 https://sites.google.com/site/dengwirda/jigsaw https://www.researchgate.net/publication/287398529_An_E fficient_Approach_for_Solving_Mesh_Optimization_Proble ms_Using_Newton%27s_Method/figures?lo=1 Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
4242 • Example: • Important note: the geometry of faces is not stored; only rendered when needed! What you see on the screen is not necessarily what you store! Polygon Mesh Representation A light-weight model composed of points and a set of topological relations among them. Points >Vertices Lines > Edges Polygons > Faces Mesh T1: {0,4,3} T2:{0,1,4} Q1:{1,2,5,4} Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
4343 • The geometry of a Mesh can be represented by its points (known as [geometrical] vertices in Rhino), i.e. a ‘list’ of 3D points in ℝ3 • The topology of a Mesh can be represented based on its [topological] vertices, referring to a ‘set’ of 3D points in ℝ3 , there are multiple ways to describe how these vertices are spatially related (connected or adjacent) to one another and also to edges and faces of the Mesh • Same topology and different geometries: Mesh Mesh Geometry versus Mesh Topology T1: {0,4,3} T2:{0,1,4} Q1:{1,2,5,4}T1 T2 Q1 T1 T2 Q1 0 3 1 4 5 2 0 3 1 4 5 2 Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
4444 Triangulated Meshes Some preliminary definitions: fan, star, strip A triangle strip=:ABCDEF  ABC, CBD, CDE, and EDF A B C D E F A Closed Triangle Fan or a ‘star’=:ABCDEF A B C D E F  ABC, ACD, ADE, and AEF A Triangle Fan=: ABCDE AB C D E  ABC, ACD, and ADE Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
4545 Mesh Boundary Some preliminary definitions: free* vertices, free* edges • If an edge has less than two faces adjacent to it then it is considered free; • If a vertex is part of such an edge it is considered as free too. * In Rhinocommon free vertices/edges are referred to as naked. Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
4646 Mesh Border denoted as 𝝏 (𝒑𝒓𝒐𝒏𝒐𝒖𝒏𝒄𝒆𝒅 𝒂𝒔 ′ 𝒏𝒂𝒃𝒍𝒂′ 𝒐𝒓 ′𝒅𝒆𝒍′) • A 1D border is the set of connected edges incident to only one face of a 2- manifold, i.e. composed of free edges. We can conclude that: • If every vertex has a closed fan, or there is no edge of valence (number of neighbors) less than 2, the given manifold has no border. Example: a box! non-manifold border manifold borderno [1D] border Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
4747 Images courtesy of Dr. Ching-Kuang Shene from Michigan Technological University • if a Mesh is supposed to be a 2D manifold then it should meet these criteria: 1. each edge is incident to only one or two faces; and 2. the faces incident to a vertex form a closed or an open fan. Non manifold Mesh examples: (note why!) Manifold Mesh A 2-manifold [everywhere] locally resembles a flat surface • if a Mesh is supposed to be orientable then, it should be possible to find ‘compatible’ orientations for any two adjacent faces; in which, for each pair of adjacent faces, the common edge of the two faces has opposite orders. • Example: Möbius band is a 2D manifold Mesh that is non-printable. Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
4848 Image Source: http://prateekvjoshi.com/2014/11/16/homomorphism-vs homeomorphism/ Mesh Topology: Homeomorphism clay models that are all topologically equal! Images courtesy of Dr. Ching-Kuang Shene from Michigan Technological University Two 2-manifold Meshes A and B are homeomorphic if their surfaces can be transformed to one another by topological transformations (bending, twisting, stretching, scaling, etc.) without cutting and gluing. Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
4949 Mesh Topology: Homeomorphism Euler-Poincare Characteristic: key to homeomorphism Only for 2D boundary representations* Images courtesy of Dr. Ching-Kuang Shene from Michigan Technological University 𝜒 𝑀 = 𝑉 − 𝐸 + 𝐹 = 2 1 − 𝑔 − 𝜕 • 𝛿 is the number of [connected] borders • 𝑔 is the number of “genera” (pl. of genus) or holes • Irrespective of tessellation! 𝜒 𝑀 = 𝑉 − 𝐸 + 𝐹 = 2 1 − 2 − 0 = −2 * For 3D BReps the Euler characteristic equation will take the form of V-E+F-B Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
5050 Mesh Topology: Homeomorphism Euler-Poincare Characteristic: key to homeomorphism • Irrespective of tessellation! 𝜒 𝑀 = 𝑉 − 𝐸 + 𝐹 = −2 𝑓𝑜𝑟 𝑑𝑜𝑢𝑏𝑙𝑒 𝑡𝑜𝑟𝑖(ℎ𝑜𝑚𝑒𝑜𝑚𝑜𝑟𝑝ℎ𝑖𝑐 𝑡𝑜 𝑠𝑜𝑙𝑖𝑑𝑠 𝑤𝑖𝑡ℎ 2 ℎ𝑜𝑙𝑒𝑠) 𝜒 𝑀 = 𝑉 − 𝐸 + 𝐹 = 0 𝑓𝑜𝑟 𝑡𝑜𝑟𝑖 (ℎ𝑜𝑚𝑒𝑜𝑚𝑜𝑟𝑝ℎ𝑖𝑐 𝑡𝑜 𝑠𝑜𝑙𝑖𝑑𝑠 𝑤𝑖𝑡ℎ 1 ℎ𝑜𝑙𝑒) 𝜒 𝑀 = 𝑉 − 𝐸 + 𝐹 = 1 𝑓𝑜𝑟 𝑑𝑖𝑠𝑘𝑠 (ℎ𝑜𝑚𝑒𝑜𝑚𝑜𝑟𝑝ℎ𝑖𝑐 𝑡𝑜 𝑠𝑢𝑟𝑓𝑎𝑐𝑒𝑠) 𝜒 𝑀 = 𝑉 − 𝐸 + 𝐹 = 2 𝑓𝑜𝑟 𝑠𝑝ℎ𝑒𝑟𝑒𝑠 (ℎ𝑜𝑚𝑒𝑜𝑚𝑜𝑟𝑝ℎ𝑖𝑐 𝑡𝑜 𝑠𝑜𝑙𝑖𝑑𝑠 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 ℎ𝑜𝑙𝑒𝑠) Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
5151 Mesh Topology: Homeomorphism Euler-Poincare Characteristic: key to homeomorphism • Irrespective of tessellation! Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
5252 Valid Mesh (often expected to be a 2–Manifold Surface) Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
5353 Mesh Topology: Homeomorphism Euler-Poincare Characteristic: key to homeomorphism • N-Manifold/Non-Manifold: Each point of an n-dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension n • Riddle: The above definition implies that for mapping some local geographic features 2D maps are fine. What about the whole globe? That is not homoeomorphic to a rectangular surface! How do we do it then? Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
5454 Poincare Duality a pairing between k-dimensional features and dual features of dimension n-k in ℝ 𝑛 PRIMAL DUAL 0D vertex (e.g. as a point) 1D edge 1D edge (e.g. as a line segment) 0D vertex PRIMAL DUAL 0D vertex (e.g. a point) 2D face 1D edge (e.g. a line segment) 1D edge 2D face (e.g. a triangle or a pixel) 0D vertex PRIMAL DUAL 0D vertex (e.g. a point) 3D body 1D edge (e.g. a line segment) 2D face 2D face (e.g. a triangle or a pixel) 1D edge 3D body (e.g. a tetrahedron or a voxel) 0D vertex ℝ ℝ3 ℝ2 Configraphics: Graph Theoretical Methods of Design and Analysis of Spatial Configurations, Nourian, P, 2016 Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
5555 Poincare Duality a pairing between k-dimensional features and dual features of dimension n-k in ℝ 𝑛 PRIMAL DUAL 0D vertex (e.g. as a point) 1D edge 1D edge (e.g. as a line segment) 0D vertex ℝ A hypothetical street network (a), a Junction-to-Junction adjacency graph (b) versus a Street-to-Street adjacency graph (c), both ‘undirected’, after Batty (Batty, 2004): red dots represent graph nodes, and blue arcs represent graph links. Batty, M., 2004. A New Theory of Space Syntax. CASA Working Paper Series, March. Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
5656 Poincare Duality a pairing between k-dimensional features and dual features of dimension n-k in ℝ 𝑛 PRIMAL DUAL 0D vertex (e.g. a point) 2D face 1D edge (e.g. a line segment) 1D edge 2D face (e.g. a triangle or a pixel) 0D vertex ℝ2 A Voronoi tessellation of 2D space and its dual that is a Delaunay triangulation Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
5757 Poincare Duality a pairing between k-dimensional features and dual features of dimension n-k in ℝ 𝑛 PRIMAL DUAL 0D vertex (e.g. a point) 3D body 1D edge (e.g. a line segment) 2D face 2D face (e.g. a triangle or a pixel) 1D edge 3D body (e.g. a tetrahedron or a voxel) 0D vertex ℝ3 representing adjacencies between 3D cells or bodies via their dual vertices (Lee, 2001) Lee, J., 2001. 3D Data Model for Representing Topological Relations of Urban Features. San Diego, CA, s.n. Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
5858 Mesh Topological Structure • Mesh Topological Data Models: Face-Vertex: e.g. 𝐹0 = {𝑉0, 𝑉5, 𝑉4} & 𝑉0 ∈ 𝐹0, 𝐹1, 𝐹12, 𝐹15, 𝐹7 Images courtesy of David Dorfman, from Wikipedia Face-Vertex (as implemented in Rhinoceros) Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
5959 Vertex-Vertex: e.g. 𝑉0~{𝑉1, 𝑉4, 𝑉3} Face-Vertex: e.g. 𝐹0 = 𝑉0, 𝑉5, 𝑉4 , 𝑉0 ∈ 𝐹0, 𝐹1, 𝐹12, 𝐹15, 𝐹7 Winged-Edge: e.g. 𝐹0 = 𝐸4, 𝐸8, 𝐸9 , 𝐸0 = 𝑉0, 𝑉1 , 𝐸0~ 𝐹1, 𝐹12 , 𝐸0~ 𝐸9, 𝐸23, 𝐸10, 𝐸20 Half-Edge: each half edge has a twin edge in opposite direction, a previous and a next Mesh Topological Structures Image courtesy of David Dorfman, from Wikipedia What is explicitly stored as topology of a Mesh: E.g. Face- Vertex (as implemented in Rhinoceros) http://doc.cgal.org/latest/HalfedgeDS/index.html Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
6060 Mesh Topological Structures Which topological relations are explicitly stored in the data model V E F V VV VE VF E EV EE EF F FV FE FF Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
6161 Mesh Topological Structure • Mesh Topological Data Models: Face-Vertex Example: http://4.rhino3d.com/5/rhinocommon/html/AllMembers_T_Rhino_Geometry_Collections_MeshTopologyVertexList.htm Face-Vertex (as implemented in Rhinoceros) Name Description ConnectedFaces Gets all faces that are connected to a given vertex. ConnectedTopologyVertices(Int32) Gets all topological vertices that are connected to a given vertex. ConnectedTopologyVertices(Int32, Boolean) Gets all topological vertices that are connected to a given vertex. The MeshTopologyVertexList type exposes the following members. Half-Edge: Half-Edge Mesh Data Structure, example implementation by Daniel Piker: http://www.grasshopper3d.com/group/plankton Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
6262 Mesh Geometry • Polygon vs Face • Triangulate • Quadrangulate Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
6363 Mesh Intersections • Boolean operation on Meshes: 1. 𝐴 ∪ 𝐵: Boolean Union 2. 𝐴 − 𝐵: Boolean Difference 3. 𝐴 ∩ 𝐵: Boolean Intersection Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
6464 Normal Vectors of a Mesh • Topological Vertices versus Geometrical Points • Joining Mesh objects: What is a Mesh box? • Welding Meshes: how does it work? • Face Normal versus Vertex Normal Where do they come from and what do they represent? Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
6565 How to Compute Mesh Normals? 00, 00 ),( vvuuv p u p vun              1 ( ) ( ) 0 ( )( ) N x i next i i next i i N y y z z      1 ( ) ( ) 0 ( )( ) N z i next i i next i i N x x y y      1 ( ) ( ) 0 ( )( ) N y i next i i next i i N z z x x      Martin Newell at Utah (remember the teapot?) Why? Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
6666 Repairing/Reconstructing Geometry • Example: Coons’ Patch • Code it and get bonus points! Image courtesy of CVG Lab Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
6767 Mesh Smoothing for (int k = 0; k <= L - 1; k++) { List<Point3d> SmoothV = new List<Point3d>(); for (int i = 0; i <= M.Vertices.Count - 1; i++) { List<Point3d> Neighbours = Neighbors(M, i); Point3d NVertex = new Point3d(0, 0, 0); foreach (point3d neighbor in Neighbours ) { NVertex = NVertex + neighbor; } NVertex = (1 / Ngh.Count) * NVertex; SmoothV.Add(NVertex); } M.Vertices.Clear(); M.Vertices.AddVertices(SmoothV); A = M; } Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
6868 Topological Transformation of Meshes http://www.idz.ro/grasshopper-mesh-relaxation-component-update/ Euclidean to Hyperbolic By Dynamic Relaxation Algorithms Euclidean to Elliptical By Subdivision Algorithms https://www.renderosity.com/mod/forumpro/?thread_id=2651940 Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
6969 • Geometry: Point (0D), Line (1D), Plane(2D), Hyper-Plane(3D) (Euclidean sub-spaces) • Geometry: Point (0D), Curve(1D), Surface(2D), Solid (3D) [differential] • Geometry: Point (0D), Line-Segment(1D), Polygon(2D), Polyhedron (3D) [piecewise linear] • Geometry: Point(0D), Line-Segment(1D), Triangle(2D), Tetrahedron(3D) [simplexes] • Topology: Node(0D), Arc(1D), Disk(2D), Ball(3D) [general/point-set] • Topology: Vertex(0D), Edge(1D), Face(2D), Body(3D) [combinatorial/algebraic] • Graph Theory: Object, Link, (and n-Cliques) • Fields: Pixel, Voxel, Raster2D, Raster 3D [scalar] On terminology (CAD, CG, GIS, Math, etc.) Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
7070 On terminology n-D features Graph Theory Topology Geometry 0D Object Vertex Point 1D Link Edge Line (Curve) 2D Cycle* Face Polygon (Surface) 3D Clique* Body Polyhedron (Solid) Configraphics: Graph Theoretical Methods of Design and Analysis of Spatial Configurations, Nourian, P, 2016 Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
7171 Questions: p.nourian@tudelft.nl

Recommended

Ar1 twf030 lecture2.1: Geometry and Topology in Computational Design
Ar1 twf030 lecture2.1: Geometry and Topology in Computational DesignAr1 twf030 lecture2.1: Geometry and Topology in Computational Design
Ar1 twf030 lecture2.1: Geometry and Topology in Computational DesignPirouz Nourian
 
Mesh final pzn_geo1004_2015_f3_2017
Mesh final pzn_geo1004_2015_f3_2017Mesh final pzn_geo1004_2015_f3_2017
Mesh final pzn_geo1004_2015_f3_2017Pirouz Nourian
 
Polygon Mesh Representation
Polygon Mesh RepresentationPolygon Mesh Representation
Polygon Mesh RepresentationPirouz Nourian
 
On NURBS Geometry Representation in 3D modelling
On NURBS Geometry Representation in 3D modellingOn NURBS Geometry Representation in 3D modelling
On NURBS Geometry Representation in 3D modellingPirouz Nourian
 
Polygon mesh
Polygon meshPolygon mesh
Polygon meshKABILESH RAMAR
 
object 3d(1)
object 3d(1)object 3d(1)
object 3d(1)HiteshJain007
 
ppt of solid modeling for cad
ppt of solid modeling for cadppt of solid modeling for cad
ppt of solid modeling for cadAyush Upadhyay
 
Solid Modeling Schemes CAM
Solid Modeling Schemes CAMSolid Modeling Schemes CAM
Solid Modeling Schemes CAMCDO
 

Recommended

Ar1 twf030 lecture2.1: Geometry and Topology in Computational Design
Ar1 twf030 lecture2.1: Geometry and Topology in Computational DesignAr1 twf030 lecture2.1: Geometry and Topology in Computational Design
Ar1 twf030 lecture2.1: Geometry and Topology in Computational DesignPirouz Nourian
 
Mesh final pzn_geo1004_2015_f3_2017
Mesh final pzn_geo1004_2015_f3_2017Mesh final pzn_geo1004_2015_f3_2017
Mesh final pzn_geo1004_2015_f3_2017Pirouz Nourian
 
Polygon Mesh Representation
Polygon Mesh RepresentationPolygon Mesh Representation
Polygon Mesh RepresentationPirouz Nourian
 
On NURBS Geometry Representation in 3D modelling
On NURBS Geometry Representation in 3D modellingOn NURBS Geometry Representation in 3D modelling
On NURBS Geometry Representation in 3D modellingPirouz Nourian
 
Polygon mesh
Polygon meshPolygon mesh
Polygon meshKABILESH RAMAR
 
object 3d(1)
object 3d(1)object 3d(1)
object 3d(1)HiteshJain007
 
ppt of solid modeling for cad
ppt of solid modeling for cadppt of solid modeling for cad
ppt of solid modeling for cadAyush Upadhyay
 
Solid Modeling Schemes CAM
Solid Modeling Schemes CAMSolid Modeling Schemes CAM
Solid Modeling Schemes CAMCDO
 
presentation on solid manipulation in computer aided design
presentation on solid manipulation in computer aided designpresentation on solid manipulation in computer aided design
presentation on solid manipulation in computer aided designRakshit vadi
 
Solid modelling cg
Solid modelling cgSolid modelling cg
Solid modelling cgNareek
 
427lects
427lects427lects
427lectsPraveen Kumar
 
On mesh
On meshOn mesh
On meshThenraja Vettivelraj
 
Solid modeling
Solid modelingSolid modeling
Solid modelingDhruv Shah
 
CSG
CSGCSG
CSGselvakumar948
 
IMPROVEMENTS OF THE ANALYSIS OF HUMAN ACTIVITY USING ACCELERATION RECORD OF E...
IMPROVEMENTS OF THE ANALYSIS OF HUMAN ACTIVITY USING ACCELERATION RECORD OF E...IMPROVEMENTS OF THE ANALYSIS OF HUMAN ACTIVITY USING ACCELERATION RECORD OF E...
IMPROVEMENTS OF THE ANALYSIS OF HUMAN ACTIVITY USING ACCELERATION RECORD OF E...sipij
 
5 geometric-modeling-ppt-university-of-victoria
5 geometric-modeling-ppt-university-of-victoria5 geometric-modeling-ppt-university-of-victoria
5 geometric-modeling-ppt-university-of-victoriaRaghu Gadde
 
Solid modeling
Solid modelingSolid modeling
Solid modelingKRvEsL
 
Multi-legged Robot Walking Strategies, with an Emphasis on Image-based Methods
Multi-legged Robot Walking Strategies, with an Emphasis on Image-based MethodsMulti-legged Robot Walking Strategies, with an Emphasis on Image-based Methods
Multi-legged Robot Walking Strategies, with an Emphasis on Image-based MethodsKazi Mostafa
 
1288
12881288
1288Jean-Dominique FAVREAU
 
Template Matching - Pattern Recognition
Template Matching - Pattern RecognitionTemplate Matching - Pattern Recognition
Template Matching - Pattern RecognitionMustafa Salam
 
Enhancing the Design pattern Framework of Robots Object Selection Mechanism -...
Enhancing the Design pattern Framework of Robots Object Selection Mechanism -...Enhancing the Design pattern Framework of Robots Object Selection Mechanism -...
Enhancing the Design pattern Framework of Robots Object Selection Mechanism -...INFOGAIN PUBLICATION
 
Unit 3 visual realism
Unit 3 visual realismUnit 3 visual realism
Unit 3 visual realismJavith Saleem
 
Template matching03
Template matching03Template matching03
Template matching03amitkhanna1991
 
A survey on graph kernels
A survey on graph kernelsA survey on graph kernels
A survey on graph kernelsvincyy
 
M ESH S IMPLIFICATION V IA A V OLUME C OST M EASURE
M ESH S IMPLIFICATION V IA A V OLUME C OST M EASUREM ESH S IMPLIFICATION V IA A V OLUME C OST M EASURE
M ESH S IMPLIFICATION V IA A V OLUME C OST M EASUREijcga
 
Lec10 matching
Lec10 matchingLec10 matching
Lec10 matchingSuravet Konsetthee
 
1422798749.2779lecture 5
1422798749.2779lecture 51422798749.2779lecture 5
1422798749.2779lecture 5SRM UNIVERSITY, RAMAPURAM
 
WEFT_ResearchBook
WEFT_ResearchBookWEFT_ResearchBook
WEFT_ResearchBookToon Van Mieghem
 
Mod Two Homology and Cohomology (Jean Claude)
Mod Two Homology and Cohomology (Jean Claude)Mod Two Homology and Cohomology (Jean Claude)
Mod Two Homology and Cohomology (Jean Claude)Justin Hoang
 
Paracompact Space Literature Survey
Paracompact Space Literature SurveyParacompact Space Literature Survey
Paracompact Space Literature Surveysumitbajpeyee
 

More Related Content

What's hot

presentation on solid manipulation in computer aided design
presentation on solid manipulation in computer aided designpresentation on solid manipulation in computer aided design
presentation on solid manipulation in computer aided designRakshit vadi
 
Solid modelling cg
Solid modelling cgSolid modelling cg
Solid modelling cgNareek
 
Solid modeling
Solid modelingSolid modeling
Solid modelingDhruv Shah
 
IMPROVEMENTS OF THE ANALYSIS OF HUMAN ACTIVITY USING ACCELERATION RECORD OF E...
IMPROVEMENTS OF THE ANALYSIS OF HUMAN ACTIVITY USING ACCELERATION RECORD OF E...IMPROVEMENTS OF THE ANALYSIS OF HUMAN ACTIVITY USING ACCELERATION RECORD OF E...
IMPROVEMENTS OF THE ANALYSIS OF HUMAN ACTIVITY USING ACCELERATION RECORD OF E...sipij
 
5 geometric-modeling-ppt-university-of-victoria
5 geometric-modeling-ppt-university-of-victoria5 geometric-modeling-ppt-university-of-victoria
5 geometric-modeling-ppt-university-of-victoriaRaghu Gadde
 
Solid modeling
Solid modelingSolid modeling
Solid modelingKRvEsL
 
Multi-legged Robot Walking Strategies, with an Emphasis on Image-based Methods
Multi-legged Robot Walking Strategies, with an Emphasis on Image-based MethodsMulti-legged Robot Walking Strategies, with an Emphasis on Image-based Methods
Multi-legged Robot Walking Strategies, with an Emphasis on Image-based MethodsKazi Mostafa
 
Template Matching - Pattern Recognition
Template Matching - Pattern RecognitionTemplate Matching - Pattern Recognition
Template Matching - Pattern RecognitionMustafa Salam
 
Enhancing the Design pattern Framework of Robots Object Selection Mechanism -...
Enhancing the Design pattern Framework of Robots Object Selection Mechanism -...Enhancing the Design pattern Framework of Robots Object Selection Mechanism -...
Enhancing the Design pattern Framework of Robots Object Selection Mechanism -...INFOGAIN PUBLICATION
 
Unit 3 visual realism
Unit 3 visual realismUnit 3 visual realism
Unit 3 visual realismJavith Saleem
 
A survey on graph kernels
A survey on graph kernelsA survey on graph kernels
A survey on graph kernelsvincyy
 
M ESH S IMPLIFICATION V IA A V OLUME C OST M EASURE
M ESH S IMPLIFICATION V IA A V OLUME C OST M EASUREM ESH S IMPLIFICATION V IA A V OLUME C OST M EASURE
M ESH S IMPLIFICATION V IA A V OLUME C OST M EASUREijcga
 

What's hot (19)

presentation on solid manipulation in computer aided design
presentation on solid manipulation in computer aided designpresentation on solid manipulation in computer aided design
presentation on solid manipulation in computer aided design
 
Solid modelling cg
Solid modelling cgSolid modelling cg
Solid modelling cg
 
427lects
427lects427lects
427lects
 
On mesh
On meshOn mesh
On mesh
 
Solid modeling
Solid modelingSolid modeling
Solid modeling
 
CSG
CSGCSG
CSG
 
IMPROVEMENTS OF THE ANALYSIS OF HUMAN ACTIVITY USING ACCELERATION RECORD OF E...
IMPROVEMENTS OF THE ANALYSIS OF HUMAN ACTIVITY USING ACCELERATION RECORD OF E...IMPROVEMENTS OF THE ANALYSIS OF HUMAN ACTIVITY USING ACCELERATION RECORD OF E...
IMPROVEMENTS OF THE ANALYSIS OF HUMAN ACTIVITY USING ACCELERATION RECORD OF E...
 
5 geometric-modeling-ppt-university-of-victoria
5 geometric-modeling-ppt-university-of-victoria5 geometric-modeling-ppt-university-of-victoria
5 geometric-modeling-ppt-university-of-victoria
 
Solid modeling
Solid modelingSolid modeling
Solid modeling
 
Multi-legged Robot Walking Strategies, with an Emphasis on Image-based Methods
Multi-legged Robot Walking Strategies, with an Emphasis on Image-based MethodsMulti-legged Robot Walking Strategies, with an Emphasis on Image-based Methods
Multi-legged Robot Walking Strategies, with an Emphasis on Image-based Methods
 
1288
12881288
1288
 
Template Matching - Pattern Recognition
Template Matching - Pattern RecognitionTemplate Matching - Pattern Recognition
Template Matching - Pattern Recognition
 
Enhancing the Design pattern Framework of Robots Object Selection Mechanism -...
Enhancing the Design pattern Framework of Robots Object Selection Mechanism -...Enhancing the Design pattern Framework of Robots Object Selection Mechanism -...
Enhancing the Design pattern Framework of Robots Object Selection Mechanism -...
 
Unit 3 visual realism
Unit 3 visual realismUnit 3 visual realism
Unit 3 visual realism
 
Template matching03
Template matching03Template matching03
Template matching03
 
A survey on graph kernels
A survey on graph kernelsA survey on graph kernels
A survey on graph kernels
 
M ESH S IMPLIFICATION V IA A V OLUME C OST M EASURE
M ESH S IMPLIFICATION V IA A V OLUME C OST M EASUREM ESH S IMPLIFICATION V IA A V OLUME C OST M EASURE
M ESH S IMPLIFICATION V IA A V OLUME C OST M EASURE
 
Lec10 matching
Lec10 matchingLec10 matching
Lec10 matching
 
1422798749.2779lecture 5
1422798749.2779lecture 51422798749.2779lecture 5
1422798749.2779lecture 5
 

Similar to Geo1004 lecture 1_topology&amp;topological_datamodels_final

Mod Two Homology and Cohomology (Jean Claude)
Mod Two Homology and Cohomology (Jean Claude)Mod Two Homology and Cohomology (Jean Claude)
Mod Two Homology and Cohomology (Jean Claude)Justin Hoang
 
Paracompact Space Literature Survey
Paracompact Space Literature SurveyParacompact Space Literature Survey
Paracompact Space Literature Surveysumitbajpeyee
 
IEEE CEC 2013 Tutorial on Geometry of Evolutionary Algorithms
IEEE CEC 2013 Tutorial on Geometry of Evolutionary AlgorithmsIEEE CEC 2013 Tutorial on Geometry of Evolutionary Algorithms
IEEE CEC 2013 Tutorial on Geometry of Evolutionary AlgorithmsAlbertoMoraglio
 
ODSC India 2018: Topological space creation &amp; Clustering at BigData scale
ODSC India 2018: Topological space creation &amp; Clustering at BigData scaleODSC India 2018: Topological space creation &amp; Clustering at BigData scale
ODSC India 2018: Topological space creation &amp; Clustering at BigData scaleKuldeep Jiwani
 
Selected papers-2014 Topology Conference
Selected papers-2014 Topology ConferenceSelected papers-2014 Topology Conference
Selected papers-2014 Topology ConferenceSpiros Louvros
 
Analysis and Enhancement of Algorithms in Computational Geometry
Analysis and Enhancement of Algorithms in Computational GeometryAnalysis and Enhancement of Algorithms in Computational Geometry
Analysis and Enhancement of Algorithms in Computational GeometryKasun Ranga Wijeweera
 
Coates: topological approximations for spatial representation
Coates: topological approximations for spatial representationCoates: topological approximations for spatial representation
Coates: topological approximations for spatial representationArchiLab 7
 
Conceptual Spaces for Cognitive Architectures: A Lingua Franca for Different ...
Conceptual Spaces for Cognitive Architectures: A Lingua Franca for Different ...Conceptual Spaces for Cognitive Architectures: A Lingua Franca for Different ...
Conceptual Spaces for Cognitive Architectures: A Lingua Franca for Different ...Antonio Lieto
 
Aitf 2014 pem_introduction_presentation_feb28_ram_version2
Aitf 2014 pem_introduction_presentation_feb28_ram_version2Aitf 2014 pem_introduction_presentation_feb28_ram_version2
Aitf 2014 pem_introduction_presentation_feb28_ram_version2Bob MacMillan
 
Lexie.IGARSS11.pptx
Lexie.IGARSS11.pptxLexie.IGARSS11.pptx
Lexie.IGARSS11.pptxgrssieee
 
Lexie.IGARSS11.pptx
Lexie.IGARSS11.pptxLexie.IGARSS11.pptx
Lexie.IGARSS11.pptxgrssieee
 
Lexie.IGARSS11.v3.pptx
Lexie.IGARSS11.v3.pptxLexie.IGARSS11.v3.pptx
Lexie.IGARSS11.v3.pptxgrssieee
 
DigitalGeometry.ppt
DigitalGeometry.pptDigitalGeometry.ppt
DigitalGeometry.pptSushmaG48
 

Similar to Geo1004 lecture 1_topology&amp;topological_datamodels_final (20)

WEFT_ResearchBook
WEFT_ResearchBookWEFT_ResearchBook
WEFT_ResearchBook
 
Mod Two Homology and Cohomology (Jean Claude)
Mod Two Homology and Cohomology (Jean Claude)Mod Two Homology and Cohomology (Jean Claude)
Mod Two Homology and Cohomology (Jean Claude)
 
Paracompact Space Literature Survey
Paracompact Space Literature SurveyParacompact Space Literature Survey
Paracompact Space Literature Survey
 
IEEE CEC 2013 Tutorial on Geometry of Evolutionary Algorithms
IEEE CEC 2013 Tutorial on Geometry of Evolutionary AlgorithmsIEEE CEC 2013 Tutorial on Geometry of Evolutionary Algorithms
IEEE CEC 2013 Tutorial on Geometry of Evolutionary Algorithms
 
TOPOLOGY
TOPOLOGYTOPOLOGY
TOPOLOGY
 
ODSC India 2018: Topological space creation &amp; Clustering at BigData scale
ODSC India 2018: Topological space creation &amp; Clustering at BigData scaleODSC India 2018: Topological space creation &amp; Clustering at BigData scale
ODSC India 2018: Topological space creation &amp; Clustering at BigData scale
 
Selected papers-2014 Topology Conference
Selected papers-2014 Topology ConferenceSelected papers-2014 Topology Conference
Selected papers-2014 Topology Conference
 
Calculus volume 1
Calculus volume 1Calculus volume 1
Calculus volume 1
 
Locus ppt .pptx
Locus ppt .pptxLocus ppt .pptx
Locus ppt .pptx
 
Analysis and Enhancement of Algorithms in Computational Geometry
Analysis and Enhancement of Algorithms in Computational GeometryAnalysis and Enhancement of Algorithms in Computational Geometry
Analysis and Enhancement of Algorithms in Computational Geometry
 
Coates: topological approximations for spatial representation
Coates: topological approximations for spatial representationCoates: topological approximations for spatial representation
Coates: topological approximations for spatial representation
 
Conceptual Spaces for Cognitive Architectures: A Lingua Franca for Different ...
Conceptual Spaces for Cognitive Architectures: A Lingua Franca for Different ...Conceptual Spaces for Cognitive Architectures: A Lingua Franca for Different ...
Conceptual Spaces for Cognitive Architectures: A Lingua Franca for Different ...
 
10.1.1.17.1245
10.1.1.17.124510.1.1.17.1245
10.1.1.17.1245
 
Aitf 2014 pem_introduction_presentation_feb28_ram_version2
Aitf 2014 pem_introduction_presentation_feb28_ram_version2Aitf 2014 pem_introduction_presentation_feb28_ram_version2
Aitf 2014 pem_introduction_presentation_feb28_ram_version2
 
Lexie.IGARSS11.pptx
Lexie.IGARSS11.pptxLexie.IGARSS11.pptx
Lexie.IGARSS11.pptx
 
Lexie.IGARSS11.pptx
Lexie.IGARSS11.pptxLexie.IGARSS11.pptx
Lexie.IGARSS11.pptx
 
hodge-feec-pres-defense
hodge-feec-pres-defensehodge-feec-pres-defense
hodge-feec-pres-defense
 
Variograms
VariogramsVariograms
Variograms
 
Lexie.IGARSS11.v3.pptx
Lexie.IGARSS11.v3.pptxLexie.IGARSS11.v3.pptx
Lexie.IGARSS11.v3.pptx
 
DigitalGeometry.ppt
DigitalGeometry.pptDigitalGeometry.ppt
DigitalGeometry.ppt
 

More from Pirouz Nourian

Ar1 twf030 lecture3.1: Design Optimization
Ar1 twf030 lecture3.1: Design OptimizationAr1 twf030 lecture3.1: Design Optimization
Ar1 twf030 lecture3.1: Design OptimizationPirouz Nourian
 
Syntactic space syntax4generativedesign
Syntactic space syntax4generativedesignSyntactic space syntax4generativedesign
Syntactic space syntax4generativedesignPirouz Nourian
 
Tudelft stramien 16_9_on_optimization
Tudelft stramien 16_9_on_optimizationTudelft stramien 16_9_on_optimization
Tudelft stramien 16_9_on_optimizationPirouz Nourian
 
Intro computational design_mega2016_1_with_recommendedplugins
Intro computational design_mega2016_1_with_recommendedpluginsIntro computational design_mega2016_1_with_recommendedplugins
Intro computational design_mega2016_1_with_recommendedpluginsPirouz Nourian
 
Point Cloud Segmentation for 3D Reconstruction
Point Cloud Segmentation for 3D ReconstructionPoint Cloud Segmentation for 3D Reconstruction
Point Cloud Segmentation for 3D ReconstructionPirouz Nourian
 
Preliminaries of Analytic Geometry and Linear Algebra 3D modelling
Preliminaries of Analytic Geometry and Linear Algebra 3D modellingPreliminaries of Analytic Geometry and Linear Algebra 3D modelling
Preliminaries of Analytic Geometry and Linear Algebra 3D modellingPirouz Nourian
 
Point Cloud Processing: Estimating Normal Vectors and Curvature Indicators us...
Point Cloud Processing: Estimating Normal Vectors and Curvature Indicators us...Point Cloud Processing: Estimating Normal Vectors and Curvature Indicators us...
Point Cloud Processing: Estimating Normal Vectors and Curvature Indicators us...Pirouz Nourian
 

More from Pirouz Nourian (10)

Ar1 twf030 lecture3.1: Design Optimization
Ar1 twf030 lecture3.1: Design OptimizationAr1 twf030 lecture3.1: Design Optimization
Ar1 twf030 lecture3.1: Design Optimization
 
Ar1 twf030 lecture2.2
Ar1 twf030 lecture2.2Ar1 twf030 lecture2.2
Ar1 twf030 lecture2.2
 
Ar1 twf030 lecture1.1
Ar1 twf030 lecture1.1Ar1 twf030 lecture1.1
Ar1 twf030 lecture1.1
 
Ar1 twf030 lecture1.2
Ar1 twf030 lecture1.2Ar1 twf030 lecture1.2
Ar1 twf030 lecture1.2
 
Syntactic space syntax4generativedesign
Syntactic space syntax4generativedesignSyntactic space syntax4generativedesign
Syntactic space syntax4generativedesign
 
Tudelft stramien 16_9_on_optimization
Tudelft stramien 16_9_on_optimizationTudelft stramien 16_9_on_optimization
Tudelft stramien 16_9_on_optimization
 
Intro computational design_mega2016_1_with_recommendedplugins
Intro computational design_mega2016_1_with_recommendedpluginsIntro computational design_mega2016_1_with_recommendedplugins
Intro computational design_mega2016_1_with_recommendedplugins
 
Point Cloud Segmentation for 3D Reconstruction
Point Cloud Segmentation for 3D ReconstructionPoint Cloud Segmentation for 3D Reconstruction
Point Cloud Segmentation for 3D Reconstruction
 
Preliminaries of Analytic Geometry and Linear Algebra 3D modelling
Preliminaries of Analytic Geometry and Linear Algebra 3D modellingPreliminaries of Analytic Geometry and Linear Algebra 3D modelling
Preliminaries of Analytic Geometry and Linear Algebra 3D modelling
 
Point Cloud Processing: Estimating Normal Vectors and Curvature Indicators us...
Point Cloud Processing: Estimating Normal Vectors and Curvature Indicators us...Point Cloud Processing: Estimating Normal Vectors and Curvature Indicators us...
Point Cloud Processing: Estimating Normal Vectors and Curvature Indicators us...
 

Recently uploaded

Effects of rheological properties on mixing
Effects of rheological properties on mixingEffects of rheological properties on mixing
Effects of rheological properties on mixingviprabot1
 
Biology for Computer Engineers Course Handout.pptx
Biology for Computer Engineers Course Handout.pptxBiology for Computer Engineers Course Handout.pptx
Biology for Computer Engineers Course Handout.pptxDeepakSakkari2
 
VICTOR MAESTRE RAMIREZ - Planetary Defender on NASA's Double Asteroid Redirec...
VICTOR MAESTRE RAMIREZ - Planetary Defender on NASA's Double Asteroid Redirec...VICTOR MAESTRE RAMIREZ - Planetary Defender on NASA's Double Asteroid Redirec...
VICTOR MAESTRE RAMIREZ - Planetary Defender on NASA's Double Asteroid Redirec...VICTOR MAESTRE RAMIREZ
 
Electronically Controlled suspensions system .pdf
Electronically Controlled suspensions system .pdfElectronically Controlled suspensions system .pdf
Electronically Controlled suspensions system .pdfme23b1001
 
Churning of Butter, Factors affecting .
Churning of Butter, Factors affecting .Churning of Butter, Factors affecting .
Churning of Butter, Factors affecting .Satyam Kumar
 
Introduction to Machine Learning Unit-3 for II MECH
Introduction to Machine Learning Unit-3 for II MECHIntroduction to Machine Learning Unit-3 for II MECH
Introduction to Machine Learning Unit-3 for II MECHC Sai Kiran
 
Decoding Kotlin - Your guide to solving the mysterious in Kotlin.pptx
Decoding Kotlin - Your guide to solving the mysterious in Kotlin.pptxDecoding Kotlin - Your guide to solving the mysterious in Kotlin.pptx
Decoding Kotlin - Your guide to solving the mysterious in Kotlin.pptxJoão Esperancinha
 
Risk Assessment For Installation of Drainage Pipes.pdf
Risk Assessment For Installation of Drainage Pipes.pdfRisk Assessment For Installation of Drainage Pipes.pdf
Risk Assessment For Installation of Drainage Pipes.pdfROCENODodongVILLACER
 
Call Us ≽ 8377877756 ≼ Call Girls In Shastri Nagar (Delhi)
Call Us ≽ 8377877756 ≼ Call Girls In Shastri Nagar (Delhi)Call Us ≽ 8377877756 ≼ Call Girls In Shastri Nagar (Delhi)
Call Us ≽ 8377877756 ≼ Call Girls In Shastri Nagar (Delhi)dollysharma2066
 
Software and Systems Engineering Standards: Verification and Validation of Sy...
Software and Systems Engineering Standards: Verification and Validation of Sy...Software and Systems Engineering Standards: Verification and Validation of Sy...
Software and Systems Engineering Standards: Verification and Validation of Sy...VICTOR MAESTRE RAMIREZ
 
Gfe Mayur Vihar Call Girls Service WhatsApp -> 9999965857 Available 24x7 ^ De...
Gfe Mayur Vihar Call Girls Service WhatsApp -> 9999965857 Available 24x7 ^ De...Gfe Mayur Vihar Call Girls Service WhatsApp -> 9999965857 Available 24x7 ^ De...
Gfe Mayur Vihar Call Girls Service WhatsApp -> 9999965857 Available 24x7 ^ De...srsj9000
 
CCS355 Neural Network & Deep Learning Unit II Notes with Question bank .pdf
CCS355 Neural Network & Deep Learning Unit II Notes with Question bank .pdfCCS355 Neural Network & Deep Learning Unit II Notes with Question bank .pdf
CCS355 Neural Network & Deep Learning Unit II Notes with Question bank .pdfAsst.prof M.Gokilavani
 
pipeline in computer architecture design
pipeline in computer architecture designpipeline in computer architecture design
pipeline in computer architecture designssuser87fa0c1
 
Call Girls Narol 7397865700 Independent Call Girls
Call Girls Narol 7397865700 Independent Call GirlsCall Girls Narol 7397865700 Independent Call Girls
Call Girls Narol 7397865700 Independent Call Girlsssuser7cb4ff
 
Call Girls Delhi {Jodhpur} 9711199012 high profile service
Call Girls Delhi {Jodhpur} 9711199012 high profile serviceCall Girls Delhi {Jodhpur} 9711199012 high profile service
Call Girls Delhi {Jodhpur} 9711199012 high profile servicerehmti665
 
Oxy acetylene welding presentation note.
Oxy acetylene welding presentation note.Oxy acetylene welding presentation note.
Oxy acetylene welding presentation note.eptoze12
 
CCS355 Neural Networks & Deep Learning Unit 1 PDF notes with Question bank .pdf
CCS355 Neural Networks & Deep Learning Unit 1 PDF notes with Question bank .pdfCCS355 Neural Networks & Deep Learning Unit 1 PDF notes with Question bank .pdf
CCS355 Neural Networks & Deep Learning Unit 1 PDF notes with Question bank .pdfAsst.prof M.Gokilavani
 

Recently uploaded (20)

Effects of rheological properties on mixing
Effects of rheological properties on mixingEffects of rheological properties on mixing
Effects of rheological properties on mixing
 
Biology for Computer Engineers Course Handout.pptx
Biology for Computer Engineers Course Handout.pptxBiology for Computer Engineers Course Handout.pptx
Biology for Computer Engineers Course Handout.pptx
 
VICTOR MAESTRE RAMIREZ - Planetary Defender on NASA's Double Asteroid Redirec...
VICTOR MAESTRE RAMIREZ - Planetary Defender on NASA's Double Asteroid Redirec...VICTOR MAESTRE RAMIREZ - Planetary Defender on NASA's Double Asteroid Redirec...
VICTOR MAESTRE RAMIREZ - Planetary Defender on NASA's Double Asteroid Redirec...
 
Electronically Controlled suspensions system .pdf
Electronically Controlled suspensions system .pdfElectronically Controlled suspensions system .pdf
Electronically Controlled suspensions system .pdf
 
Call Us -/9953056974- Call Girls In Vikaspuri-/- Delhi NCR
Call Us -/9953056974- Call Girls In Vikaspuri-/- Delhi NCRCall Us -/9953056974- Call Girls In Vikaspuri-/- Delhi NCR
Call Us -/9953056974- Call Girls In Vikaspuri-/- Delhi NCR
 
Churning of Butter, Factors affecting .
Churning of Butter, Factors affecting .Churning of Butter, Factors affecting .
Churning of Butter, Factors affecting .
 
Introduction to Machine Learning Unit-3 for II MECH
Introduction to Machine Learning Unit-3 for II MECHIntroduction to Machine Learning Unit-3 for II MECH
Introduction to Machine Learning Unit-3 for II MECH
 
Decoding Kotlin - Your guide to solving the mysterious in Kotlin.pptx
Decoding Kotlin - Your guide to solving the mysterious in Kotlin.pptxDecoding Kotlin - Your guide to solving the mysterious in Kotlin.pptx
Decoding Kotlin - Your guide to solving the mysterious in Kotlin.pptx
 
Risk Assessment For Installation of Drainage Pipes.pdf
Risk Assessment For Installation of Drainage Pipes.pdfRisk Assessment For Installation of Drainage Pipes.pdf
Risk Assessment For Installation of Drainage Pipes.pdf
 
Call Us ≽ 8377877756 ≼ Call Girls In Shastri Nagar (Delhi)
Call Us ≽ 8377877756 ≼ Call Girls In Shastri Nagar (Delhi)Call Us ≽ 8377877756 ≼ Call Girls In Shastri Nagar (Delhi)
Call Us ≽ 8377877756 ≼ Call Girls In Shastri Nagar (Delhi)
 
Software and Systems Engineering Standards: Verification and Validation of Sy...
Software and Systems Engineering Standards: Verification and Validation of Sy...Software and Systems Engineering Standards: Verification and Validation of Sy...
Software and Systems Engineering Standards: Verification and Validation of Sy...
 
Gfe Mayur Vihar Call Girls Service WhatsApp -> 9999965857 Available 24x7 ^ De...
Gfe Mayur Vihar Call Girls Service WhatsApp -> 9999965857 Available 24x7 ^ De...Gfe Mayur Vihar Call Girls Service WhatsApp -> 9999965857 Available 24x7 ^ De...
Gfe Mayur Vihar Call Girls Service WhatsApp -> 9999965857 Available 24x7 ^ De...
 
CCS355 Neural Network & Deep Learning Unit II Notes with Question bank .pdf
CCS355 Neural Network & Deep Learning Unit II Notes with Question bank .pdfCCS355 Neural Network & Deep Learning Unit II Notes with Question bank .pdf
CCS355 Neural Network & Deep Learning Unit II Notes with Question bank .pdf
 
young call girls in Rajiv Chowk🔝 9953056974 🔝 Delhi escort Service
young call girls in Rajiv Chowk🔝 9953056974 🔝 Delhi escort Serviceyoung call girls in Rajiv Chowk🔝 9953056974 🔝 Delhi escort Service
young call girls in Rajiv Chowk🔝 9953056974 🔝 Delhi escort Service
 
pipeline in computer architecture design
pipeline in computer architecture designpipeline in computer architecture design
pipeline in computer architecture design
 
Call Girls Narol 7397865700 Independent Call Girls
Call Girls Narol 7397865700 Independent Call GirlsCall Girls Narol 7397865700 Independent Call Girls
Call Girls Narol 7397865700 Independent Call Girls
 
Call Girls Delhi {Jodhpur} 9711199012 high profile service
Call Girls Delhi {Jodhpur} 9711199012 high profile serviceCall Girls Delhi {Jodhpur} 9711199012 high profile service
Call Girls Delhi {Jodhpur} 9711199012 high profile service
 
Oxy acetylene welding presentation note.
Oxy acetylene welding presentation note.Oxy acetylene welding presentation note.
Oxy acetylene welding presentation note.
 
POWER SYSTEMS-1 Complete notes examples
POWER SYSTEMS-1 Complete notes examplesPOWER SYSTEMS-1 Complete notes examples
POWER SYSTEMS-1 Complete notes examples
 
CCS355 Neural Networks & Deep Learning Unit 1 PDF notes with Question bank .pdf
CCS355 Neural Networks & Deep Learning Unit 1 PDF notes with Question bank .pdfCCS355 Neural Networks & Deep Learning Unit 1 PDF notes with Question bank .pdf
CCS355 Neural Networks & Deep Learning Unit 1 PDF notes with Question bank .pdf
 

Geo1004 lecture 1_topology&amp;topological_datamodels_final

  • 1. 11 Topology On Topology and Topological Data Models in Geometric Modeling of Space Dr.ir. Pirouz Nourian Assistant Professor of Design Informatics Department of Architectural Engineering & Technology Faculty of Architecture and Built Environment Lecture notes of the course Geo1004: 3D Modelling of Built Environment, MSc Geomatics, TU Delft
  • 2. 22 • Geographical Modelling, replicating space using geometric objects with global coordinates • Geometrical Modelling, replicating space using geometric objects with local coordinates • Topological Modelling, replicating space using topological constructs such as simplexes • Graphical Modelling, replicating space using space objects and proximity links • Spectral Modelling, replicating space using Eigen space of differential operators Spectral Modelling for Spatial Network Analysis Abstractions of Space in Modelling Built Environments Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 3. 33 • Geographical Modelling, replicating space using geometric objects with global coordinates • Geometrical Modelling, replicating space using geometric objects with local coordinates • Topological Modelling, replicating space using topological constructs such as simplexes • Graphical Modelling, replicating space using space objects and proximity links • Spectral Modelling, replicating space using Eigen space of differential operators Spectral Modelling for Spatial Network Analysis Abstractions of Space in Modelling Built Environments Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 4. 44 0D: Point (the space therein everywhere resembles a 0-dimensional Euclidean Space) 1D: Curve (the space therein everywhere resembles a 1-dimensional Euclidean Space) 2D: Surface (the space therein everywhere resembles a 2-dimensional Euclidean Space) 3D: Solid (the space therein everywhere resembles a 3-dimensional Euclidean Space) Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 5. 55 Discrete Models of Manifold Spaces Mesh Graph/Network Grid Discrete Representations of Continuous Space Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 6. 66 Topology could be described as the study of topological invariants/properties of spatial objects, such as connectedness, continuity, closeness and dimension which remain invariant while applying topological two-directional continuous one-to-one or homoeomorphic transformations, such as those applied to a rubber-sheet or a piece of dough to change their shapes without tearing them apart, making cuts, making holes, gluing pieces. See also a glossary here: http://www.solitaryroad.com/c775.html http://www.gmanetwork.com/news/scitech/science/583886/when-is-a-coffee-mug-a-donut-topology-explains-it/story/ Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 7. 77 a riddle: Can you wear a piece of dough as a T-Shirt? What do we do with the dough to wear it? Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 8. 88 a riddle: Can you wear a piece of dough as a T-Shirt? What do we do with the dough to wear it? Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 9. 99 a riddle: Can you wear a piece of dough as a T-Shirt? What do we do with the dough to wear it? Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 10. 1010 a riddle: Can you wear a piece of dough as a T-Shirt? What do we do with the dough to wear it? Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 11. 1111 Geometry without Dimensions / Rubber-Sheet Geometry Image from Encyclopaedia Britannica Why is topology important in spatial modelling?Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 12. 1212 Geometry without Dimensions / Rubber-Sheet Geometry Escher’s Balcony, 1945 Escher’s Print Gallery, 1956Image from Encyclopaedia Britannica Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 13. 1313 Here is a formal definition of what qualifies as a topological model: Think of it as an abstraction of Euclidean space in which neighbourhoods are open balls [open sets, the space inside the balls without the ball surfaces] around points. Concretely, a topology [can be loosely considered as a topological model] on a point set 𝕏 is a collection 𝒰 [that is e.g. a geometric model whose inner space is abstracted as a collection of such neighbourhoods] of subsets of 𝕏, called open sets, such that: (i) 𝕏 is open and the empty set ∅ is open; (ii) if 𝑈1 and 𝑈2 are open, then 𝑈1 ∩ 𝑈2 is open; //[i.e. the intersection of two open sets is open] (iii) if 𝑈𝑖 is open for all 𝑖 in some possibly infinite, possible uncountable, index set, then the union of all 𝑈𝑖 is open. //[i.e. the unison of open sets is open] From Computational Topology, An Introduction, Edelsbrunner, Herbert and Harer, John, 2010, p.15, annotations in brackets from Pirouz Nourian. Here is an example of what this definition could mean e.g. in case of modelling with point clouds: A User’s Guide to Topological Data Analysis Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 14. 1414 Topology is a very broad subject area, in which at least two fields are relevant to us: Point-Set Topology (a.k.a. General Topology) Algebraic Topology (f.k.a. Combinatorial Topology) Point-Set Topology: Point-Set topology, also called set-theoretic topology or general topology, is the study of the general abstract nature of continuity or "closeness" on spaces. Basic point-set topological notions are ones like continuity, dimension, and connectedness. In the context of Point Set Topology we discuss: Nodes, Arcs, Disks, Balls (topological models use to describe the space inside Point Neighbourhoods, Curves, Surfaces, and Solids) Loosely speaking, this sense of topology is evident in the global structure of a geometric model. Algebraic Topology: Algebraic Topology, which includes Combinatorial Topology, is the study of “topological invariants” or ”topological properties” of combinatorial maps or decompositions of spaces such as [but not limited to] simplicial complexes. In the context of Algebraic Topology we discuss: Vertices, Edges, Faces, Bodies, (topological models used to tessellate/approximate/decompose Points, Curves, Surfaces, and Solids) Loosely speaking, this sense of topology is evident in the local structure of a geometric model. We shall see how both of these senses come together in one equation describing topological invariants. Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 15. 1515 https://www.nature.com/articles/nphoton.2015.244 Rubber-Sheet Geometry? Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 16. 1616 “If it looks like a duck, swims like a duck, and quacks like a duck, then it probably is a duck.” Image: DUCK: GETTY Images; ILLUSTRATION: MARTIN O'NEILL, from http://www.nature.com/nature/journal/v484/n7395/full/484451a.html?message-global=remove how do we model a curve in 3D space? Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 17. 1717 1D 2D 3D Parametric Modelling: modelling objects explicitly using their homeomorphic topological space Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 18. 1818 • 1D: Curves (t parameter) • 2D: Surfaces (u & v parameters) • (3D?): B-Reps (each face is a surface) NURBS Objects Non Uniform Rational Basis Splines are used for accurately modeling free-form geometries Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 19. 1919 NURBS Representation Non Uniform Rational Basis Splines (NURBS) are used for modeling free-form geometries accurately Image courtesy of http://www.boatdesign.netImage courtesy of Wikimedia • An elegant mathematical description of a physical drafting aid as a (set of) parametric equation(s). Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 20. 2020 Parametric Curves in General How do numeric weights correspond to physical weights? Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 21. 2121 Splines in Computer Graphics All types of curves can be modeled as splines • Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 22. 2222 Bezier interpolationModelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 23. 2323 NURBS equations All from a summary by Markus Altmann: http://web.cs.wpi.edu/~matt/courses/cs563/talks/nurbs.html Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 24. 2424 NURBS Representation Non Uniform Rational Basis Splines (NURBS) are used for modeling free-form geometries accurately • An elegant mathematical description of a physical drafting aid as a (set of) parametric equation(s). • offer one common mathematical form for both, standard analytical shapes (e.g. conics) and free form shapes; • provide the flexibility to design a large variety of shapes; • can be evaluated reasonably fast by numerically stable and accurate algorithms; • are invariant under affine as well as perspective transformations; • are generalizations of non-rational B-splines and non-rational and rational Bezier curves and surfaces. Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 25. 2525 NURBS Surfaces Rectangular Patches Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 26. 2626 Parametric Space Images courtesy of David Rutten, from Rhinoscript 101 Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 27. 2727 Parametric Locations:Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 28. 2828 Bilbao Guggenheim Museum Bus stop near Sebastiaansbrug Delft Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 29. 2929 Curvature 𝐾 = 1 𝑅 One over the radius of the osculation (kissing) circle Why did people [for such a long time] believe that the Earth was flat? Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 30. 3030 2D Curvature: Gaussian Images courtesy of Raja Issa, Essential Mathematics for Computational Design 𝐾 𝐺 = 𝑲 𝑚𝑖𝑛. 𝑲 𝑚𝑎𝑥 Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 31. 3131 Gaussian Curvature 𝐾 𝐺 = 𝐾 𝑚𝑖𝑛 𝐾 𝑚𝑎𝑥 Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 32. 3232 Continuity Images courtesy of Raja Issa, Essential Mathematics for Computational Design Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 33. 3333 2D Curvature: Mean 𝐻 = 𝑲 𝑚𝑖𝑛 +𝑲 𝑚𝑎𝑥 2 Anish Kapoor, Marsyas, 2002, installation view, Tate Modern https://en.wikipedia.org/wiki/Mean_curvature Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 34. 3434 Boundary RepresentationsModelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 35. 3535 Boundary Representations • 0D: Point • 1D: Curve (represented by boundary Points) • 2D: Surface (represented by boundary Curves) • 3D: Solid (represented by boundary Surfaces) Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 36. 3636 Imagine the Boundary of Manifolds • 0D: Point (the space therein everywhere resembles a 0-dimensional Euclidean Space) • 1D: Curve (the space therein everywhere resembles a 1-dimensional Euclidean Space) • 2D: Surface (the space therein everywhere resembles a 2-dimensional Euclidean Space) • 3D: Solid (the space therein everywhere resembles a 3-dimensional Euclidean Space) Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 37. 3737 Boundary Representations • 0D: Point • 1D: Curve (represented by boundary Points) • 2D: Surface (represented by boundary Curves) • 3D: Solid (represented by boundary Surfaces) • Free-Form Parametric surfaces • Piecewise Linear: Polygon Meshes Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 38. 3838 A simplex cell [actually the space inside it] is defined as the locus of points defined by linear interpolations of the corner vertices with barycentric coordinates denoted by 𝛼𝑖, as follows: 𝑣0, 𝑣1, … , 𝑣 𝑘 ∈ ℝ 𝑛 𝐶 = {𝛼0 𝑣0 + 𝛼1 𝑣1 + ⋯ . +𝛼 𝑘 𝑣 𝑘|𝛼𝑖 ≥ 0, 0 ≤ 𝑖 ≤ 𝑘, ෍ 𝑖 𝛼𝑖 = 1} • 0-simplex: Point • 1-simplex: Line • 2-simplex: Triangle • 3-simplex: Tetrahedron Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 39. 3939 A simplex cell [actually the space inside it] is defined as the locus of points defined by linear interpolations of the corner vertices with barycentric coordinates denoted by 𝛼𝑖, as follows: 𝑣0, 𝑣1, … , 𝑣 𝑘 ∈ ℝ 𝑛 𝐶 = {𝛼0 𝑣0 + 𝛼1 𝑣1 + ⋯ . +𝛼 𝑘 𝑣 𝑘|𝛼𝑖 ≥ 0, 0 ≤ 𝑖 ≤ 𝑘, ෍ 𝑖 𝛼𝑖 = 1} http://beachpackagingdesign.com/boxvox/the-tetra-pak-t Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 40. 4040 A simplex cell is defined as the locus of points defined by linear interpolations of the corner vertices with barycentric coordinates denoted by 𝛼𝑖, as follows: 𝑣0, 𝑣1, … , 𝑣 𝑘 ∈ ℝ 𝑛 𝐶 = {𝛼0 𝑣0 + 𝛼1 𝑣1 + ⋯ . +𝛼 𝑘 𝑣 𝑘|𝛼𝑖 ≥ 0, 0 ≤ 𝑖 ≤ 𝑘, ෍ 𝑖 𝛼𝑖 = 1} http://beachpackagingdesign.com/boxvox/the-tetra-pak-t A simplicial complex is a collection of simplices (simplexes) 𝐾 in ℝ 𝑛 such that: 1. Every face of a simplex of 𝐾 is in 𝐾, and 2. The intersection of any two simplexes of 𝐾 is a [generalized] face of each of them [with dimension 𝑘 − 1] Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 41. 4141 https://sites.google.com/site/dengwirda/jigsaw https://www.researchgate.net/publication/287398529_An_E fficient_Approach_for_Solving_Mesh_Optimization_Proble ms_Using_Newton%27s_Method/figures?lo=1 Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 42. 4242 • Example: • Important note: the geometry of faces is not stored; only rendered when needed! What you see on the screen is not necessarily what you store! Polygon Mesh Representation A light-weight model composed of points and a set of topological relations among them. Points >Vertices Lines > Edges Polygons > Faces Mesh T1: {0,4,3} T2:{0,1,4} Q1:{1,2,5,4} Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 43. 4343 • The geometry of a Mesh can be represented by its points (known as [geometrical] vertices in Rhino), i.e. a ‘list’ of 3D points in ℝ3 • The topology of a Mesh can be represented based on its [topological] vertices, referring to a ‘set’ of 3D points in ℝ3 , there are multiple ways to describe how these vertices are spatially related (connected or adjacent) to one another and also to edges and faces of the Mesh • Same topology and different geometries: Mesh Mesh Geometry versus Mesh Topology T1: {0,4,3} T2:{0,1,4} Q1:{1,2,5,4}T1 T2 Q1 T1 T2 Q1 0 3 1 4 5 2 0 3 1 4 5 2 Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 44. 4444 Triangulated Meshes Some preliminary definitions: fan, star, strip A triangle strip=:ABCDEF  ABC, CBD, CDE, and EDF A B C D E F A Closed Triangle Fan or a ‘star’=:ABCDEF A B C D E F  ABC, ACD, ADE, and AEF A Triangle Fan=: ABCDE AB C D E  ABC, ACD, and ADE Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 45. 4545 Mesh Boundary Some preliminary definitions: free* vertices, free* edges • If an edge has less than two faces adjacent to it then it is considered free; • If a vertex is part of such an edge it is considered as free too. * In Rhinocommon free vertices/edges are referred to as naked. Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 46. 4646 Mesh Border denoted as 𝝏 (𝒑𝒓𝒐𝒏𝒐𝒖𝒏𝒄𝒆𝒅 𝒂𝒔 ′ 𝒏𝒂𝒃𝒍𝒂′ 𝒐𝒓 ′𝒅𝒆𝒍′) • A 1D border is the set of connected edges incident to only one face of a 2- manifold, i.e. composed of free edges. We can conclude that: • If every vertex has a closed fan, or there is no edge of valence (number of neighbors) less than 2, the given manifold has no border. Example: a box! non-manifold border manifold borderno [1D] border Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 47. 4747 Images courtesy of Dr. Ching-Kuang Shene from Michigan Technological University • if a Mesh is supposed to be a 2D manifold then it should meet these criteria: 1. each edge is incident to only one or two faces; and 2. the faces incident to a vertex form a closed or an open fan. Non manifold Mesh examples: (note why!) Manifold Mesh A 2-manifold [everywhere] locally resembles a flat surface • if a Mesh is supposed to be orientable then, it should be possible to find ‘compatible’ orientations for any two adjacent faces; in which, for each pair of adjacent faces, the common edge of the two faces has opposite orders. • Example: Möbius band is a 2D manifold Mesh that is non-printable. Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 48. 4848 Image Source: http://prateekvjoshi.com/2014/11/16/homomorphism-vs homeomorphism/ Mesh Topology: Homeomorphism clay models that are all topologically equal! Images courtesy of Dr. Ching-Kuang Shene from Michigan Technological University Two 2-manifold Meshes A and B are homeomorphic if their surfaces can be transformed to one another by topological transformations (bending, twisting, stretching, scaling, etc.) without cutting and gluing. Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 49. 4949 Mesh Topology: Homeomorphism Euler-Poincare Characteristic: key to homeomorphism Only for 2D boundary representations* Images courtesy of Dr. Ching-Kuang Shene from Michigan Technological University 𝜒 𝑀 = 𝑉 − 𝐸 + 𝐹 = 2 1 − 𝑔 − 𝜕 • 𝛿 is the number of [connected] borders • 𝑔 is the number of “genera” (pl. of genus) or holes • Irrespective of tessellation! 𝜒 𝑀 = 𝑉 − 𝐸 + 𝐹 = 2 1 − 2 − 0 = −2 * For 3D BReps the Euler characteristic equation will take the form of V-E+F-B Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 50. 5050 Mesh Topology: Homeomorphism Euler-Poincare Characteristic: key to homeomorphism • Irrespective of tessellation! 𝜒 𝑀 = 𝑉 − 𝐸 + 𝐹 = −2 𝑓𝑜𝑟 𝑑𝑜𝑢𝑏𝑙𝑒 𝑡𝑜𝑟𝑖(ℎ𝑜𝑚𝑒𝑜𝑚𝑜𝑟𝑝ℎ𝑖𝑐 𝑡𝑜 𝑠𝑜𝑙𝑖𝑑𝑠 𝑤𝑖𝑡ℎ 2 ℎ𝑜𝑙𝑒𝑠) 𝜒 𝑀 = 𝑉 − 𝐸 + 𝐹 = 0 𝑓𝑜𝑟 𝑡𝑜𝑟𝑖 (ℎ𝑜𝑚𝑒𝑜𝑚𝑜𝑟𝑝ℎ𝑖𝑐 𝑡𝑜 𝑠𝑜𝑙𝑖𝑑𝑠 𝑤𝑖𝑡ℎ 1 ℎ𝑜𝑙𝑒) 𝜒 𝑀 = 𝑉 − 𝐸 + 𝐹 = 1 𝑓𝑜𝑟 𝑑𝑖𝑠𝑘𝑠 (ℎ𝑜𝑚𝑒𝑜𝑚𝑜𝑟𝑝ℎ𝑖𝑐 𝑡𝑜 𝑠𝑢𝑟𝑓𝑎𝑐𝑒𝑠) 𝜒 𝑀 = 𝑉 − 𝐸 + 𝐹 = 2 𝑓𝑜𝑟 𝑠𝑝ℎ𝑒𝑟𝑒𝑠 (ℎ𝑜𝑚𝑒𝑜𝑚𝑜𝑟𝑝ℎ𝑖𝑐 𝑡𝑜 𝑠𝑜𝑙𝑖𝑑𝑠 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 ℎ𝑜𝑙𝑒𝑠) Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 51. 5151 Mesh Topology: Homeomorphism Euler-Poincare Characteristic: key to homeomorphism • Irrespective of tessellation! Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 52. 5252 Valid Mesh (often expected to be a 2–Manifold Surface) Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 53. 5353 Mesh Topology: Homeomorphism Euler-Poincare Characteristic: key to homeomorphism • N-Manifold/Non-Manifold: Each point of an n-dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension n • Riddle: The above definition implies that for mapping some local geographic features 2D maps are fine. What about the whole globe? That is not homoeomorphic to a rectangular surface! How do we do it then? Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 54. 5454 Poincare Duality a pairing between k-dimensional features and dual features of dimension n-k in ℝ 𝑛 PRIMAL DUAL 0D vertex (e.g. as a point) 1D edge 1D edge (e.g. as a line segment) 0D vertex PRIMAL DUAL 0D vertex (e.g. a point) 2D face 1D edge (e.g. a line segment) 1D edge 2D face (e.g. a triangle or a pixel) 0D vertex PRIMAL DUAL 0D vertex (e.g. a point) 3D body 1D edge (e.g. a line segment) 2D face 2D face (e.g. a triangle or a pixel) 1D edge 3D body (e.g. a tetrahedron or a voxel) 0D vertex ℝ ℝ3 ℝ2 Configraphics: Graph Theoretical Methods of Design and Analysis of Spatial Configurations, Nourian, P, 2016 Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 55. 5555 Poincare Duality a pairing between k-dimensional features and dual features of dimension n-k in ℝ 𝑛 PRIMAL DUAL 0D vertex (e.g. as a point) 1D edge 1D edge (e.g. as a line segment) 0D vertex ℝ A hypothetical street network (a), a Junction-to-Junction adjacency graph (b) versus a Street-to-Street adjacency graph (c), both ‘undirected’, after Batty (Batty, 2004): red dots represent graph nodes, and blue arcs represent graph links. Batty, M., 2004. A New Theory of Space Syntax. CASA Working Paper Series, March. Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 56. 5656 Poincare Duality a pairing between k-dimensional features and dual features of dimension n-k in ℝ 𝑛 PRIMAL DUAL 0D vertex (e.g. a point) 2D face 1D edge (e.g. a line segment) 1D edge 2D face (e.g. a triangle or a pixel) 0D vertex ℝ2 A Voronoi tessellation of 2D space and its dual that is a Delaunay triangulation Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 57. 5757 Poincare Duality a pairing between k-dimensional features and dual features of dimension n-k in ℝ 𝑛 PRIMAL DUAL 0D vertex (e.g. a point) 3D body 1D edge (e.g. a line segment) 2D face 2D face (e.g. a triangle or a pixel) 1D edge 3D body (e.g. a tetrahedron or a voxel) 0D vertex ℝ3 representing adjacencies between 3D cells or bodies via their dual vertices (Lee, 2001) Lee, J., 2001. 3D Data Model for Representing Topological Relations of Urban Features. San Diego, CA, s.n. Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 58. 5858 Mesh Topological Structure • Mesh Topological Data Models: Face-Vertex: e.g. 𝐹0 = {𝑉0, 𝑉5, 𝑉4} & 𝑉0 ∈ 𝐹0, 𝐹1, 𝐹12, 𝐹15, 𝐹7 Images courtesy of David Dorfman, from Wikipedia Face-Vertex (as implemented in Rhinoceros) Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 59. 5959 Vertex-Vertex: e.g. 𝑉0~{𝑉1, 𝑉4, 𝑉3} Face-Vertex: e.g. 𝐹0 = 𝑉0, 𝑉5, 𝑉4 , 𝑉0 ∈ 𝐹0, 𝐹1, 𝐹12, 𝐹15, 𝐹7 Winged-Edge: e.g. 𝐹0 = 𝐸4, 𝐸8, 𝐸9 , 𝐸0 = 𝑉0, 𝑉1 , 𝐸0~ 𝐹1, 𝐹12 , 𝐸0~ 𝐸9, 𝐸23, 𝐸10, 𝐸20 Half-Edge: each half edge has a twin edge in opposite direction, a previous and a next Mesh Topological Structures Image courtesy of David Dorfman, from Wikipedia What is explicitly stored as topology of a Mesh: E.g. Face- Vertex (as implemented in Rhinoceros) http://doc.cgal.org/latest/HalfedgeDS/index.html Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 60. 6060 Mesh Topological Structures Which topological relations are explicitly stored in the data model V E F V VV VE VF E EV EE EF F FV FE FF Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 61. 6161 Mesh Topological Structure • Mesh Topological Data Models: Face-Vertex Example: http://4.rhino3d.com/5/rhinocommon/html/AllMembers_T_Rhino_Geometry_Collections_MeshTopologyVertexList.htm Face-Vertex (as implemented in Rhinoceros) Name Description ConnectedFaces Gets all faces that are connected to a given vertex. ConnectedTopologyVertices(Int32) Gets all topological vertices that are connected to a given vertex. ConnectedTopologyVertices(Int32, Boolean) Gets all topological vertices that are connected to a given vertex. The MeshTopologyVertexList type exposes the following members. Half-Edge: Half-Edge Mesh Data Structure, example implementation by Daniel Piker: http://www.grasshopper3d.com/group/plankton Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 62. 6262 Mesh Geometry • Polygon vs Face • Triangulate • Quadrangulate Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 63. 6363 Mesh Intersections • Boolean operation on Meshes: 1. 𝐴 ∪ 𝐵: Boolean Union 2. 𝐴 − 𝐵: Boolean Difference 3. 𝐴 ∩ 𝐵: Boolean Intersection Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 64. 6464 Normal Vectors of a Mesh • Topological Vertices versus Geometrical Points • Joining Mesh objects: What is a Mesh box? • Welding Meshes: how does it work? • Face Normal versus Vertex Normal Where do they come from and what do they represent? Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 65. 6565 How to Compute Mesh Normals? 00, 00 ),( vvuuv p u p vun              1 ( ) ( ) 0 ( )( ) N x i next i i next i i N y y z z      1 ( ) ( ) 0 ( )( ) N z i next i i next i i N x x y y      1 ( ) ( ) 0 ( )( ) N y i next i i next i i N z z x x      Martin Newell at Utah (remember the teapot?) Why? Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 66. 6666 Repairing/Reconstructing Geometry • Example: Coons’ Patch • Code it and get bonus points! Image courtesy of CVG Lab Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 67. 6767 Mesh Smoothing for (int k = 0; k <= L - 1; k++) { List<Point3d> SmoothV = new List<Point3d>(); for (int i = 0; i <= M.Vertices.Count - 1; i++) { List<Point3d> Neighbours = Neighbors(M, i); Point3d NVertex = new Point3d(0, 0, 0); foreach (point3d neighbor in Neighbours ) { NVertex = NVertex + neighbor; } NVertex = (1 / Ngh.Count) * NVertex; SmoothV.Add(NVertex); } M.Vertices.Clear(); M.Vertices.AddVertices(SmoothV); A = M; } Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 68. 6868 Topological Transformation of Meshes http://www.idz.ro/grasshopper-mesh-relaxation-component-update/ Euclidean to Hyperbolic By Dynamic Relaxation Algorithms Euclidean to Elliptical By Subdivision Algorithms https://www.renderosity.com/mod/forumpro/?thread_id=2651940 Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 69. 6969 • Geometry: Point (0D), Line (1D), Plane(2D), Hyper-Plane(3D) (Euclidean sub-spaces) • Geometry: Point (0D), Curve(1D), Surface(2D), Solid (3D) [differential] • Geometry: Point (0D), Line-Segment(1D), Polygon(2D), Polyhedron (3D) [piecewise linear] • Geometry: Point(0D), Line-Segment(1D), Triangle(2D), Tetrahedron(3D) [simplexes] • Topology: Node(0D), Arc(1D), Disk(2D), Ball(3D) [general/point-set] • Topology: Vertex(0D), Edge(1D), Face(2D), Body(3D) [combinatorial/algebraic] • Graph Theory: Object, Link, (and n-Cliques) • Fields: Pixel, Voxel, Raster2D, Raster 3D [scalar] On terminology (CAD, CG, GIS, Math, etc.) Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology
  • 70. 7070 On terminology n-D features Graph Theory Topology Geometry 0D Object Vertex Point 1D Link Edge Line (Curve) 2D Cycle* Face Polygon (Surface) 3D Clique* Body Polyhedron (Solid) Configraphics: Graph Theoretical Methods of Design and Analysis of Spatial Configurations, Nourian, P, 2016 Modelling Space Manifolds Topology Point Set Topology Algebraic Topology Non-Euclidean Space Geometric Models Parametric Models Boundary Representation Simplexes Polyhedral Meshes Polygon Meshes Manifold Mesh Euler-Poincare Equation Poincare Duality Topological Models Normal Vectors Meshing Mesh Transform Terminology