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GEOMETRY-TOPOLOGY

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Pirouz Nourian

The document discusses various topics in geometry and topology used in computer graphics and design, including: 1) Different data models used to represent geometry like NURBS (non-uniform rational basis splines) and boundary representations. 2) Concepts of dimensionality where a 0D object is a point, 1D is a curve, 2D is a surface, and 3D is a solid. 3) Terminology used in different fields, with graph theory using objects and links, topology using vertices and edges, and geometry using points and lines.

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11 Geometry & Topology in Computer Geometry & Design Dr.ir. Pirouz Nourian Assistant Professor of Design Informatics Department of Architectural Engineering & Technology Faculty of Architecture and Built Environment
22 A Line is something that looks like a line & works like a line! β€œIf 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 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
33 What you see on the screen is only a rasterized image of the object! What you see is not what you get! Image: RenΓ© Magritte, ceci n'est pas une pipe (this is not a pipe) Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
44 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
55 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
66 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
77 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Escher’s Balcony
88 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling ℝ ℝ 𝟐 ℝ πŸ‘ 1D 2D 3D http://paulbourke.net/geometry/hyperspace/
99 Cartesian Product of Sets ℝ 𝟏 = ℝ ℝ 𝟐 = ℝ Γ— ℝ ℝ πŸ‘ = ℝ Γ— ℝ Γ— ℝ 1D 2D 3D http://paulbourke.net/geometry/hyperspace/ Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 𝐴 = {π‘Ž, 𝑏, 𝑐} 𝐡 = {1,2} 𝐴 Γ— 𝐡 = { π‘Ž, 1 , π‘Ž, 2 , 𝑏, 1 , 𝑏, 2 , 𝑐, 1 , (𝑐, 2)}
1010 ℝ 𝟏 = ℝ ℝ 𝟐 = ℝ Γ— ℝ ℝ πŸ‘ = ℝ Γ— ℝ Γ— ℝ 1D 2D 3D http://paulbourke.net/geometry/hyperspace/ 𝐴 = {π‘Ž, 𝑏, 𝑐} 𝐡 = {1,2} 𝐴 Γ— 𝐡 = { π‘Ž, 1 , π‘Ž, 2 , 𝑏, 1 , 𝑏, 2 , 𝑐, 1 , (𝑐, 2)} Cartesian Product of Sets Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling http://ndp.jct.ac.il/tutorials/discrete/node28.html
1111 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 1D 2D 3D Curved Spaces
1212 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 1D 2D 3D
1313 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 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)
1414 On terminology β€’ Geometry: Point (0D), Curve(1D), Surface(2D), Solid (3D) [free-form] β€’ Geometry: Point (0D), Line(1D), Polygon(2D), Polyhedron (3D) [piecewise linear] β€’ Topology: Vertex(0D), Edge(1D), Face(2D), Body(3D) β€’ Graph Theory: Object, Link, (and n-Cliques) Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
1515 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) Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Configraphics: Graph Theoretical Methods of Design and Analysis of Spatial Configurations, Nourian, P, 2016
1616 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling β€’ 1D: Curves (t parameter) β€’ 2D: Surfaces (u & v parameters) β€’ 3D: B-Reps (each face is a surface) NURBS Objects Non Uniform Rational Basis Splines are used for accurately modeling free-form geometries
1717 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 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 β€’ An elegant mathematical description of a physical drafting aid as a (set of) parametric equation(s).
1818 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Parametric Curves in General How do numeric weights correspond to physical weights?
1919 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Splines in Computer Graphics All types of curves can be modeled as splines β€’
2020 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Bezier interpolation
2121 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling NURBS equations All from a summary by Markus Altmann: http://web.cs.wpi.edu/~matt/courses/cs563/talks/nurbs.html
2222 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling NURBS Representation Non Uniform Rational Basis Splines (NURBS) are used for modeling free-form geometries accurately β€’ An elegant mathematical description of a physical drafting aid as a (set of) parametric equation(s). β€’ offer one common mathematical form for both, standard analytical shapes (e.g. conics) and free form shapes; β€’ provide the flexibility to design a large variety of shapes; β€’ can be evaluated reasonably fast by numerically stable and accurate algorithms; β€’ are invariant under affine as well as perspective transformations; β€’ are generalizations of non-rational B-splines and non-rational and rational Bezier curves and surfaces.
2323 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling NURBS Surfaces Rectangular Patches
2424 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Parametric Space Images courtesy of David Rutten, from Rhinoscript 101
2525 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Parametric Locations:
2626 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Bilbao Guggenheim Museum Bus stop near Sebastiaansbrug Delft
2727 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 1D 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?
2828 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 2D Curvature: Gaussian Images courtesy of Raja Issa, Essential Mathematics for Computational Design 𝐾 𝐺 = 𝐾 π‘šπ‘–π‘› 𝐾 π‘šπ‘Žπ‘₯
2929 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 2D Curvature: Gaussian 𝐾 𝐺 = 𝐾 π‘šπ‘–π‘› 𝐾 π‘šπ‘Žπ‘₯
3030 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Blending Surfaces & Continuity Images courtesy of Raja Issa, Essential Mathematics for Computational Design
3131 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 2D Curvature: Mean 𝐾 𝐺 = 𝐾 π‘šπ‘–π‘› +𝐾 π‘šπ‘Žπ‘₯ 2 Anish Kapoor, Marsyas, 2002, installation view, Tate Modern https://en.wikipedia.org/wiki/Mean_curvature
3232 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Boundary Representations
3333 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Boundary Representations β€’ 0D: Point β€’ 1D: Curve (represented by boundary Points) β€’ 2D: Surface (represented by boundary Curves) β€’ 3D: Solid (represented by boundary Surfaces)
3434 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Imagine the Boundary of Manifolds β€’ 0D: Point (the space therein everywhere resembles a 0-dimensional Euclidean Space) β€’ 1D: Curve (the space therein everywhere resembles a 1-dimensional Euclidean Space) β€’ 2D: Surface (the space therein everywhere resembles a 2-dimensional Euclidean Space) β€’ 3D: Solid (the space therein everywhere resembles a 3-dimensional Euclidean Space)
3535 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Boundary Representations β€’ 0D: Point β€’ 1D: Curve (represented by boundary Points) β€’ 2D: Surface (represented by boundary Curves) β€’ 3D: Solid (represented by boundary Surfaces) β€’ Free-Form Parametric surfaces β€’ Piecewise Linear: Polygon Meshes
3636 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling The 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 π‘˜ ∈ ℝ 𝑛 𝐢 = {𝛼0 𝑣0 + 𝛼1 𝑣1 + β‹― . +𝛼 π‘˜ 𝑣 π‘˜|𝛼𝑖 β‰₯ 0, 0 ≀ 𝑖 ≀ π‘˜, ෍ 𝑖 𝛼𝑖 = 1} β€’ 0-simplex: Point β€’ 1-simplex: Line β€’ 2-simplex: Triangle β€’ 3-simplex: Tetrahedron
3737 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 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
3838 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling β€’ Example: β€’ 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}
3939 β€’ The geometry of a Mesh can be represented by its points (known as [geometrical] vertices in Rhino), i.e. a β€˜list’ of 3D points in ℝ3 β€’ The topology of a Mesh can be represented based on its [topological] vertices, referring to a β€˜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 β€’ 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 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
4040 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 β€˜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 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
4141 Mesh Boundary Some preliminary definitions: free* vertices, free* edges β€’ If an edge has less than two faces adjacent to it then it is considered free; β€’ 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. Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
4242 Mesh Border denoted as 𝝏 (𝒑𝒓𝒐𝒏𝒐𝒖𝒏𝒄𝒆𝒅 𝒂𝒔 β€² 𝒏𝒂𝒃𝒍𝒂′ 𝒐𝒓 ′𝒅𝒆𝒍′) β€’ 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: β€’ 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 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
4343 Images courtesy of Dr. Ching-Kuang Shene from Michigan Technological University β€’ 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 β€’ if a Mesh is supposed to be orientable then, it should be possible to find β€˜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. β€’ Example: MΓΆbius band is a 2D manifold Mesh that is non-printable. Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
4444 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. Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
4545 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 πœ’ 𝑀 = 𝑉 βˆ’ 𝐸 + 𝐹 = 2 1 βˆ’ 𝑔 βˆ’ πœ• β€’ 𝛿 is the number of [connected] borders β€’ 𝑔 is the number of β€œgenera” (pl. of genus) or holes β€’ Irrespective of tessellation! πœ’ 𝑀 = 𝑉 βˆ’ 𝐸 + 𝐹 = 2 1 βˆ’ 2 βˆ’ 0 = βˆ’2 * For 3D BReps the Euler characteristic equation will take the form of V-E+F-B Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
4646 Mesh Topology: Homeomorphism Euler-Poincare Characteristic: key to homeomorphism β€’ Irrespective of tessellation! πœ’ 𝑀 = 𝑉 βˆ’ 𝐸 + 𝐹 = βˆ’2 π‘“π‘œπ‘Ÿ π‘‘π‘œπ‘’π‘π‘™π‘’ π‘‘π‘œπ‘Ÿπ‘–(β„Žπ‘œπ‘šπ‘’π‘œπ‘šπ‘œπ‘Ÿπ‘β„Žπ‘–π‘ π‘‘π‘œ π‘ π‘œπ‘™π‘–π‘‘π‘  π‘€π‘–π‘‘β„Ž 2 β„Žπ‘œπ‘™π‘’π‘ ) πœ’ 𝑀 = 𝑉 βˆ’ 𝐸 + 𝐹 = 0 π‘“π‘œπ‘Ÿ π‘‘π‘œπ‘Ÿπ‘– (β„Žπ‘œπ‘šπ‘’π‘œπ‘šπ‘œπ‘Ÿπ‘β„Žπ‘–π‘ π‘‘π‘œ π‘ π‘œπ‘™π‘–π‘‘π‘  π‘€π‘–π‘‘β„Ž 1 β„Žπ‘œπ‘™π‘’) πœ’ 𝑀 = 𝑉 βˆ’ 𝐸 + 𝐹 = 1 π‘“π‘œπ‘Ÿ π‘‘π‘–π‘ π‘˜π‘  (β„Žπ‘œπ‘šπ‘’π‘œπ‘šπ‘œπ‘Ÿπ‘β„Žπ‘–π‘ π‘‘π‘œ π‘ π‘’π‘Ÿπ‘“π‘Žπ‘π‘’π‘ ) πœ’ 𝑀 = 𝑉 βˆ’ 𝐸 + 𝐹 = 2 π‘“π‘œπ‘Ÿ π‘ π‘β„Žπ‘’π‘Ÿπ‘’π‘  (β„Žπ‘œπ‘šπ‘’π‘œπ‘šπ‘œπ‘Ÿπ‘β„Žπ‘–π‘ π‘‘π‘œ π‘ π‘œπ‘™π‘–π‘‘π‘  π‘€π‘–π‘‘β„Žπ‘œπ‘’π‘‘ β„Žπ‘œπ‘™π‘’π‘ ) Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
4747 Mesh Topology: Homeomorphism Euler-Poincare Characteristic: key to homeomorphism β€’ Irrespective of tessellation! Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
4848 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Valid Mesh (often expected to be a 2–Manifold Surface)
4949 Mesh Topology: Homeomorphism Euler-Poincare Characteristic: key to homeomorphism β€’ N-Manifold/Non-Manifold: Each point of an n-dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension n β€’ 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? Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
5050 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 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
5151 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 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 β€˜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.
5252 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 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
5353 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 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)
5454 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Mesh Topological Structure β€’ 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)
5555 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 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
5656 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Mesh Topological Structure β€’ 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
5757 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Mesh Geometry β€’ Polygon vs Face β€’ Triangulate β€’ Quadrangulate
5858 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Mesh Intersections β€’ Boolean operation on Meshes: 1. 𝐴 βˆͺ 𝐡: Boolean Union 2. 𝐴 βˆ’ 𝐡: Boolean Difference 3. 𝐴 ∩ 𝐡: Boolean Intersection
5959 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Normal Vectors of a Mesh β€’ Topological Vertices versus Geometrical Points β€’ Joining Mesh objects: What is a Mesh box? β€’ Welding Meshes: how does it work? β€’ Face Normal versus Vertex Normal Where do they come from and what do they represent?
6060 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling How to Compute Mesh Normals? 00, 00 ),( vvuuv p u p vun ο€½ο€½ οƒ· οƒΈ οƒΆ    ο‚Ά ο‚Ά ο‚΄ ο‚Ά ο‚Ά ο€½ 1 ( ) ( ) 0 ( )( ) N x i next i i next i i N y y z z ο€­ ο€½ ο€½ ο€­ οƒ₯ 1 ( ) ( ) 0 ( )( ) N z i next i i next i i N x x y y ο€­ ο€½ ο€½ ο€­ οƒ₯ 1 ( ) ( ) 0 ( )( ) N y i next i i next i i N z z x x ο€­ ο€½ ο€½ ο€­ οƒ₯ Martin Newell at Utah (remember the teapot?) Why?
6161 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Repairing/Reconstructing Geometry β€’ Example: Coons’ Patch β€’ Code it and get bonus points! Image courtesy of CVG Lab
6262 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 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; }
6363 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Mesh Smoothing (Relaxation) Kangaroo Physics: https://vimeo.com/27484394
6464 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
6565 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Literal Boundary Representation A strategy for design modeling/3D sketching Curves Surfaces Solids Curves From Objects Points http://forums.newtek.com/showthread.php?143633-Can-Lightwave-Model-an-airplane-boat-Hull-Like-the-way-they-do-in-this-C4D-tutorial
6666 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Transformative Modeling A different strategy for design modeling/3D sketching Solids Topological Transformations Transformed Solids Curves From Objects https://3dprint.com/88049/3d-printed-topology-joke/
6767 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Subdivision Modelling A strategy for design modeling/3D sketching Model a Simple BRep Subdivide to Smoothen http://slodive.com/design/3ds-max-modeling-tutorials/
6868 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Implicit modelling as level sets A strategy for design modeling/3D sketching Model a Scalar Field Get Iso-surfaces
6969 Questions: p.nourian@tudelft.nl

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GEOMETRY-TOPOLOGY

  • 1. 11 Geometry & Topology in Computer Geometry & Design Dr.ir. Pirouz Nourian Assistant Professor of Design Informatics Department of Architectural Engineering & Technology Faculty of Architecture and Built Environment
  • 2. 22 A Line is something that looks like a line & works like a line! β€œIf 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 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
  • 3. 33 What you see on the screen is only a rasterized image of the object! What you see is not what you get! Image: RenΓ© Magritte, ceci n'est pas une pipe (this is not a pipe) Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
  • 4. 44 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
  • 5. 55 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
  • 6. 66 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
  • 7. 77 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Escher’s Balcony
  • 8. 88 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling ℝ ℝ 𝟐 ℝ πŸ‘ 1D 2D 3D http://paulbourke.net/geometry/hyperspace/
  • 9. 99 Cartesian Product of Sets ℝ 𝟏 = ℝ ℝ 𝟐 = ℝ Γ— ℝ ℝ πŸ‘ = ℝ Γ— ℝ Γ— ℝ 1D 2D 3D http://paulbourke.net/geometry/hyperspace/ Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 𝐴 = {π‘Ž, 𝑏, 𝑐} 𝐡 = {1,2} 𝐴 Γ— 𝐡 = { π‘Ž, 1 , π‘Ž, 2 , 𝑏, 1 , 𝑏, 2 , 𝑐, 1 , (𝑐, 2)}
  • 10. 1010 ℝ 𝟏 = ℝ ℝ 𝟐 = ℝ Γ— ℝ ℝ πŸ‘ = ℝ Γ— ℝ Γ— ℝ 1D 2D 3D http://paulbourke.net/geometry/hyperspace/ 𝐴 = {π‘Ž, 𝑏, 𝑐} 𝐡 = {1,2} 𝐴 Γ— 𝐡 = { π‘Ž, 1 , π‘Ž, 2 , 𝑏, 1 , 𝑏, 2 , 𝑐, 1 , (𝑐, 2)} Cartesian Product of Sets Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling http://ndp.jct.ac.il/tutorials/discrete/node28.html
  • 11. 1111 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 1D 2D 3D Curved Spaces
  • 12. 1212 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 1D 2D 3D
  • 13. 1313 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 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)
  • 14. 1414 On terminology β€’ Geometry: Point (0D), Curve(1D), Surface(2D), Solid (3D) [free-form] β€’ Geometry: Point (0D), Line(1D), Polygon(2D), Polyhedron (3D) [piecewise linear] β€’ Topology: Vertex(0D), Edge(1D), Face(2D), Body(3D) β€’ Graph Theory: Object, Link, (and n-Cliques) Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
  • 15. 1515 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) Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Configraphics: Graph Theoretical Methods of Design and Analysis of Spatial Configurations, Nourian, P, 2016
  • 16. 1616 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling β€’ 1D: Curves (t parameter) β€’ 2D: Surfaces (u & v parameters) β€’ 3D: B-Reps (each face is a surface) NURBS Objects Non Uniform Rational Basis Splines are used for accurately modeling free-form geometries
  • 17. 1717 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 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 β€’ An elegant mathematical description of a physical drafting aid as a (set of) parametric equation(s).
  • 18. 1818 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Parametric Curves in General How do numeric weights correspond to physical weights?
  • 19. 1919 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Splines in Computer Graphics All types of curves can be modeled as splines β€’
  • 20. 2020 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Bezier interpolation
  • 21. 2121 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling NURBS equations All from a summary by Markus Altmann: http://web.cs.wpi.edu/~matt/courses/cs563/talks/nurbs.html
  • 22. 2222 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling NURBS Representation Non Uniform Rational Basis Splines (NURBS) are used for modeling free-form geometries accurately β€’ An elegant mathematical description of a physical drafting aid as a (set of) parametric equation(s). β€’ offer one common mathematical form for both, standard analytical shapes (e.g. conics) and free form shapes; β€’ provide the flexibility to design a large variety of shapes; β€’ can be evaluated reasonably fast by numerically stable and accurate algorithms; β€’ are invariant under affine as well as perspective transformations; β€’ are generalizations of non-rational B-splines and non-rational and rational Bezier curves and surfaces.
  • 23. 2323 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling NURBS Surfaces Rectangular Patches
  • 24. 2424 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Parametric Space Images courtesy of David Rutten, from Rhinoscript 101
  • 25. 2525 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Parametric Locations:
  • 26. 2626 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Bilbao Guggenheim Museum Bus stop near Sebastiaansbrug Delft
  • 27. 2727 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 1D 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?
  • 28. 2828 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 2D Curvature: Gaussian Images courtesy of Raja Issa, Essential Mathematics for Computational Design 𝐾 𝐺 = 𝐾 π‘šπ‘–π‘› 𝐾 π‘šπ‘Žπ‘₯
  • 29. 2929 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 2D Curvature: Gaussian 𝐾 𝐺 = 𝐾 π‘šπ‘–π‘› 𝐾 π‘šπ‘Žπ‘₯
  • 30. 3030 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Blending Surfaces & Continuity Images courtesy of Raja Issa, Essential Mathematics for Computational Design
  • 31. 3131 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 2D Curvature: Mean 𝐾 𝐺 = 𝐾 π‘šπ‘–π‘› +𝐾 π‘šπ‘Žπ‘₯ 2 Anish Kapoor, Marsyas, 2002, installation view, Tate Modern https://en.wikipedia.org/wiki/Mean_curvature
  • 32. 3232 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Boundary Representations
  • 33. 3333 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Boundary Representations β€’ 0D: Point β€’ 1D: Curve (represented by boundary Points) β€’ 2D: Surface (represented by boundary Curves) β€’ 3D: Solid (represented by boundary Surfaces)
  • 34. 3434 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Imagine the Boundary of Manifolds β€’ 0D: Point (the space therein everywhere resembles a 0-dimensional Euclidean Space) β€’ 1D: Curve (the space therein everywhere resembles a 1-dimensional Euclidean Space) β€’ 2D: Surface (the space therein everywhere resembles a 2-dimensional Euclidean Space) β€’ 3D: Solid (the space therein everywhere resembles a 3-dimensional Euclidean Space)
  • 35. 3535 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Boundary Representations β€’ 0D: Point β€’ 1D: Curve (represented by boundary Points) β€’ 2D: Surface (represented by boundary Curves) β€’ 3D: Solid (represented by boundary Surfaces) β€’ Free-Form Parametric surfaces β€’ Piecewise Linear: Polygon Meshes
  • 36. 3636 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling The 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 π‘˜ ∈ ℝ 𝑛 𝐢 = {𝛼0 𝑣0 + 𝛼1 𝑣1 + β‹― . +𝛼 π‘˜ 𝑣 π‘˜|𝛼𝑖 β‰₯ 0, 0 ≀ 𝑖 ≀ π‘˜, ෍ 𝑖 𝛼𝑖 = 1} β€’ 0-simplex: Point β€’ 1-simplex: Line β€’ 2-simplex: Triangle β€’ 3-simplex: Tetrahedron
  • 37. 3737 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 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
  • 38. 3838 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling β€’ Example: β€’ 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}
  • 39. 3939 β€’ The geometry of a Mesh can be represented by its points (known as [geometrical] vertices in Rhino), i.e. a β€˜list’ of 3D points in ℝ3 β€’ The topology of a Mesh can be represented based on its [topological] vertices, referring to a β€˜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 β€’ 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 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
  • 40. 4040 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 β€˜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 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
  • 41. 4141 Mesh Boundary Some preliminary definitions: free* vertices, free* edges β€’ If an edge has less than two faces adjacent to it then it is considered free; β€’ 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. Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
  • 42. 4242 Mesh Border denoted as 𝝏 (𝒑𝒓𝒐𝒏𝒐𝒖𝒏𝒄𝒆𝒅 𝒂𝒔 β€² 𝒏𝒂𝒃𝒍𝒂′ 𝒐𝒓 ′𝒅𝒆𝒍′) β€’ 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: β€’ 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 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
  • 43. 4343 Images courtesy of Dr. Ching-Kuang Shene from Michigan Technological University β€’ 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 β€’ if a Mesh is supposed to be orientable then, it should be possible to find β€˜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. β€’ Example: MΓΆbius band is a 2D manifold Mesh that is non-printable. Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
  • 44. 4444 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. Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
  • 45. 4545 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 πœ’ 𝑀 = 𝑉 βˆ’ 𝐸 + 𝐹 = 2 1 βˆ’ 𝑔 βˆ’ πœ• β€’ 𝛿 is the number of [connected] borders β€’ 𝑔 is the number of β€œgenera” (pl. of genus) or holes β€’ Irrespective of tessellation! πœ’ 𝑀 = 𝑉 βˆ’ 𝐸 + 𝐹 = 2 1 βˆ’ 2 βˆ’ 0 = βˆ’2 * For 3D BReps the Euler characteristic equation will take the form of V-E+F-B Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
  • 46. 4646 Mesh Topology: Homeomorphism Euler-Poincare Characteristic: key to homeomorphism β€’ Irrespective of tessellation! πœ’ 𝑀 = 𝑉 βˆ’ 𝐸 + 𝐹 = βˆ’2 π‘“π‘œπ‘Ÿ π‘‘π‘œπ‘’π‘π‘™π‘’ π‘‘π‘œπ‘Ÿπ‘–(β„Žπ‘œπ‘šπ‘’π‘œπ‘šπ‘œπ‘Ÿπ‘β„Žπ‘–π‘ π‘‘π‘œ π‘ π‘œπ‘™π‘–π‘‘π‘  π‘€π‘–π‘‘β„Ž 2 β„Žπ‘œπ‘™π‘’π‘ ) πœ’ 𝑀 = 𝑉 βˆ’ 𝐸 + 𝐹 = 0 π‘“π‘œπ‘Ÿ π‘‘π‘œπ‘Ÿπ‘– (β„Žπ‘œπ‘šπ‘’π‘œπ‘šπ‘œπ‘Ÿπ‘β„Žπ‘–π‘ π‘‘π‘œ π‘ π‘œπ‘™π‘–π‘‘π‘  π‘€π‘–π‘‘β„Ž 1 β„Žπ‘œπ‘™π‘’) πœ’ 𝑀 = 𝑉 βˆ’ 𝐸 + 𝐹 = 1 π‘“π‘œπ‘Ÿ π‘‘π‘–π‘ π‘˜π‘  (β„Žπ‘œπ‘šπ‘’π‘œπ‘šπ‘œπ‘Ÿπ‘β„Žπ‘–π‘ π‘‘π‘œ π‘ π‘’π‘Ÿπ‘“π‘Žπ‘π‘’π‘ ) πœ’ 𝑀 = 𝑉 βˆ’ 𝐸 + 𝐹 = 2 π‘“π‘œπ‘Ÿ π‘ π‘β„Žπ‘’π‘Ÿπ‘’π‘  (β„Žπ‘œπ‘šπ‘’π‘œπ‘šπ‘œπ‘Ÿπ‘β„Žπ‘–π‘ π‘‘π‘œ π‘ π‘œπ‘™π‘–π‘‘π‘  π‘€π‘–π‘‘β„Žπ‘œπ‘’π‘‘ β„Žπ‘œπ‘™π‘’π‘ ) Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
  • 47. 4747 Mesh Topology: Homeomorphism Euler-Poincare Characteristic: key to homeomorphism β€’ Irrespective of tessellation! Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
  • 48. 4848 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Valid Mesh (often expected to be a 2–Manifold Surface)
  • 49. 4949 Mesh Topology: Homeomorphism Euler-Poincare Characteristic: key to homeomorphism β€’ N-Manifold/Non-Manifold: Each point of an n-dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension n β€’ 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? Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
  • 50. 5050 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 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
  • 51. 5151 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 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 β€˜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.
  • 52. 5252 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 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
  • 53. 5353 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 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)
  • 54. 5454 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Mesh Topological Structure β€’ 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)
  • 55. 5555 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 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
  • 56. 5656 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Mesh Topological Structure β€’ 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
  • 57. 5757 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Mesh Geometry β€’ Polygon vs Face β€’ Triangulate β€’ Quadrangulate
  • 58. 5858 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Mesh Intersections β€’ Boolean operation on Meshes: 1. 𝐴 βˆͺ 𝐡: Boolean Union 2. 𝐴 βˆ’ 𝐡: Boolean Difference 3. 𝐴 ∩ 𝐡: Boolean Intersection
  • 59. 5959 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Normal Vectors of a Mesh β€’ Topological Vertices versus Geometrical Points β€’ Joining Mesh objects: What is a Mesh box? β€’ Welding Meshes: how does it work? β€’ Face Normal versus Vertex Normal Where do they come from and what do they represent?
  • 60. 6060 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling How to Compute Mesh Normals? 00, 00 ),( vvuuv p u p vun ο€½ο€½ οƒ· οƒΈ οƒΆ    ο‚Ά ο‚Ά ο‚΄ ο‚Ά ο‚Ά ο€½ 1 ( ) ( ) 0 ( )( ) N x i next i i next i i N y y z z ο€­ ο€½ ο€½ ο€­ οƒ₯ 1 ( ) ( ) 0 ( )( ) N z i next i i next i i N x x y y ο€­ ο€½ ο€½ ο€­ οƒ₯ 1 ( ) ( ) 0 ( )( ) N y i next i i next i i N z z x x ο€­ ο€½ ο€½ ο€­ οƒ₯ Martin Newell at Utah (remember the teapot?) Why?
  • 61. 6161 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Repairing/Reconstructing Geometry β€’ Example: Coons’ Patch β€’ Code it and get bonus points! Image courtesy of CVG Lab
  • 62. 6262 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling 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; }
  • 63. 6363 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Mesh Smoothing (Relaxation) Kangaroo Physics: https://vimeo.com/27484394
  • 64. 6464 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling
  • 65. 6565 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Literal Boundary Representation A strategy for design modeling/3D sketching Curves Surfaces Solids Curves From Objects Points http://forums.newtek.com/showthread.php?143633-Can-Lightwave-Model-an-airplane-boat-Hull-Like-the-way-they-do-in-this-C4D-tutorial
  • 66. 6666 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Transformative Modeling A different strategy for design modeling/3D sketching Solids Topological Transformations Transformed Solids Curves From Objects https://3dprint.com/88049/3d-printed-topology-joke/
  • 67. 6767 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Subdivision Modelling A strategy for design modeling/3D sketching Model a Simple BRep Subdivide to Smoothen http://slodive.com/design/3ds-max-modeling-tutorials/
  • 68. 6868 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Polyhedral Meshes Polygon Meshes Topology Poincare Duality Topological Model Normal Vectors Meshing Mesh Smoothing Mesh Subdivision Design Modelling Implicit modelling as level sets A strategy for design modeling/3D sketching Model a Scalar Field Get Iso-surfaces