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- 1. 1Challenge the future Boundary Representations 1: Fundamentals of NURBS surface representations Ir. Pirouz Nourian PhD candidate & Instructor, chair of Design Informatics, since 2010 MSc in Architecture 2009 BSc in Control Engineering 2005 MSc Geomatics, GEO1004, Directed by Dr. Sisi Zlatanova
- 2. 2Challenge the future [Geometric, Topologic] Spatial Data Models Representations • Computer Graphics (mainly concerned with visualization) • Computational Geometry (algorithmic geometry) • AEC {CAD, CAM, BIM} (architectural engineering and construction) • CAD=: Computer Aided Design • CAM=: Computer Aided Manufacturing • BIM=: Building Information Modeling • GIS: how can we represent geometric objects in large scale properly and consistently? Different terminologies and jargons! Some common grounds
- 3. 3Challenge the future Categories of 3D Geometry Representations • Volume Representation: 1. Tetrahedral Meshes 2. Voxel Models • Boundary Representation: [AKA Surface Representation] 1. Polygon Mesh Models (Simple Brep) 2. [complex] B-rep* Models (NURBS patches) Interior included or only the closure ? A note on our terminology * B-rep here refers to a specific class of boundary representations composed of advanced faces (as implemented in Rhino): • ISO 10303-514 Advanced boundary representation, a solid defining a volume with possible voids that is composed by advanced faces • ISO 10303-511 Topologically bounded surface, definition of an advanced face, that is a bounded surface where the surface is of type elementary (plane, cylindrical, conical, spherical or toroidal), or a swept surface, or b spline surface. The boundaries are defined by lines, conics, polylines, surface curves, or b spline curves
- 4. 4Challenge the future Boundary Representation Representing high dimensional objects with lower dimensional primitives 1. Polygon Mesh ≡ Simple B-rep=composed of straight/flat elements We will discuss them later in depth 1. NURBS patches ≡[complex] B-rep*=composed of curved elements
- 5. 5Challenge the future Why NURBS? • Known and used in AEC {CAD, BIM} • In GIS ? advantages and disadvantages for GIS? Bilbao Guggenheim Museum Bus stop near Sebastiaansbrug Delft
- 6. 6Challenge the future 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).
- 7. 7Challenge the future Splines in Computer Graphics All types of curves can be modeled as splines •
- 8. 8Challenge the future NURBS Representation Non Uniform Rational Basis Splines (NURBS) are used for modeling free-form geometries accurately • 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. From: http://web.cs.wpi.edu/~matt/courses/cs563/talks/nurbs.html
- 9. 9Challenge the future Parametric Curves in General How do numeric weights correspond to physical weights? 𝑃 𝑡 = 𝑥(𝑡) 𝑦(𝑡) 𝑧(𝑡) Example 1: 𝑌 = 𝑋 + 1 Example 1: 𝑋2 + 𝑌2 = 1 → 𝑃 𝑡 = 𝑡 𝑡 + 1 0 → 𝑃 𝑡 = 𝐶𝑜𝑠(𝑡) 𝑆𝑖𝑛(𝑡) 0 𝐶𝑜𝑠2(𝑡) + 𝑆𝑖𝑛2(𝑡) = 1
- 10. 10Challenge the future NURBS equations All from a summary by Markus Altmann: http://web.cs.wpi.edu/~matt/courses/cs563/talks/nurbs.html 𝐶 𝑢 = 𝑊𝑖.𝑛 𝑖=0 𝑃𝑖. 𝑁𝑖,𝑘(𝑢) 𝑊𝑖.𝑛 𝑖=0 𝑁𝑖,𝑘(𝑢) 𝑡ℎ𝑒 𝑐𝑢𝑟𝑣𝑒 𝑒𝑣𝑎𝑙𝑢𝑎𝑡𝑒𝑑 𝑎𝑡 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑢 where 𝑊𝑖 : weights 𝑃𝑖 : control points (vector) 𝑁𝑖,𝑘 : normalized B-spline basis functions of degree k These B-splines are defined recursively as: 𝑁𝑖,𝑘 𝑢 = 𝑢 − 𝑡𝑖 𝑡𝑖+𝑘 − 𝑡𝑖 × 𝑁𝑖,𝑘−1 𝑢 + 𝑡𝑖+𝑘+1 − 𝑢 𝑡𝑖+𝑘+1 − 𝑡𝑖+1 × 𝑁𝑖+1,𝑘−1 𝑢 and 𝑁𝑖,0 𝑢 = 1, 𝑖𝑓 𝑡𝑖 ≤ 𝑢 < 𝑡𝑖+1 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Where 𝑡𝑖 are the knots forming a knot vector 𝑈 = {𝑡0, 𝑡1, … , 𝑡 𝑚} Note: 𝑘 enumerates 0 to degree, 𝑚 = 𝑛 + 𝑘 + 1
- 11. 11Challenge the future NURBS interpolation All contenet from Raja Issa [Essential Mathematics for Computational Design]
- 12. 12Challenge the future A weighted NURBS curve How do numeric weights correspond to physical weights?
- 13. 13Challenge the future • 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
- 14. 14Challenge the future Parametric Space Images courtesy of David Rutten, from Rhinoscript 101
- 15. 15Challenge the future Parametric Locations: 1D Objects (Curves): u parameter (AKA as t parameter) • Point at that address (𝐶(𝑢)) • Tangent vector • Derivatives (𝐶′(𝑢), 𝐶′′(𝑢)) • Curvature 2D Objects (Surfaces): u,v parameters • Point at that address (𝑆(𝑢, 𝑣)) • Normal vector (𝑁(𝑢, 𝑣)) • Curvature
- 16. 16Challenge the future 1D Curvature: Vector or Scalar? •
- 17. 17Challenge the future Continuity • G0 (Position continuous) • G1 (Tangent continuous) • G2 ( Curvature Continuous) Images courtesy of Raja Issa, Essential Mathematics for Computational Design
- 18. 18Challenge the future (1D Curvature Analysis) • Discretization Segments • Measurement at the middle of each segment • Attribution to each segment
- 19. 19Challenge the future Surface Curvature • Images courtesy of Raja Issa, Essential Mathematics for Computational Design
- 20. 20Challenge the future Surface Continuity: Zebra Analysis • Open question: How can we measure curvature on meshes? Images courtesy of Raja Issa, Essential Mathematics for Computational Design
- 21. 21Challenge the future (2D Curvature Analysis: NURBS surface) • Discretization Sub-surfaces • Measurement At UV points • Attribution To sub-surfaces
- 22. 22Challenge the future Questions?p.nourian@tudelft.nl We will now see a NURBS data model…

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