Ar1 twf030 lecture2.1

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  1. 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. 2. 22 A Line is something that looks like a line & works like a line! 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 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. 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: Ren 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. 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. 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. 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. 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 Eschers Balcony
  8. 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 1D 2D 3D http://paulbourke.net/geometry/hyperspace/
  9. 9. 99 Cartesian Product of Sets = = = 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 = {, , } = {1,2} = { , 1 , , 2 , , 1 , , 2 , , 1 , (, 2)}
  10. 10. 1010 = = = 1D 2D 3D http://paulbourke.net/geometry/hyperspace/ = {, , } = {1,2} = { , 1 , , 2 , , 1 , , 2 , , 1 , (, 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. 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. 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. 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. 14. 1414 On terminology Geometry: Point (0D), Curve(1D), Surface(2D), Solid (3D) [free-form] Geometry: Point (0D), Line(1D), Polygon(2D), Polyhedron (3D) [piecewise linear] Topology: Vertex(0D), Edge(1D), Face(2D), Body(3D) 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. 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. 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 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
  17. 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 An elegant mathematical description of a physical drafting aid as a (set of) parametric equation(s).
  18. 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. 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
  20. 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. 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. 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 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.
  23. 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. 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. 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. 26. 2626 Data Models Types of Geometry Dimensionality Manifolds Terminology NURBS B-Reps Simplexes Po