CS 468, Spring 2013 Differential Geometry for Computer Science Justin Solomon and Adrian Butscher

Transcription

1 CS 468, Spring 2013 Differential Geometry for Computer Science Justin Solomon and Adrian Butscher

2 µ R 3 µ R 2 Not suitable for implementation

3 What is a discrete surface? How do you store it?

4 1. Each edge is incident to one or two faces 2. Faces incident to a vertex form a closed or open fan

5 1. Each edge is incident to one or two faces 2. Faces incident to a vertex form a closed or open fan

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7 f f(t) P f(t + h)

8 f(t ) f f(t) P f(t + h)

9 O(h 2 ) f(t ) f f(t) P f(t + h)

10 Piecewise linear faces are reasonable building blocks.

11 Simple to render Arbitrary topology possible Basis for subdivision, refinement

12 Topology [tuh-pol-uh-jee]: The study of geometric properties that remain invariant under certain transformations

13 Geometry: This vertex is at (x,y,z).

14 Topology: These vertices are connected.

15 V = fv 1 ; v 2 ; : : : ; v n g ½ R n E = fe 1 ; e 2 ; : : : ; e k g ½ V V F = ff 1 ; f 2 ; : : : ; f m g ½ V V V Easy to generalize to non-triangles

16 Valence = 6

17 V E + F = Â Â = 2 2g g = 0 g = 1 g = 2

18 V E + F = Â Â = 2 2g g = 0 g = 1 g = 2

19 V E + F = Â Each edge is adjacent to two faces. Each face has three edges. 2E = 3F Closed mesh: Easy estimates!

20 V 1 2 F = Â Each edge is adjacent to two faces. Each face has three edges. 2E = 3F Closed mesh: Easy estimates!

21 V 1 2 F = Â F ¼ 2V Closed mesh: Easy estimates!

22 E ¼ 3V F ¼ 2V average valence ¼ 6 General estimates

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24 Normal field isn t continuous

25 Normal field isn t continuous

26 Must represent geometry and topology of surface.

27 x1 y1 z1 / x2 y2 z2 / x3 y3 z3 x1 y1 z1 / x2 y2 z2 / x3 y3 z3 x1 y1 z1 / x2 y2 z2 / x3 y3 z3 x1 y1 z1 / x2 y2 z2 / x3 y3 z3 x1 y1 z1 / x2 y2 z2 / x3 y3 z3 CS (M. Ben-Chen), other slides Triangle soup

28 x1 y1 z1 / x2 y2 z2 / x3 y3 z3 x1 y1 z1 / x2 y2 z2 / x3 y3 z3 x1 y1 z1 / x2 y2 z2 / x3 y3 z3 x1 y1 z1 / x2 y2 z2 / x3 y3 z3 x1 y1 z1 / x2 y2 z2 / x3 y3 z3 glbegin(gl_triangles) CS (M. Ben-Chen), other slides Triangle soup

29 f f v v v 5 v 3 v 2 f 1 f 2 v 1 CS (M. Ben-Chen), other slides Shared vertex structure

30 for i=1 to n for each vertex v v =.5*v +.5*(average of neighbors);

31 Neighboring vertices to a vertex Neighboring faces to an edge Edges adjacent to a face Edges adjacent to a vertex Mostly localized

32 Neighboring vertices to a vertex Neighboring faces to an edge Edges adjacent to a face Edges adjacent to a vertex Mostly localized

33 Vertices Faces Half-edges Structure tuned for meshes

34 Oriented edge

35 Vertex stores: Arbitrary outgoing halfedge

36 Face stores: Arbitrary adjacent halfedge

37 Halfedge stores: Flip Next Face Vertex

38 Iterate(v): startedge = v.out; e = startedge; do process(e.flip.from) e = e.flip.next while e!= startedge

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40 Face Dimension 2 Edge Dimension 1 Vertex Dimension 0

41 Face Dimension 2 Edge Dimension 1 Vertex Dimension 0

42 Face Dimension 2 Edge Dimension 1 Vertex Dimension 0

43 @ Face Dimension 2 Edge Dimension 1 Vertex Dimension 0

44 f :! R Map points to real numbers

45 f 2 R jv j Map vertices to real numbers

46 What is the integral of f? Z M f da

47 Use hat functions to interpolate

48 v i i Discrete version of da

49 v i i Z i f da = f i j i j Discrete version of da

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54 ????

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56 e! Rot! Rot = e! Flip

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59 Complex data structures enable simpler traversal at cost of more bookkeeping.

60 ftp://ftp-sop.inria.fr/geometrica/alliez/signing.pdf Implicit surfaces

61 Smoothed-particle hydrodynamics

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63 Cleanest: Design software

64 Cleanest: Design software

65 Volumetric extraction

66 Volumetric extraction

67 Point clouds

68 Well-behaved dual mesh

69 Tangent plane Derive local triangulation from tangent projection Restricted Delaunay Usual Delaunay strategy but in smaller part of R 3 Inside/outside labeling Find inside/outside labels for tetrahedra Empty balls Require existence of sphere around triangle with no other point Delaunay Triangulation Based Surface Reconstruction: Ideas and Algorithms Cazals and Giesen 2004

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71 CS 468, Spring 2013 Differential Geometry for Computer Science Justin Solomon and Adrian Butscher