Shape Modeling. Differential Geometry Primer Smooth Definitions Discrete Theory in a Nutshell. CS 523: Computer Graphics, Spring 2011

Transcription

1 CS 523: Computer Graphics, Spring 2011 Shape Modeling Differential Geometry Primer Smooth Definitions Discrete Theory in a Nutshell 2/15/2011 1

2 Motivation Geometry processing: understand geometric characteristics, e.g. smoothness 2/15/2011 2

3 Motivation Geometry processing: understand geometric characteristics, e.g. smoothness how shapes deform 2/15/2011 3

4 Curves smooth definition 2/15/2011 4

5 Curves are 1-dimensional parameterizations p: R R d, d = 1, 2, 3, t p(t) Curves smooth definition t=0 t=0.5 Planar curve:p(t) = (x(t), y(t)) Space curve: p(t) = (x(t), y(t), z(t)) t=0.75 t=1 2/15/2011 5

6 Circle in 2D p(t) = (r cos(t), r sin(t)) t [0, 2π) Parametric Curves Examples Béziercurve n p( t) = p B n i i= 0 n i B n i ( t) i ( ) ( ) n t = t 1 t i i Curve and control polygon Basis functions 2/15/2011 6

7 Equal pace of the parameter along the curve len (p(t 1 ), p(t 2 )) = t 1 t 2 Curves arc length parameterization 2/15/2011 7

8 Secant A line through two points on the curve. 2/15/2011 8

9 Secant A line through two points on the curve. 2/15/2011 9

10 Tangent The limiting secant as the two points come together. 2/15/

11 Secant: p(t) p(s) Tangent: p (t) = (x (t), y (t), ) If tis arc-length: p (t) = 1 Secant and tangent parametric form 2/15/

12 Tangent, normal, radius of curvature r p 2/15/

13 Circle of curvature Consider the circle passing through three points on the curve 2/15/

14 Circle of curvature the limiting circle as three points come together. 2/15/

15 Radius of curvature, r 2/15/

16 Radius of curvature, r = 1/κ Curvature κ= r 1 r 1/κ 2/15/

17 Signed curvature Sense of traversal along curve. +κ κ 2/15/

18 nˆ ( p) Gauss map, n(p) Point on curve maps to point on unit circle. 2/15/

19 Curvature = change in normal direction Absolute curvature (assuming arc length t) κ= n ˆ ( t ) Parameter-free view: via the Gauss map curve Gauss map curve Gauss map 2/15/

20 Assume tis arc-length parameter Curvature normal parametric form p ( t) = κnˆ( t ) nˆ ( t) p(t) p (t) [Kobbelt and Schröder] 2/15/

21 Curvature normal parametric form Note: if the parameter has constant speed, it only changes along the normal direction In other words, p ( t) p ( t) nˆ ( t) p ( t), p ( t) = 1 / differentiate both sides p(t) p ( t), p ( t) + p ( t), p ( t) = 0 p (t) p ( t), p ( t) = 0 2/15/

22 Turning number, k Number of orbits in Gaussian image. 2/15/

23 Turning number theorem +2π κ Ω ds= 2πk 2π For a closed curve, the integral of curvature is an integer multiple of 2π. +4π 0 2/15/

24 Discrete planar curves 2/15/

25 Discrete planar curves Piecewise linear curves Not smooth at vertices Can t take derivatives Generalize notions from the smooth world for the discrete case! 2/15/

26 Tangents, normals For any point on the edge, the tangent is simply the unit vector along the edge and the normal is the perpendicular vector 2/15/

27 Tangents, normals For vertices, we have many options 2/15/

28 Tangents, normals Can choose to average the adjacent edge normals n ˆ v = n ˆ ˆ e +n nˆ e e 1 e nˆ e e 2 e 2 e 1 2/15/

29 Tangents, normals Weight by edge lengths nˆ v = e 1 nˆ e + e 2 nˆ 1 e 2 e 1 nˆ ˆ e + e2 n 1 e 2 e 2 e 1 2/15/

30 Finite number of vertices each lying on the curve, connected by straight edges. Inscribed polygon, p connection between discrete and smooth 2/15/

31 The length of a discrete curve n n+ 1 len( p) = i= 1 d i = i= 1 p i+ 1 p i Sum of edge lengths p 1 p p 3 2 d 2 p d 4 3 d 1 2/15/

32 The length of a continuous curve Length of longest of all inscribed polygons. sup len( p) ) p 2/15/

33 The length of a continuous curve or take limit over a refinement sequence lim h 0 len( p) h= max edge length 2/15/

34 The length of a continuous curve In the continuous form: b len= p ( s) s=a ds p 2 p 3 p 4 p 1 2/15/

35 The length of a continuous curve Compare: b + len= p ( s) ds len( p) = p i+ 1 s=a n 1 i=1 p i tangent length p 2 p 3 p 4 p 1 2/15/

36 The length of a continuous curve When the parameter is arc-length: l len= p ( t) dt= 1dt t = 0 t= 0 l = l 2/15/

37 Curvature of a discrete curve Curvature is the change in normal direction as we travel along the curve no change along each edge curvature is zero along edges 2/15/

38 Curvature of a discrete curve Curvature is the change in normal direction as we travel along the curve normal changes at vertices record the turning angle! 2/15/

39 Curvature of a discrete curve Curvature is the change in normal direction as we travel along the curve normal changes at vertices record the turning angle! 2/15/

40 Curvature of a discrete curve Curvature is the change in normal direction as we travel along the curve same as the turning angle between the edges 2/15/

41 Signed curvature of a discrete curve Zero along the edges Turning angle at the vertices = the change in normal direction α 1, α 2 > 0, α 3 < 0 α 1 α 2 α 3 2/15/

42 Total signed curvature n tsc( p) = α i i= 1 Sum of turning angles α 1 α 2 α 3 2/15/

43 Discrete Gauss Map Edges map to points, vertices map to arcs. 2/15/

44 Discrete Gauss Map Turning number well-defined for discrete curves. 2/15/

45 Discrete Turning Number Theorem n tsc( p) = i= 1 α i = 2πk For a closed curve, the total signed curvature is an integer multiple of 2π. proof: sum of exterior angles 2/15/

46 Structure preservation Arbitrary discrete curve total signed curvature obeys discrete turning number theorem even coarse mesh (curve) which continuous theorems to preserve? that depends on the application 2/15/

47 Convergence Consider refinement sequence length of inscribed polygon approaches length of smooth curve in general, discrete measure approaches continuous analogue which refinement sequence? depends on discrete operator pathological sequences may exist in what sense does the operator converge? (point-wise, L 2 ; linear, quadratic) 2/15/

48 Curvature normal = length gradient Can use this to define discrete curvature! 2/15/

49 Curvature normal = length gradient 2/15/

50 Curvature normal = length gradient 2/15/

51 Curvature normal = length gradient 2/15/

52 Curvature normal = length gradient + 2/15/

53 Curvature normal = length gradient 2/15/

54 Curvature normal = length gradient 2/15/

55 Recap Structurepreservation Convergence For an arbitrary (even coarse) discrete curve, the discrete measure of curvature obeys the discrete turning number theorem. In the limit of a refinement sequence, discrete measures of length and curvature agree with continuous measures. 2/15/