Justin Solomon MIT, Spring 2017

Transcription

1 Justin Solomon MIT, Spring 2017 Some materials from Stanford CS 468, spring 2013 (Butscher & Solomon)

2 What is a curve?

3 A function?

4 Not a curve

5 Jams on accelerator

6

7 A curve is a set of points with certain properties. It is not a function.

8 Set of points that locally looks like a line.

9

10 Now this is OK!

11 Trace of parameterized curve Component functions

12 Geometric measurements should be invariant to changes of parameter.

13 On the board: Effect on velocity and acceleration.

14 On the board: Independence of parameter

15 Constant-speed parameterization

16

17 Differential geometry should be coordinate-invariant. Referring to x and y is a hack! (but sometimes convenient )

18 How do you characterize shape without coordinates?

19 On the board: Use coordinates from the curve to express its shape!

20

21 Fundamental theorem of the local theory of plane curves: k(s) characterizes a planar curve up to rigid motion.

22 Fundamental theorem of the local theory of plane curves: k(s) characterizes a planar curve up to rigid motion. Statement shorter than the name!

23 Image from DDG course notes by E. Grinspun Provides intuition for curvature

24 Binormal: T N Curvature: In-plane motion Torsion: Out-of-plane motion

25 Fundamental theorem of the local theory of space curves: Curvature and torsion characterize a 3D curve up to rigid motion.

26 Suspicion: Application to time series analysis? ML? C. Jordan, 1874 Gram-Schmidt on first n derivatives

27 What do these calculations look like in software?

28 Piecewise smooth approximations

29 What is the arc length of a cubic Bézier curve?

30 What is the arc length of a cubic Bézier curve?

31 Sad fact: Closed-form expressions rarely exist. When they do exist, they usually are messy.

32

33 Piecewise linear

34 Boring differential structure

35 THEOREM: As Δh 0, [insert statement].

36 THEOREM: As Δh 0, [insert statement].

37 Convergence to continuous theory Discrete behavior

38 Examine discrete theories of differentiable curves.

39 Examine discrete theories of differentiable curves.

40 Normal map from curve to S 1

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44 A global theorem!

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49 ( ) Total change in curvature

50 ( ) Total change in curvature

51 ( ) Total change in curvature

52 Same integrated curvature

53 Same integrated curvature

54 ( ) Total change in curvature

55 )( )( )( )( )( )( )(

56 decreases length the fastest.

57

58 Same behavior in the limit

59 Does discrete curvature converge in limit?

60 Does discrete curvature converge in limit? Questions: Type of convergence? Sampling? Class of curves?

61 Different discrete behavior Same convergence

62 Curves in 3D

63

64 NMR scanner Kinked alpha helix Structure Determination of Membrane Proteins Using Discrete Frenet Frame and Solid State NMR Restraints Achuthan and Quine Discrete Mathematics and its Applications, ed. M. Sethumadhavan (2006)

65 Discrete Frenet frame Bond and torsion angles (derivatives converge to κ and τ, resp.) Discrete frame introduced in: The resultant electric moment of complex molecules Eyring, Physical Review, 39(4): , 1932.

66 Discrete construction that works for fractal curves and converges in continuum limit. Discrete Frenet Frame, Inflection Point Solitons, and Curve Visualization with Applications to Folded Proteins Hu, Lundgren, and Niemi Physical Review E 83 (2011)

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68 Discrete Elastic Rods Bergou, Wardetzky, Robinson, Audoly, and Grinspun SIGGRAPH

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70 Normal part encodes twist

71 Punish turning the steering wheel

72 Punish turning the steering wheel

73 Punish non-tangent change in material frame

74 Punish non-tangent change in material frame Swapping m 1 and m 2 does not affect E twist!

75

76 Most relaxed frame

77 No twist ( parallel transport ) Most relaxed frame

78 Degrees of freedom for elastic energy: Shape of curve Twist angle θ

79 Upper index: dual Lower index: primal

80 Tangent unambiguous on edge

81 Turning angle Yet another curvature! Integrated curvature

82 Yet another curvature! Orthogonal to osculating plane, norm κ i Darboux vector

83 Convert to pointwise and integrate

84 Map tangent to tangent Preserve binormal Orthogonal

85

86 Note θ 0 can be arbitrary

87 \omit{physics}

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90 One curve, three curvatures.

91 Easy theoretical object, hard to use.

92 Proper frames and DOFs go a long way.

93 Surfaces

94 Justin Solomon MIT, Spring 2017 Some materials from Stanford CS 468, spring 2013 (Butscher & Solomon)