Energy-minimizing Curve Design Gang Xu Zhejiang University Ouyang Building, 20-December-2006

2 ContentContent Application Background(5min) Traditional energy-minimizing curves(30min)  Internal energy  External energy Energy-minimizing curves in manifolds(15min) Energy-minimizing curve networks (15min) Our recent work(15min) Summary and outlook(5min)

3

4 Application Background(1) Energy minimizing is the favourite of nature!

5 Application Background(2) Fairing curves

6 Application Background(3) Computer vision and image processing geodesics and active contours

7 Application Background(4) Medicine (Path planning of surgical sutures)

8 Application Background (5) Robotic  Snake robots  Motion design

Traditional Energy-minimizing Curve Internal energy External energy

10 Internal energy of curves(1) Stretch energy —— length Strain (bend) energy —— spline and fairness

11 Internal energy of curves(2) Energy in tension Variation of curvature —— circle-like

12 Internal energy of curves(3) Jerk and load Energy of 3D curves

13 Interpolating curves with gradual changes in curvature(CAD,1987) H. Meier and H. Nowacki, Germany Interpolating Solve linear system

14 Method to approximate the space curve of least energy and prescribed length (CAD, 1987) M Kallay This paper presents a numerical method for computing the curves of least strain energy, given the positions and directions of the endpoints and the total length. Discrete method

15 Variational design of rational Bezier curves (CAGD, 1991) H Hagen and GP Bonneau Describe a calculus of variation approach to design the weights of a rational curve in a way as to achieve a smooth curve in the sense of an energy integral Method?

16 Minimum curvature variation curves (PhD, 1992) (1) HP. Moreton, CH. Sequin Method: numerical integration gradient descent

17 Minimum curvature variation curves (PhD, 1992) (2) Space curves

18 Variational subdivision curves (TOG, 1998) Leif Kobbelt Interpolating variational subdivision curves Approximation variational subdivision curves (Hofer and Pottmann, TVC, 2002)

19 Interpolating Method(1) Objective function (open)

20 Interpolating Method(2) Solve a linear system

21 Interpolating Method(3)

22 Approximating Method(1) Objective function

23 Approximating Method(2) Solve a linear system

24 Approximating Method(3)

25 Approximating Method(4)

26 SummarySummary Fair and smoothness Numerical method Not geometric method!

Traditional Energy-minimizing Curve Internal energy External energy

28 Interactive design of constrained variational curves (CAGD, 1995) W. Wesselink, RC. Veltkamp Motivation constrained condition energy function (global) Edit using control points Not flexible! Not variational design!

29 Solution Solution

30 ConstraintsConstraints Point interpolation Normal (tangent) interpolation

31 External energy operators(1) Director

32 External energy operators(2) Point attractor

33 External energy operators(3) Curve attractor

34 Combing the energy terms

35 Computation(1)Computation(1)

36 Computation(2)Computation(2)

37 Computation(3)Computation(3)

38 ExamplesExamples

39 LimitationLimitation

40 Modeling 3D curves of minimal energy( EG, 1995) Generalize 2D to 3D Differences(1)

41 Difference(2)Difference(2)

42 Difference(3) —— constraints Point-in-planePoint-in-plane Point-on-linePoint-on-line

43 Difference (4) Plane attractor

44 Difference (5) Director

45 Difference (6) Profiler

46 Difference (7) Point repellor

47 SummarySummary Unify of smoothness and interaction Generalization to surface is easy! Numerical method Not geometric method!

48 Energy minimizing splines in manifolds (Siggraph, 2004) Hofer, Pottmann, Wallner CharacterizationComputationApplication

49 Variational interpolation in curved geometries Find curve as solution of a variational problem Use energy functions from spline theory, but restrict curve to surface Any surface representation, dimension & co-dimens.

50 InputInput Points p 1,..., p N on d-dimensional surface S in R n, parameter values u 1,..., u N p1p1 pipi pNpN p2p2 S

51 Geodesics on surfaces Minimize L 2 norm of 1 st derivative on surface S: Shortest connecting curve c on surface traced with const. speed Pieces of c have 2 nd derivative vectors orthogonal to S S

52 Counterparts to cubic splines on surfaces Minimize L 2 norm of 2 nd derivative: Interpol. C 2 curve c on S 4 th derivative vectors of c are orthogonal to surface Existence [Bohl 1999, Wallner 2004] S

53 Counterparts to splines in tension on surfaces(1) Minimize: C 2 curve such that is orthogonal to surface S

54 Counterparts to splines in tension on surfaces(2)

55 Energy minimizing splines in manifolds CharacterizationComputationApplication

56 Computation (1) Non-linear problem Even for simple surfaces no explicit solution Numerical algorithm for various surface representations and dimensions

57 Computation (2) Discretize curve on S in R low to polygon P View P as a point X in high-dim space R high Constraint manifold  in R high is set of X‘s for which vertices of P are contained in S P R low S R high  X

58 Computation (3) Quadratic functional  Discretization  Quadratic function P... Minimizer of F in R high P*... Minimizer of F on  Matrix Q of F determines a Euclidean metric   R high P P*P*

59 Computation (4) Solution P* of our problem is normal footpoint of P on  in the metric given by Q Iterative algorithm with geometrically motivated stepsize control  P R high P*P* X0X0

60 Computation (5) Iterative algorithm with geometrically motivated stepsize control Algorithm needs: –initial value x 0 –tangent space –Projection onto surface

61 Spline curves on various surface representations parametric implicit triangle mesh point cloud

62 Energy minimizing splines in manifolds CharacterizationComputationApplication

63 Cyclic motion minimizing cubic spline energy E 2

64 Cyclic motion minimizing tension spline energy E t

65 Cyclic motion minimizing kinetic energy E 1

66 Splines avoiding obstacles in 3D / on bounded surfaces

67 Variational motion design in the presence of obstacles

68 Energy minimizing curve networks Mininum variation networks Mininum variation networks Fair webs on surface Fair webs on surface

69 Minimal variational networks (Siggraph, 1992) Moreton, Sequin Problem Method: constraints and energy minimizing

70 MotivationMotivation obtain high quality surface

71 ExamplesExamples

72 Energy minimizing curve networks Mininum variation networks Mininum variation networks Fair webs on surface Fair webs on surface

73 Fair Webs (TVC, 2007) Wallner, Pottmann, Hofer Contribution: a variational approach to the design of energy minimizing curve networks that are constrained to lie in a given surface or to avoid a given obstacle

74 InputInput Connectivity

75 EnergyEnergy

76 PropertiesProperties

77 Fair polygon networks

78 Application (1) Aesthetic remeshing

79 Application (2) Fair parameterization

80 Application (3) Surface restoration and approximation

81 Application (4) Fair surface design in the presence of obstacles

82 Thanks! Congratulations on 7th Anniversary of Macao's Return!