Curvatures, Invariants and How to Get Them Without (M)Any Derivatives

Transcription

1 Curvatures, Invariants and How to Get Them Without (M)Any Derivatives Mathieu Desbrun & Peter Schröder 1 Classical Notions Curves arclength parameterization center of osculating ( kissing ) circle (also defines osc. plane) tilt of plane is torsion Euclidian motion invariant uniquely characterizes curve! 2

2 Surfaces First fundamental form parameterized surface tangent vectors tangent space 3 Tangent Vector 4

3 Tangent Plane 5 Normal Vector N N 6

4 Metric Measure stuff angle, length, area symmetric, bilinear form 7 Metric Measuring area areas in tangent space no dependence on parameterization discrete setting easy sum areas of triangles 8

5 Metric Within each triangle the metric is obvious angle area 9 Geometry of the Normal Gauss map normal at point consider curve in surface again study its curvature at p tilting of normal along the curve 10

6 Shape Operator Derivative of Gauss map tangent space to itself second fundamental form linear map self-adjoint 11 Curvature of Curves Normal curvature curve in surface all curves with same tangent vector have same normal curvature! 12

7 Invariants Mean and Gaussian curvature determinant and trace only eigen values and (ortho) vectors 13 Invariants Mean and Gaussian curvature determinant and trace only determinant and trace only eigen values and (ortho) vectors 14

8 Curvatures Integral representations smooth setting 15 Discrete Setup? 16

9 Gaussian Curvature On a mesh can t take the limit average does make sense only makes sense as an integral, NEVER pointwise Discrete Gauss curvature at a vertex 17 A Good Definition? Gaussian curvature over a surface Gauss Bonnet Gauss-Bonnet closed, oriented discrete 18

10 A Good Definition? developable 19 Scalar Mean Curvature Integral representation variation along a vector field for constant V move out of integral 20

11 Boundary Integrals Vector area volume gradient: vector area another normal discrete version only makes sense as an integral, NEVER pointwise area weighted triangle normals 21 Boundary Integrals Area gradient vector mean curvature another normal discrete version only makes sense as an integral, NEVER pointwise 22

12 Boundary Integrals Area gradient vector mean curvature another normal discrete version only makes sense as an integral, NEVER pointwise 23 Laplace-(Beltrami) Surface over tangent plane in eigen basis in eigen basis principal curvature directions Laplace-Beltrami Laplace on the surface of the surface 25

13 Steiner Polynomial And now for a totally different view consider convex polyhedron Steiner: vertices? 26 Minimal Surface Minimum area energy Hermann Schwarz, 1890 DiMarco, Physics, Montana 30

14 Minimal Surface Minimum area energy Hermann Schwarz, 1890 DiMarco, Physics, Montana 31 Minimal Surface Minimum area energy Hermann Schwarz, 1890 DiMarco, Physics, Montana 32

15 Minimal Surface Minimum area energy Hermann Schwarz, 1890 DiMarco, Physics, Montana 33 Minimal Surface Minimum area energy Hermann Schwarz, 1890 DiMarco, Physics, Montana 34

16 Minimal Surface Minimum area energy Hermann Schwarz, 1890 DiMarco, Physics, Montana 35 Minimal Surface Minimum area energy Hermann Schwarz, 1890 DiMarco, Physics, Montana 36

17 Minimal Surface Minimum area energy Hermann Schwarz, 1890 DiMarco, Physics, Montana 37 Minimal Surface Minimum area energy Hermann Schwarz, 1890 DiMarco, Physics, Montana 38

18 Minimal Surface Minimum area energy Hermann Schwarz, 1890 DiMarco, Physics, Montana 39 Mean Curvature Flow Laplace-Beltrami Dirichlet energy on surface 40

19 Mean Curvature Flow Laplace-Beltrami Dirichlet energy on surface 41 Mean Curvature Flow Laplace-Beltrami Dirichlet energy on surface 42

20 Mean Curvature Flow Laplace-Beltrami Dirichlet energy on surface 43 Mean Curvature Flow Laplace-Beltrami Dirichlet energy on surface 44

21 Mean Curvature Flow Laplace-Beltrami Dirichlet energy on surface 45 Convergence? Can be tricky see Cohen-Steiner paper think about chinese lanterns Schwarz s example a good one to keep in mind 46

22 Parameterizations What is a parameterization? function from some region c R 2 to the embedded surface M c R 3 we go the other way around how to measure distortion? Image from Sander et al Measuring Distortion Dirichlet energy of a map [Pinkall/Polthier 93] harmonic param: minimizer is discrete harmonic angles in mesh texture coords. need to fix boundary 48

23 Harmonic Map Properties of minimizer link with area of triangle?? conditions for u to be conformal 49 Discrete Conformal Minimizer of conformal energy Fixed boundary: Dirichlet condition Free boundary: match gradient of area 50

24 Little Aside What s the param of a flat mesh? itself Notion of Barycentric Coordinates vertex can be reconstructed by a linear combo of its neighbors try it on cot formula can you think of other weights? 51 Recap Invariants as overarching theme shape does not depend on Euclidean motions metric and curvatures smooth continuous notions to discrete notions variational formulations careful: generally only as averages 52

25 Tools Operators we have now volume gradient: notion of normal area gradient: notion of normal Flow also: mean curvature smoothing, parameterization, editing (bi-laplace-beltrami) lt i) G 1 boundaries to make smooth bump Harmonic Conformal 53 Down the Line Approach so far essentially linear: PL mesh same equations can be derived with DEC: discrete exterior calculus coming right up! 54