1. CS175 2003 1 CS 175 – Week 7 Parameterization Linear Methods

2. CS175 2003 2 Overview motivation linear methods spring model discrete conformal mean value

3. CS175 2003 3 Definition triangle mesh T = {T 1, …,T n } with edges E and vertices V = V I [ V B surface Ω T = [ i=1…n T i parameter domain Ω ½ R 2 parameterization  : Ω ! Ω T inverse map  =  -1

4. CS175 2003 4 Motivation correspondence between 2- manifold Ω T and R 2 perform operations on Ω T in R 2 colouring (texture mapping) remeshing surface fitting

5. CS175 2003 5 Background flattening of surfaces into R 2 maps of the Earth orthographic (Egyptians, Greeks) stereographic (Hipparchus, 190–120 b.c.) conformal cylindric (Mercator, 1512– 1594) area-preserving (Lambert, 1728–1777) inevitable distortion

6. CS175 2003 6 1D Analogue interpret edges as springs minimize total spring energy subject to boundary conditions different spring constants uniform centripetal chord length

7. CS175 2003 7 Goals bijection linear precision invariant to rotation and translation minimize distortion

8. CS175 2003 8 2D Spring Model same approach fix boundary vertices minimize spring energy solve linear system symmetric positive definite

9. CS175 2003 9 2D Spring Model analogues of uniform, centripetal, and chord-length parameterization  bijection  invariant to rotation and translation  linear precision  minimize distortion

10. CS175 2003 10 Discrete Conformal Maps Riemann (1826–1866) any surface has a conformal map conformal = preserves angles conformal energy E C = E D – A Dirichlet energy E D (f) = 1/2 s || rf || 2 fix boundary and minimize E D gives different spring constants

11. CS175 2003 11 Discrete Conformal Maps spring constants may be negative linear system still positive definite  minimize distortion  linear precision  invariant to rotation and translation  bijection

12. CS175 2003 12 Mean Value Coordinates conformal maps are harmonic mean value theorem other property of harmonic maps discretize at vertices gives non-symmetric spring constants

13. CS175 2003 13 Mean Value Coordinates linear system not symmetric still regular  bijection  linear precision  minimize distortion  invariant to rotation and translation

14. CS175 2003 14 Next Session boundary issues non-linear methods parameterization of closed surfaces