1. Mesh Parameterization: Theory and Practice Differential Geometry Primer

2. Mesh Parameterization: Theory and Practice Differential Geometry Primer surface parameter domain mapping and Parameterization

3. Mesh Parameterization: Theory and Practice Differential Geometry Primer Example – Cylindrical Coordinates

4. Mesh Parameterization: Theory and Practice Differential Geometry Primer Example – Orthographic Projection

5. Mesh Parameterization: Theory and Practice Differential Geometry Primer Example – Stereographic Projection

6. Mesh Parameterization: Theory and Practice Differential Geometry Primer Example – Mappings of the Earth usually, surface properties get distorted orthographic ∼ 500 B.C. stereographic ∼ 150 B.C. Mercator 1569 Lambert 1772 conformal (angle-preserving) equiareal (area-preserving)

7. Mesh Parameterization: Theory and Practice Differential Geometry Primer Distortion is (almost) Inevitable Theorema Egregium (C. F. Gauß) “A general surface cannot be parameterized without distortion.” no distortion = conformal + equiareal = isometric requires surface to be developable – planes – cones – cylinders

8. Mesh Parameterization: Theory and Practice Differential Geometry Primer What is Distortion? parameter point surface point small disk around image of under shape of

9. Mesh Parameterization: Theory and Practice Differential Geometry Primer Linearization Jacobian of tangent plane at Taylor expansion of first order approximation of

10. Mesh Parameterization: Theory and Practice Differential Geometry Primer Infinitesimal Dis(k)tortion small disk around image of under shape of – ellipse – semiaxes and behavior in the limit

11. Mesh Parameterization: Theory and Practice Differential Geometry Primer Linear Map Surgery Singular Value Decomposition (SVD) of with rotations and and scale factors (singular values)

12. Mesh Parameterization: Theory and Practice Differential Geometry Primer Notion of Distortion isometric or length-preserving conformal or angle-preserving equiareal or area-preserving everything defined pointwise on

13. Mesh Parameterization: Theory and Practice Differential Geometry Primer Example – Cylindrical Coordinates ⇒ isometric

14. Mesh Parameterization: Theory and Practice Differential Geometry Primer with ⇒ Example – Orthographic Projection neither conformal nor equiareal

15. Mesh Parameterization: Theory and Practice Differential Geometry Primer with ⇒ conformal Example – Stereographic Projection

16. Mesh Parameterization: Theory and Practice Differential Geometry Primer Computing the Stretch Factors first fundamental form eigenvalues of singular values of and

17. Mesh Parameterization: Theory and Practice Differential Geometry Primer Measuring Distortion local distortion measure has minimum at – isometric measure – conformal measure overall distortion

18. Mesh Parameterization: Theory and Practice Differential Geometry Primer Piecewise Linear Parameterizations piecewise linear atomic maps distortion constant per triangle overall distortion

19. Mesh Parameterization: Theory and Practice Differential Geometry Primer Linear Methods the terms and are quadratic in the parameter points Dirichlet energy Conformal energy minimization yields linear problem [Pinkall & Polthier 1993] [Eck et al. 1995] [Lévy et al. 2002] [Desbrun et al. 2002]

20. Mesh Parameterization: Theory and Practice Differential Geometry Primer Linear Methods both result in barycentric mappings with discrete harmonic weights for interior vertices Dirichlet maps require to fix all boundary vertices Conformal maps only two – result depends on this choice – best choice → [Mullen et al. 2008] both maps not necessarily bijective

21. Mesh Parameterization: Theory and Practice Differential Geometry Primer Non-linear Methods MIPS energy Area-preserving MIPS [Degener et al. 2003] [Hormann & Greiner 2000]

22. Mesh Parameterization: Theory and Practice Differential Geometry Primer Non-linear Methods Green-Lagrange deformation tensor Stretch energies (,, and symmetric stretch) [Sander et al. 2001] [Sorkine et al. 2002] [Maillot et al. 1993]