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Siggraph Course Mesh Parameterization: Theory and Practice

Siggraph Course Mesh Parameterization: Theory and Practice. Differential Geometry Primer. Parameterization. surface parameter domain mapping and. Example – Cylindrical Coordinates. Example – Orthographic Projection. Example – Stereographic Projection. Example – Mappings of the Earth.

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Siggraph Course Mesh Parameterization: Theory and Practice

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  1. Siggraph CourseMesh Parameterization: Theory and Practice Differential Geometry Primer

  2. Parameterization • surface • parameter domain • mapping and

  3. Example – Cylindrical Coordinates

  4. Example – Orthographic Projection

  5. Example – Stereographic Projection

  6. 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. 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. What is Distortion? • parameter point • surface point • small disk around • image of under • shapeof D f(D)

  9. Linearization • Jacobianof • tangent plane at • Taylor expansion of • first order approximationof

  10. Infinitesimal Dis(k)tortion • small disk around • image of under • shapeof • ellipse • semiaxes and • behavior in the limit

  11. Linear Map Surgery • Singular Value Decomposition (SVD) of with rotations and and scale factors (singular values)

  12. Notion of Distortion • isometricor length-preserving • conformalor angle-preserving • equiarealor area-preserving • everything defined pointwise on

  13. Example – Cylindrical Coordinates • ⇒ isometric

  14. Example – Orthographic Projection • with • a⇒ neither conformalnor equiareal

  15. Example – Stereographic Projection • with • ⇒ conformal

  16. Computing the Stretch Factors • first fundamental form • eigenvalues of • singular values of and

  17. Measuring Distortion • local distortion measure • has minimum at • isometric measure • conformal measure • overall distortion

  18. Examples – Conformal Measures • Conformalenergy • MIPS energy [Pinkall & Polthier 1993][Lévy et al. 2002][Desbrun et al. 2002] [Hormann & Greiner 2000]

  19. Examples – Isometric Measures • Green-Lagrange deformation tensor • Combined energy [Maillot et al. 1993] [Degener et al. 2003]

  20. Examples – Other Measures • Dirichlet energy • Stretchenergies ( , , and symmetric stretch) [Pinkall & Polthier 1993] [Eck et al. 1995] [Sander et al. 2001] [Sorkine et al. 2002]

  21. Piecewise Linear Parameterizations • piecewise linear atomic maps • distortion constant per triangle • overall distortion

  22. Beyond Distortion • surface normal • surface area • independent of the particular parameterization • intrinsic surface properties

  23. Curvature • second fundamental form • Gaussian curvature • mean curvature

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