Siggraph Course Mesh Parameterization: Theory and Practice

Siggraph CourseMesh 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 • usually, surface properties get distorted orthographic ∼ 500 B.C. stereographic ∼ 150 B.C. Mercator 1569 Lambert 1772 conformal (angle-preserving) equiareal (area-preserving)

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

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

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

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

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

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

Example – Cylindrical Coordinates • ⇒ isometric

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

Example – Stereographic Projection • with • ⇒ conformal

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

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

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

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

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

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

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

Curvature • second fundamental form • Gaussian curvature • mean curvature

Siggraph Course Mesh Parameterization: Theory and Practice