Mesh Parameterization Theory and Practice Differential Geometry Primer

Mesh Parameterization: Theory and Practice Differential Geometry Primer

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

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

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

Example – Stereographic Projection • • 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 conformal (angle-preserving) Mesh Parameterization: Theory and Practice Differential Geometry Primer Lambert 1772 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 Mesh Parameterization: Theory and Practice Differential Geometry Primer

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

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

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

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

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

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

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

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

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

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

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

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 Mesh Parameterization: Theory and Practice Differential Geometry Primer

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

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