Why manifolds?

MotivationWe know well how to compute with planar domains

and functions many graphics and geometric modeling applications

involve domains of nontrivial topology closed surfaces, configuration spaces, light fields …

Manifolds: a tool for constructing algorithms do computations on planar domains then blend

together;how to blend smoothly?

Manifolds: a tool for understanding algorithms why do we see (or do not see) problems when

computing with complex domains?

DomainsGeometric modeling

construct smooth surfacesCan get unique combinations of properties

understand how to build smooth global parametrizations

Animation smoothly interpolate motions represent config. spaces for motion editing

Rendering assemble smooth lightfields from different

views, represent BRDFs

Constructing smooth surfaces

Can get unique combinations of properties: arbitrary smoothness, local support, flexibility; compare: even C2 subdivision is verydifficult; add local charts anywhere you want

ParametrizationGlobal parametrization

Gives us tools to get smoothness everywhere

Gu and Yau, 2003 Ray, Li, Levy, Sheffer, Alliez, 2005

ParametrizationEssential question: what is a smooth function

on a mesh?

ParametrizationWhy this one global algorithm works better

than another?

parametrization derivative approximations

Khodakovsky and Schröder

ExamplesAnimation

Configuration spaces are manifoldsRendering

Light fields are manifoldsSurface modeling and parameterization

Goals for surface modelingA high-order surface construction

Important for geometric and numerical computation

Desirable features or smoothness At least 3-flexibility at vertices Closed-form smooth local parameterizations Can handle arbitrary control meshes Good visual quality Easy to implement

Smoothness smoothness

A standard goal in CAGD important for high-accuracy computation

Computing surface properties : needed for normal : needed for curvatures, reflection lines; : needed for curvature variation;

FlexibilityAbility to represent local geometry

Property of basis function, instead of the surface Two-Flexibility: any desired curvature at any

point 1-flexible 2-flexible

Todo: replace with shaded picture

Local ParameterizationExplicit smooth local parameterization

For any point, there is an explicit formula defining the surface in a neighborhood of this point

Simplifies many tasks Defining functions on surfaces Integration over surfaces Surface-surface intersections Computing geodesics

Spline-based Approach

Construct surface patch for each faceFind smooth local parameterization for every point

Difficult to guarantee smoothness for points on patch boundaries

Manifold-based Approach

Construct overlapping charts covering the meshBuild local geometry approximating the mesh on each chartFind blending function for each chartGet the surface by blending local geometry

… …

Previous WorkHigh-order spline patches

S-patches [Loop and DeRose 1989] DMS splines [Seidel 1994] Freeform splines [Prautzsch 1997] TURBS [Reif 1998]

C2 flexible subdivision surfaces G2 subdivision [Prautzsch and Umlauf 1996]

Manifold-based approach [Grimm and Hughes 1995] [Navau and Garcia 2000] [Grimm 2002] [Ying, Zorin 2004] [Qin, Gu, He, 2005] [Grimm 2005]