Why manifolds?

Motivation We 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?

Domains Geometric modeling construct smooth surfaces Can 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 very difficult; add local charts anywhere you want

Parametrization Global parametrization Gives us tools to get smoothness everywhere Gu and Yau, 2003Ray, Li, Levy, Sheffer, Alliez, 2005

Parametrization Essential question: what is a smooth function on a mesh?

Parametrization Why this one global algorithm works better than another? parametrization derivative approximations Khodakovsky and Schröder

Animation Configuration spaces are manifolds

Rendering Light fields are manifolds