Hyperbolic Orbifold Tutte Embeddings

摘要

Tutte’s embedding is one of the most popular approaches for computingrnparameterizations of surface meshes in computer graphicsrnand geometry processing. Its popularity can be attributed to its simplicity,rnthe guaranteed bijectivity of the embedding, and its relationrnto continuous harmonic mappings.rnIn this work we extend Tutte’s embedding into hyperbolic conesurfacesrncalled orbifolds. Hyperbolic orbifolds are simple surfacesrnexhibiting different topologies and cone singularities and thereforernprovide a flexible and useful family of target domains. The hyperbolicrnOrbifold Tutte embedding is defined as a critical point of arnDirichlet energy with special boundary constraints and is proved tornbe bijective, while also satisfying a set of points-constraints. Anrnefficient algorithm for computing these embeddings is developed.rnWe demonstrate a powerful application of the hyperbolic Tutte embeddingrnfor computing a consistent set of bijective, seamless mapsrnbetween all pairs in a collection of shapes, interpolating a set ofrnuser-prescribed landmarks, in a fast and robust manner.