Free-form deformation of solid geometric models
SIGGRAPH '86 Proceedings of the 13th annual conference on Computer graphics and interactive techniques
A multisided generalization of Bézier surfaces
ACM Transactions on Graphics (TOG)
Extended free-form deformation: a sculpturing tool for 3D geometric modeling
SIGGRAPH '90 Proceedings of the 17th annual conference on Computer graphics and interactive techniques
A signal processing approach to fair surface design
SIGGRAPH '95 Proceedings of the 22nd annual conference on Computer graphics and interactive techniques
Free-form deformations with lattices of arbitrary topology
SIGGRAPH '96 Proceedings of the 23rd annual conference on Computer graphics and interactive techniques
Steinitz representations of polyhedra and the Colin de Verdiére number
Journal of Combinatorial Theory Series B
Generalized barycentric coordinates on irregular polygons
Journal of Graphics Tools
Computer Aided Geometric Design
t-FFD: free-form deformation by using triangular mesh
SM '03 Proceedings of the eighth ACM symposium on Solid modeling and applications
Globally smooth parameterizations with low distortion
ACM SIGGRAPH 2003 Papers
Fundamentals of spherical parameterization for 3D meshes
ACM SIGGRAPH 2003 Papers
Mean value coordinates for closed triangular meshes
ACM SIGGRAPH 2005 Papers
Computer Aided Geometric Design - Special issue: Geometric modelling and differential geometry
Discrete differential forms for computational modeling
SIGGRAPH '05 ACM SIGGRAPH 2005 Courses
Fluid animation with dynamic meshes
ACM SIGGRAPH 2006 Papers
Mean value coordinates for arbitrary planar polygons
ACM Transactions on Graphics (TOG)
A general geometric construction of coordinates in a convex simplicial polytope
Computer Aided Geometric Design
Spherical barycentric coordinates
SGP '06 Proceedings of the fourth Eurographics symposium on Geometry processing
On transfinite barycentric coordinates
SGP '06 Proceedings of the fourth Eurographics symposium on Geometry processing
A unified, integral construction for coordinates over closed curves
Computer Aided Geometric Design
Transfinite mean value interpolation
Computer Aided Geometric Design
Mesh parameterization: theory and practice
ACM SIGGRAPH ASIA 2008 courses
Transfinite mean value interpolation in general dimension
Journal of Computational and Applied Mathematics
Pointwise radial minimization: Hermite interpolation on arbitrary domains
SGP '08 Proceedings of the Symposium on Geometry Processing
GMP'08 Proceedings of the 5th international conference on Advances in geometric modeling and processing
Barycentric interpolation and mappings on smooth convex domains
Proceedings of the 14th ACM Symposium on Solid and Physical Modeling
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A fundamental problem in geometry processing is that of expressing a point inside a convex polyhedron as a combination of the vertices of the polyhedron. Instances of this problem arise often in mesh parameterization and 3D deformation. A related problem is to express a vector lying in a convex cone as a non-negative combination of edge rays of this cone. This problem also arises in many applications such as planar graph embedding and spherical parameterization. In this paper, we present a unified geometric construction for building these weighted combinations using the notion of polar duals. We show that our method yields a simple geometric construction for Wachspress's barycentric coordinates, as well as for constructing Colin de Verdière matrices from convex polyhedra---a critical step in Lovasz's method with applications to parameterizations.