Multivariable Curve Interpolation
Journal of the ACM (JACM)
Generalized barycentric coordinates on irregular polygons
Journal of Graphics Tools
Computer Aided Geometric Design
An intuitive framework for real-time freeform modeling
ACM SIGGRAPH 2004 Papers
Mean value coordinates for closed triangular meshes
ACM SIGGRAPH 2005 Papers
Mean value coordinates for arbitrary planar polygons
ACM Transactions on Graphics (TOG)
Image vectorization using optimized gradient meshes
ACM SIGGRAPH 2007 papers
Harmonic coordinates for character articulation
ACM SIGGRAPH 2007 papers
Automatic rigging and animation of 3D characters
ACM SIGGRAPH 2007 papers
ACM SIGGRAPH 2008 papers
Diffusion curves: a vector representation for smooth-shaded images
ACM SIGGRAPH 2008 papers
Transfinite mean value interpolation
Computer Aided Geometric Design
Coordinates for instant image cloning
ACM SIGGRAPH 2009 papers
Automatic and topology-preserving gradient mesh generation for image vectorization
ACM SIGGRAPH 2009 papers
Patch-based image vectorization with automatic curvilinear feature alignment
ACM SIGGRAPH Asia 2009 papers
Pointwise radial minimization: Hermite interpolation on arbitrary domains
SGP '08 Proceedings of the Symposium on Geometry Processing
Maximum entropy coordinates for arbitrary polytopes
SGP '08 Proceedings of the Symposium on Geometry Processing
Controllable conformal maps for shape deformation and interpolation
ACM SIGGRAPH 2010 papers
Bounded biharmonic weights for real-time deformation
ACM SIGGRAPH 2011 papers
Positive Gordon-Wixom coordinates
Computer-Aided Design
Computer Graphics Forum
Surface- and volume-based techniques for shape modeling and analysis
SIGGRAPH Asia 2013 Courses
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We present a new method for interpolating both boundary values and gradients over a 2D polygonal domain. Despite various previous efforts, it remains challenging to define a closed-form interpolant that produces natural-looking functions while allowing flexible control of boundary constraints. Our method builds on an existing transfinite interpolant over a continuous domain, which in turn extends the classical mean value interpolant. We re-derive the interpolant from the mean value property of biharmonic functions, and prove that the interpolant indeed matches the gradient constraints when the boundary is piece-wise linear. We then give closed-form formula (as generalized barycentric coordinates) for boundary constraints represented as polynomials up to degree 3 (for values) and 1 (for normal derivatives) over each polygon edge. We demonstrate the flexibility and efficiency of our coordinates in two novel applications, smooth image deformation using curved cage networks and adaptive simplification of gradient meshes.