Abstract
Barycentric coordinates are very popular for interpolating data values on polyhedral domains. It has been recently shown that expressing them as complex functions has various advantages when interpolating two-dimensional data in the plane, and in particular for holomorphic maps. We extend and generalize these results by investigating the complex representation of real-valued barycentric coordinates, when applied to planar domains. We show how the construction for generating real-valued barycentric coordinates from a given weight function can be applied to generating complex-valued coordinates, thus deriving complex expressions for the classical barycentric coordinates: Wachspress, mean value, and discrete harmonic. Furthermore, we show that a complex barycentric map admits the intuitive interpretation as a complex-weighted combination of edge-to-edge similarity transformations, allowing the design of "home-made" barycentric maps with desirable properties. Thus, using the tools of complex analysis, we provide a methodology for analyzing existing barycentric mappings, as well as designing new ones.
Original language | English (US) |
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Pages (from-to) | 1533-1542 |
Number of pages | 10 |
Journal | Eurographics Symposium on Geometry Processing |
Volume | 30 |
Issue number | 5 |
DOIs | |
State | Published - 2011 |
Externally published | Yes |
All Science Journal Classification (ASJC) codes
- Modeling and Simulation
- Geometry and Topology
Keywords
- Categories and subject descriptors (according to ACM CCS)
- G.1.1 [Numerical analysis]
- I.3.3 [Computer graphics]
- Interpolation-Interpolation formulas
- Picture/Image generation-line and curve generation