Complex barycentric coordinates (Technion)
Traditionally, barycentric coordinates are required to satisfy a number of key properties, such as smoothness and positivity. In two-dimensional space, the coordinates are defined based on a closed contour - usually a polygon. Each point in space is assigned a set of real number coordinates such that one coordinate is assigned with respect to each vertex of the surrounding polygon. Taking linear combinations of the data values given at the polygon vertices and using the coordinates as coefficients leads to smooth propagation of the data over the plane. A major limitation of barycentric coordinates is that they reproduce affine transformations. In addition, some types of barycentric coordinates are hard to compute mainly because they do not have a closed-form expression and numerical solutions need to be found.We propose to relax the requirement that the barycentric coordinates be real. Instead, we allow them to be complex numbers. In addition, we define a new type of barycentric coordinates, called point-to-point coordinates, which are based on an enclosing contour but also based on a set of internal “control points”. Such coordinates can be derived from any other coordinates (even real ones) under most circumstances. The computation is done by adding a regularity condition as well as interpolation constraints at the selected points and solving a linear system. In contrast to real barycentric coordinates, complex barycentric coordinates are not affine invariant. However, they are still translation, rotation and scale invariant, which are desirable features. The types of coordinates that we construct lead to smooth conformal mappings from a planar source domain to a planar target domain.
• A more elegant and compact formulation which leads to more simple and efficient mathematical expressions. In addition, our formulation is general enough to allow us to apply the coordinates to any complex-valued data rather than using them for shape deformation only.
• The mappings produced conform better to the given data values through more accurate results.
• The number of point-to-point coordinates needed in a typical scenario is significantly smaller than the number of vertices in an enclosing contour.
• Fewer coordinates means easier control for the user with faster, more accurate data manipulation and reduced usage of computer resources.
• Computation can be done using programmable graphics hardware in a straightforward manner.
• Computer animation
• Data interpolation
• Image and 2-D geometry deformation
• Color interpolation
• Parameterization
COM-1103
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