Michael S. Floater

Mean value coordinates

We derive a generalization of barycentric coordinates which allows a vertex in a planar triangulation to be expressed as a convex combination of its neighbouring vertices. The coordinates are motivated by the Mean Value Theorem for harmonic functions and can be used to simplify and improve methods for parameterization and morphing.

Surface Parameterization: a Tutorial and Survey

This paper provides a tutorial and survey of methods for parameterizing surfaces with a view to applications in geometric modelling and computer graphics. We gather various concepts from differential geometry which are relevant to surface mapping and use them to understand the strengths and weaknesses of the many methods for parameterizing piecewise linear surfaces and their relationship to one another.

Parametrization and smooth approximation of surface triangulations

A method based on graph theory is investigated for creating global parametrizations for surface triangulations for the purpose of smooth surface fitting. The parametrizations, which are planar triangulations, are the solutions of linear systems based on convex combinations. A particular parametrization, called shape-preserving, is found to lead to visually smooth surface approximations.

Mathematical methods for curves and surfaces

We will deal with the translation surfaces which are the shapes generated by translating one curve along another one. We focus on the geometry of translation surfaces generated by two algebraic curves in space and study their properties, especially those useful for geometric modelling purposes. It is a classical result that each minimal surface may be obtained as a translation surface generated by an isotropic curve and its complex conjugate. Thus, we can study the minimal surfaces as special instances of translation surfaces. All the results about translation surfaces will be directly applied also to minimal surfaces. Finally, we present a construction of rational isotropic curves with a prescribed tangent field which leads to the description of all rational minimal surfaces. A close relation to surfaces with Pythagorean normals will be also discussed.

Multistep scattered data interpolation using compactly supported radial basis functions

A hierarchical scheme is presented for smoothly interpolating scattered data with radial basis functions of compact support. A nested sequence of subsets of the data is computed efficiently using successive Delaunay triangulations. The scale of the basis function at each level is determined from the current density of the points using information from the triangulation. The method is rotationally invariant and has good reproduction properties. Moreover the solution can be calculated and evaluated in acceptable computing time.

Mean value coordinates in 3D

Barycentric coordinates can be used both to express a point inside a tetrahedron as a convex combination of the four vertices and to linearly interpolate data given at the vertices. In this paper we generalize these coordinates to convex polyhedra and the kernels of star-shaped polyhedra. These coordinates generalize in a natural way a recently constructed set of coordinates for planar polygons, called mean value coordinates.

Advances in Multiresolution for Geometric Modelling

Compression.- Recent Advances in Compression of 3D Meshes.- Shape Compression using Spherical Geometry Images.- Data Structures.- A Survey on Data Structures for Level-of-Detail Models.- An Algorithm for Decomposing Multi-dimensional Non-manifold Objects into Nearly Manifold Components.- Encoding Level-of-Detail Tetrahedral Meshes.- Multi-Scale Geographic Maps.- Modelling.- Constrained Multiresolution Geometric Modelling.- Multi-scale and Adaptive CS-RBFs for Shape Reconstruction from Clouds of Points.- Parameterization.- Surface Parameterization: a Tutorial and Survey.- Variations on Angle Based Flattening.- Subdivision.- Recent Progress in Subdivision: a Survey.- Optimising 3D Triangulations: Improving the Initial Triangulation for the Butterfly Subdivision Scheme.- Simple Computation of the Eigencomponents of a Subdivision Matrix in the Fourier Domain.- Subdivision as a Sequence of Sampled Cp Surfaces.- Reverse Subdivision.-

A general construction of barycentric coordinates over convex polygons

\sqrt 5