# Centroid Triangulations to bivariate BSplines

CREATED: 200702090252 Speaker: Jack Snoeyink Collaborator: Yuanxin Liu ** Motivation

• representation of surfaces (irregular point set) ** triangulations ** B-splines
• join between different surfaces, i.e. in model of tea pot ** Splines
• piecewise polynomials
• B-spline: linear combination of basic functions
• B-spline of degree k: defined for any k+2 points ** can represent any polynomial of degree k ** local support ** optimal smoothness ** Multivariate splines
• Tensor product, sub division, box splines
• Properties ** no restriction on knot properties ** properties of 1D B splines
• Simplex splines ** projection from a polyhedron in $R^n$
• key to multivariate splines is in choosing the right configurations as the basis simplex splines ** Voronoi/Delaunay
• order k: set of points with the same k closest points
• Lee’s algorithm ** order k -> order k+1 ** computing centroids and flipping, computing delaunay in holes ** Centroid triangulations
• generalize Lee’s algorithm, use some other data dependent triangulations instead of Delaunay