# 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