Computer graphics and geometric modeling using Beta-splines
Computer graphics and geometric modeling using Beta-splines
An implementation of multivariate B-spline surfaces over arbitrary triangulations
Proceedings of the conference on Graphics interface '92
Curves and surfaces for CAGD: a practical guide
Curves and surfaces for CAGD: a practical guide
What is the natural generalization of univariate splines to higher dimensions?
Mathematical Methods for Curves and Surfaces
Rational Beta-Splines for Representing Curves and Surfaces
IEEE Computer Graphics and Applications
Local control of bias and tension in beta-splines
SIGGRAPH '83 Proceedings of the 10th annual conference on Computer graphics and interactive techniques
Alpha-spline: A C2 Continuous Spline with Weights and Tension Control
SMI '99 Proceedings of the International Conference on Shape Modeling and Applications
2D piecewise algebraic splines for implicit modeling
ACM Transactions on Graphics (TOG)
Computer Aided Geometric Design
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A complex geometric shape is often a composition of a set of simple ones, which may differ from each other in terms of their mathematical representations and the ways in which they are constructed. One of the necessary requirements in combining these simple shapes is that their original shapes can be preserved as much as possible. In this paper, a set of partial shape-preserving (PSP) spline basis functions is introduced to smoothly combine a collection of shape primitives with flexible blending range control. These spline basis functions can be considered as a kind of generalization of traditional B-spline basis functions, where the shape primitives used are control points or control polygons. The PSP-spline basis functions have all the advantages of the conventional B-spline technique in the sense that they are nonnegative, piecewise polynomial and of property of partition of unity. However, PSP-spline is a more powerful freeform geometric shape design technique in the sense that it is also a kind of shape-preserving spline. In addition, the PSP-spline technique implicitly integrates the weights of shape control primitives into its basis functions, which allows users to design a required geometric shape based on weighted control primitives. Though its basis functions are simply piecewise polynomial functions, it has the same shape design strengths as the rational piecewise polynomial based spline techniques such as NURBS. In particular, when control shape primitives are specified as a set of control points, PSP-spline behaves like a polygon smoother, with which a shape can be designed to approximate the specified control polygon or control mesh smoothly with any required precision. Consequently, a richer set of geometric shapes can be built using a relatively smaller set of control points.