Bivariate Simplex Spline Derived From Algebraic Curve Theory

The study of the spline function began in the Mid-20th century. Combined with computer aided design, it has been successfully applied shape design in the last few years. Spline has become a powerful tool in approximation. With the increasing widely applications, it not only have been widely used in industrial but also closely related with basic math, such as algebraic geometry, differential equation, combinatorial mathematics, discrete geometry and so on. On the one hand, some scholars use spline to solve some problems of these areas, on the other hand, there also exists some scholars try to use the knowledge of these areas to process some problems of spline, these method makes the spline have better development. Our paper employs algebraic geometry to derive the bivariate simplex spline in complete partition. The main content is as follows:(1) Summarize some methods and important results about the Pascal theorem and Morgan-Scott triangulation.(2) Make the projective geometry and algebraic geometry combined with complete parti-tion, then prove the existence and uniqueness of the simplex spline in Sμ+1μ(ΔCμ)(2≤μ≤5), by using Chasles theorem.(3) Generalize the Chasles theorem, B?ezout theorem, Cayley-Bacharach theorem, Noether theorem to complete partition of two-dimensional plane and prove that the theorem is also established.