Representing rational curve segments and surface patches using semi-algebraic sets

We provide a framework for representing segments of rational planar curves or patches of rational tensor product surfaces with no singularities using semi-algebraic sets. Given a rational planar curve segment or a rational tensor product surface patch with no singularities, we find the implicit equation of the corresponding unbounded curve or surface and then construct an algebraic box defined by some additional equations and inequalities associated to the implicit equation. This algebraic box is proved to include only the given curve segment or surface patch without any extraneous parts of the unbounded curve or surface. We also explain why it is difficult to construct such an algebraic box if the curve segment or surface patch includes some singular points such as self-intersections. In this case, we show how to isolate a neighborhood of these special points from the corresponding curve segment or surface patch and to represent these special points with small curve segments or surface patches. This framework allows us to dispense with expensive approximation methods such as voxels for representing surface patches.National Natural Science Foundation of ChinaMinisterio de Ciencia, Innovación y Universidade

A Comparison of Three Curve Intersection Algorithms

An empirical comparison is made between three algorithms for computing the points of intersection of two planar Bezier curves. The algorithms compared are: the well known Bezier subdivision algorithm, which is discussed in Lane 80; a subdivision algorithm based on interval analysis due to Koparkar and Mudur; and an algorithm due to Sederberg, Anderson and Goldman which reduces the problem to one of finding the roots of a univariate polynomial. The details of these three algorithms are presented in their respective references

Determining Critical Points of Handwritten Mathematical Symbols Represented as Parametric Curves

We consider the problem of computing critical points of plane curves represented in a finite orthogonal polynomial basis. This is motivated by an approach to the recognition of hand-written mathematical symbols in which the initial data is in such an orthogonal basis and it is desired to avoid ill-conditioned basis conversions. Our main contribution is to assemble the relevant mathematical tools to perform all the necessary operations in the orthogonal polynomial basis. These include implicitization, differentiation, root finding and resultant computation