Interval Polynomials And Interval Implicitization

In this paper, zeros of interval polynomials and interval implicitization methods are studied. First we explain the importance of error control in geometric computation. We review the research history and status of these two problems.The roots of univariate interval polynomial equations and multivariate interval polynomial systems are discussed, including real roots and complex roots. We prove that a degree n univariate polynomial has at most n zero intervals. We also prove that it has n complex zeros if multiplicities are counted. The boundaries of complex zeros are given. We also give numeric algorithms to find the zero intervals and complex zero regions. The greatest common divisors of univariate interval polynomials are studied.For multivariate polynomial systems, we prove the convergence of the zero sets when all the interval coefficients are convergent to some constants. We mainly discuss the zeros of bivariate polynomial systems in the projective space. The conclusions are mostly parallel to univariate interval polynomials, except that we cannot give the number of real zero regions. We give the boundary of real and complex zero regions, and the number of complex zero regions. We also give numeric algorithms to find the real and complex zero regions of bivariate polynomial systems. These contents are not hard to be extended to higher dimension interval polynomial systems.At last we discuss the interval implicitization problem. We mainly discuss the interval implicitization of rational B-spline curves. We solve the problem in two steps: finding an approximate implicit B-spline curve and finding the bounding implieit curves by solving some linear problems. We also give algorithm and examples to demonstrate the application of interval impli(?)itization in intersection calculation of two parametric curves.