Thomas Garrity

On computing the intersection of a pair of algebraic surfaces

Abstract The problem of finding the intersection of a pair of surfaces arises in a wide range of applications in geometric modeling. One difficulty in computing the intersection is that there has been no data structure suitable for representing arbitrary algebraic space curves. In this work, we describe such a data structure for the intersection of surfaces. This data structure represents an irreducible space curve as an irreducible algebraic plane curve plus a birational map between the plane curve and the space curve. Furthermore, we give an algorithm for constructing such a representation for the space curve formed by two intersecting algebraic surfaces. Finally we show how this data structure may be used in algorithms to answer important questions about space curves.

Geometric continuity

The area of geometric design has coined the concept of geometric continuity (GC) to characterize when two point sets join smoothly independent of the parameterization – that is, a measure of continuity that treats parameterizations as tools for describing curves or surfaces, without introducing parametric artifacts. While computer-aided geometric design (CAGD) has relied heavily on mathematical descriptions of point sets based on parametric functions in recent years, geometric continuity for parametric curves and surfaces actually needs a notion different from the direct matching of Taylor expansions, which is used to define the continuity of piecewise functions.

Factoring rational polynomials over the complex numbers

NC algorithms are given for determining the number and degrees of the factors, irreducible over the complex numbers

A generalized family of multidimensional continued fractions: TRIP Maps

{\bf C}

INVARIANTS OF VECTOR-VALUED BILINEAR AND SESQUILINEAR FORMS

, of a multivariate polynomial with rational coefficients and for approximating each irreducible factor. NC is the class of functions computable by logspace-uniform boolean circuits of polynomial size and polylogarithmic depth. The measures of size of the input polynomial are its degree, coefficient length, number of variables (d, c, and n, respectively). If n is fixed, we give a deterministic NC algorithm. If the number of variables is not fixed, we give a random (Monte-Carlo) NC algorithm in these input measures to find the number and degree of each irreducible factor.After reducing to the two-variable, square-free case, we apply the classical algebraic geometry fact that the absolute irreducible factors of

On relations of invariants for vector-valued forms

(P(z_1 ,z_2 ) = 0)

On the Applications of Multi-Equational Resultants

correspond to the connected components of the real surface (or complex curve)

On Periodic Sequences for Algebraic Numbers

P(z_1 ,z_2 ) = 0

A two-dimensional Minkowski ?(x) function

minus its singular points. In finding the number of connec...

A dual approach to triangle sequences: a multidimensional continued fraction algorithm.

Most well-known multidimensional continued fractions, including the Monkemeyer map and the triangle map, are generated by repeatedly subdividing triangles. This paper constructs a family of multidimensional continued fractions by permuting the vertices of these triangles before and after each subdivision. We obtain an even larger class of multidimensional continued fractions by composing the maps in the family. These include the algorithms of Brun, Parry-Daniels and Guting. We give criteria for when multidimensional continued fractions associate sequences to unique points, which allows us to determine when periodicity of the corresponding multidimensional continued fraction corresponds to pairs of real numbers being cubic irrationals in the same number field.