Toric surface patches have two significant geometric properties: they are multi-sided and they are generalizations of both the triangular and rectangular Bezier surface patches. They also have a very nice algebraic property: their implicit equations are closely related to the Dixon determinant. In particular, for bi-cubic toric patches without base points, their implicit equation can always be obtained very conveniently using the recently discovered Dixon quotients. In this paper, we explain the relevance of monomial corner cutting to toric patches, and how this leads to their efficient implicitization by the Dixon quotient. Many examples are given to illustrate the simplicity and power of this approach.

An inversion equation takes the Cartesian coordinates of a point on a parametric curve or surface and returns the parameter value(s) of that point. A 2D curve inversion equation has the form t = f (x, y)/g(x, y). This paper shows that practical insight into inversion can be obtained by studying the geometry of the implicit curves f (x, y) = 0 and g(x, y) = 0. For example, the relationship between the singular locus of the parametric curve and the lowest possible degree of an inversion equation can be understood in this way. Also, insight is given into what parameter value will be returned if an inversion equation is fed the Cartesian coordinates of a point that does not lie on the curve. The standard method of devising curve and surface inversion equations is a by-product of the implicitization process. This paper presents a new method for finding inversion equations, which allows us to create new inversion equations that have attractive properties. For example, we can create an inversion equation that, to first order approximation, will return the parameter value of the nearest point on the curve if given a point that does not lie precisely on a parametric curve.

It is well-known that the Dixon resultant implicitizes exactly a general tensor product surface. We show that a minor of the Dixon resultant matrix can also implicitize exactly. This occurs when the monomial support of the surface parametrization is a rectangle missing at most one sub-rectangle at each of its its corners. Unlike the Sylvester dialytic method, this way of finding the implicit equation in determinant form is automatic, in bracket form, and uses much smaller matrices.

Macaulay's concise but explicit expression for nmltivariate resultants has many potential applications in computer-aided geometric design.Here we describe its use in solid modeling for finding the intersections of three implicit quadric surfaces.By B6zout's theorem, three quadric surfaces have either at most eight or intlnitely many intersections.Our method finds the intersections, when there are finitely many, by generating a polynomial of degree at most eight whose roots are the intersection coordinates along an appropriate axis.Only addition, subtraction, and multiplication are required to find the polynomial.But when there are pmsibilities of extraneous roots, division and greatest common divisor computations are necessary to identify and remove them.