Ron Goldman

Implicit representation of parametric curves and surfaces

Abstract The following two problems are shown to have closed-form solutions requiring only the arithmetic operations of addition, subtraction, multiplication and division: (1) Given a curve or surface defined parametrically in terms of rational polynomials, find an implicit polynomial equation which defines the same curve or surface. (2) Given the Cartesian coordinates of a point on such a curve or surface, find the parameter(s) corresponding to that point. It is shown that a two-dimensional curve defined parametrically in terms of rational degree n polynomials in t can be expressed implicitly as a degree n polynomial in z and y. It is also demonstrated that a “bi-m-ic” parametric surface (where e.g., m = 3 for bicubic) can be expressed implicitly as a polynomial in x, y, z of degree 2m2. The degree of a rational bi-m-ic surface is also shown to be 2m2. The application of these results to finding curve and surface intersections is discussed.

Curvature formulas for implicit curves and surfaces

Curvature formulas for implicit curves and surfaces are derived from the classical curvature formulas in Differential Geometry for parametric curves and surfaces. These closed formulas include curvature for implicit planar curves, curvature and torsion for implicit space curves, and mean and Gaussian curvature for implicit surfaces. Some extensions of these curvature formulas to higher dimensions are also provided.

Functional composition algorithms via blossoming

In view of the fundamental role that functional composition plays in mathematics, it is not surprising that a variety of problems in geometric modeling can be viewed as instances of the following composition problem: given representations for two functions <italic>F</italic> and <italic>G</italic>, compute a representation of the function <italic>H</italic> = <italic>F o G</italic>. We examine this problem in detail for the case when <italic>F</italic> and <italic>G</italic> are given in either Be´zier or B-spline form. Blossoming techniques are used to gain theoretical insight into the structure of the solution which is then used to develop efficient, tightly codable algorithms. From a practical point of view, if the composition algorithms are implemented as library routines, a number of geometric-modeling problems can be solved with a small amount of additional software.

Vector elimination: A technique for the implicitization, inversion, and intersection of planar parametric rational polynomial curves

In this paper vector techniques and elimination methods are combined to help resolve some classical problems in computer aided geometric design. Vector techniques are applied to derive the Bezout resultant for two polynomials in one variable. This resultant is then used to solve the following two geometric problems: Given a planar parametric rational polynomial curve, (a) find the implicit polynomial equation of the curve (implicitization); (b) find the parameter value(s) corresponding to the coordinates of a point known to lie on the curve (inversion). The solutions to these two problems are closed form and, in general, require only the arithmetic operations of addition, subtraction, multiplication, and division. These closed form solutions lead to a simple, non-iterative, analytic algorithm for computing the intersection points of two planar parametric rational polynomial curves. Extensions of these techniques to planar rational Bezier curves are also discussed.

On the Validity of Implicitization by Moving Quadrics for Rational Surfaces with No Base Points

Techniques from algebraic geometry and commutative algebra are adopted to establish sufficient polynomial conditions for the validity of implicitization by the method of moving quadrics both for rectangular tensor product surfaces of bi-degree (m, n) and for triangular surfaces of total degree n in the absence of base points.

Knot insertion and deletion algorithms for B-spline curves and surfaces

1. An Introduction to Blossoming Phillip J. Barry 2. Algorithms for Progressive Curves: Extending B-Spline and Blossoming Techniques to the Monomial, Power, and Newton Dual Bases Phillip J. Barry and Ronald N. Goldman 3. Factored Knot Insertion Phillip J. Barry and Ronald N. Goldman 4. Knot Insertion Algorithms Phillip J. Barry and Ronald N. Goldman 5. Conversion Between B-Spline Bases Using the Generalized Oslo Algorithm Tom Lyche, Knut Morken and Kyrre Strom 6. How Much Can the Size of the B-Spline Coefficients be Reduced by Inserting One Knot? Tom Lyche and Knut Morken 7. An Envelope Approach to a Sketching Editor for Hierarchical Free-form Curve Design and Modification Michael J. Banks, Elaine Cohen and Timothy I. Mueller Index.

Implicitizing rational curves by the method of moving algebraic curves

Abstract A function F(x,y,t) that assigns to each parameter t an algebraic curve F(x,y,t)=0 is called a moving curve. A moving curve F(x,y,t) is said to follow a rational curve x=x(t)/w(t) , y=y(t)/w(t) if F(x(t)/w(t), y(t)/w(t),t) is identically zero. A new technique for finding the implicit equation of a rational curve based on the notion of moving conics that follow the curve is investigated. For rational curves of degree 2 n with no base points the method of moving conics generates the implicit equation as the determinant of an n × n matrix, where each entry is a quadratic polynomial in x and y , whereas standard resultant methods generate the implicit equation as the determinant of a 2 n × 2 n matrix where each entry is a linear polynomial in x and y . Thus implicitization using moving conics yields more compact representations for the implicit equation than standard resultant techniques, and these compressed expressions may lead to faster evaluation algorithms. Moreover whereas resultants fail in the presence of base points, the method of moving conics actually simplifies, because when base points are present some of the moving conics reduce to moving lines.

Subdivision algorithms for Bézier triangles

Abstract In an earlier paper the author studied subdivision algorithms for Bezier curves. These algorithms were generated by constructing degenerate Bezier triangles and tetrahedra and using their edges to subdivide the original curve. In this paper we take up the study of subdivision algorithms for the triangular Bezier surfaces first introduced by Sabin. It will be shown that such algorithms can be generated by constructing higher dimensional degenerate Bezier simplices and using their triangular subsimplices to subdivide the original triangular surface.

Degree, multiplicity, and inversion formulas for rational surfaces using u -resultants

Abstract Rational surface patches are popular in computer aided geometric design. But for any given rational parametrization, many questions arise: “Are there base points?”, “What is the degree of the surface?”, “How to do inversion?”, “Is the representation redundant?”, “Is a point exceptional?” These and many other problems can be solved in one stroke by using the u -resultant. With a computer algebra system, the implementation of our algorithms is straightforward. A method to speed up the computation is also provided.

Geometric algorithms for detecting and calculating all conic sections in the intersection of any two natural quadric surfaces

Abstract One of the most challenging aspects of the surface-surface intersection problem is the proper disposition of degenerate configurations. Even in the domain of quadric surfaces, this problem has proven to be quite difficult. The topology of the intersection as well as the basic geometric representation of the curve itself is often at stake. By Bezout′s Theorem, two quadric surfaces always intersect in a degree four curve in complex projective space. This degree four curve is degenerate if it splits into two (possibly degenerate) conic sections. In theory the presence of such degeneracies can be detected using classical algebraic geometry. Unfortunately in practice it has proven to be extremely difficult to make computer implementations of such methods reliable numerically. Here, we present geometric algorithms that detect the presence of these degeneracies and compute the resulting planar intersections. The theoretical basis of these algorithms-in particular, proofs of correctness and completeness-are extremely long and tedious. We briefly outline the approach, but present only the results of the analysis as embodied in the geometric algorithms. Interested readers are referred to R. N. Goldman and J. R. Miller (Detecting and calculating conic sections in the intersection of two natural quadric surfaces, part I: Theoretical analysis; and Detecting and calculating conic sections in the intersection of two natural quadric surfaces, part II: Geometric constructions for detection and calculation, Technical Reports TR-93-1 and TR-93-2, Department of Computer Science, University of Kansas, January 1993) for details.