Ronald N. Goldman

Rice University · Department of Computer Science

We establish the uniform convergence of the control polygons generated by repeated degree elevation of q-Bézier curves (i.e., polynomial curves represented in the q-Bernstein bases of increasing degrees) on [0,1], q>1, to a piecewise linear curve with vertices on the original curve. A similar result is proved for q<1, but surprisingly the limit ver...

We provide a probabilistic approach using renewal theory to derive some novel identities involving Eulerian numbers and uniform B-splines. The renewal perspective leads to a unified treatment for the normalized binomial coefficients and the normalized Eulerian numbers when studying their limits of sums, as well as their associated distributions --...

We investigate the problem of recognizing a generalization of surfaces of revolution appearing in the field of affine differential geometry, namely affine rotation surfaces. By using some notions from affine differential geometry, we determine how to detect whether or not a given implicit algebraic surface is an affine rotation surface. These resul...

A surface of revolution with moving axes and angles is a rational tensor product surface generated from two rational space curves by rotating one curve (the directrix) around vectors and angles generated by the other curve (the director). Here we introduce these new kinds of rational generalized surfaces of revolution, provide some interesting exam...

We construct a q-analog of the blossom for analytic functions, the analytic q-blossom. This q-analog also extends the notion of q-blossoming from polynomials to analytic functions. We then apply this analytic q-blossom to derive identities for analytic functions represented in terms of the q-Poisson basis, including q-versions of the Marsden identi...

Affine rotation surfaces, which appear in the context of affine differential geometry, are generalizations of surfaces of revolution. These affine rotation surfaces can be classified into three different families: elliptic, hyperbolic and parabolic. In this paper we investigate some properties of algebraic parabolic affine rotation surfaces, i.e. p...

We introduce a blossoming procedure for polynomials related to the Askey–Wilson operator. This new blossom is symmetric, multiaffine, and reduces to the complex representation of the polynomial on a certain diagonal. This Askey–Wilson blossom can be used to find the Askey–Wilson derivative of a polynomial of any order. We also introduce a correspon...

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 surfa...

We investigate some variants of the linear Lane-Riesenfeld algorithm in the functional setting, generated by replacing the standard binary arithmetic averages between real numbers by non-linear, binary, symmetric averages between real numbers. For certain classes of non-linear averages we show that such generalized r-th order Lane-Riesenfeld algori...

We study generalizations of the classical Bernstein operators, the Lototsky–Bernstein operators, within the framework of non-polynomial spaces Un. It is shown that for any strictly increasing function p1(x)∈C[0,1] with p1(0)=0,p1(1)=1, there exist Lototsky–Bernstein operators that can fix p1(x) and still approximate all continuous functions uniform...

We present an algorithm to implicitize rational tensor product surfaces that works correctly and efficiently even in the presence of base points by combining three complementary approaches, some classical and some novel, to implicitization. One straightforward method is to implicitize these surfaces using the Dixon A-resultant of three obvious syzy...

Affine rotation surfaces are a generalization of the well-known surfaces of revolution. Affine rotation surfaces arise naturally within the framework of affine differential geometry, a field started by Blaschke in the first decades of the past century. Affine rotations are the affine equivalents of Euclidean rotations, and include certain shears as...

Complex rational curves have been used to represent circular splines as well as many classical curves including epicycloids, cardioids, Joukowski profiles, and the lemniscate of Bernoulli. Complex rational curves are known to have low degree (typically half the degree of the corresponding rational planar curve), circular precision, invariance with...

We investigate necessary and sufficient conditions under which the de Boor algorithm can be applied to evaluate points on a T-spline surface. We compare these de Boor-suitable (DS) T-splines to the standard analysis-suitable (AS) T-splines. We also develop an algorithm to search in a T-mesh for the appropriate control points and blend them using th...

By establishing an identity between a sequence of Bernstein-type operators and a sequence of Szász-Mirakyan operators, we prove that the convergence of Bernstein-type operators is related to convergence with respect to Szàsz-Mirakyan operators. As one application of this identity, we prove that whenever the parameters are conveniently chosen, if f...

A class of generalized Bernstein operators, the Lototsky-Bernstein operators, is investigated. It is shown that for any strictly increasing function p 1 (x) ∈ C[0, 1] with p(0) = 0, p(1) = 1, there exist Lototsky-Bernstein operators that can fix p 1 (x) and still approximate all continuous functions uniformly on [0, 1]. We study these special Lotot...

We flesh out the affine geometry of R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {R}}^3}$$\end{document} represented inside the Clifford algebra R(4,4)\d...

A ruled translational surface is a rational tensor product surface generated by translating a rational space curve along a straight line or equivalently translating a straight line along a rational space curve. We show how to compute the implicit equation of a ruled translational surface from two linearly independent vectors that are perpendicular...

A translational surface is a rational tensor product surface generated from two rational space curves by translating one curve along the other curve. Translational surfaces are invariant under rigid motions: translating and rotating the two generating curves translates and rotates the translational surface by the same amount. We construct three spe...

The main goal of this paper is to study shape preserving properties of univariate Lototsky-Bernstein operators L n (f) based on Lototsky-Bernstein basis functions. The Lototsky-Bernstein basis functions b n,k (x) (0 ≤ k ≤ n) of order n are constructed by replacing x in the i th factor of the generating function for the classical Bernstein basis fun...

Implicitizing rational surfaces is a fundamental computational task in Computer Graphics and Computer Aided Design. Ray tracing, collision detection, and solid modeling all benefit from implicitization procedures for rational surfaces. The univariate resultant of two moving lines generated by a μ-basis for a rational curve represents the implicit e...

Fractals are common in nature, and can be used as well for both art and engineering. We classify those fractals that can be represented by line segments into several types: tree-based fractals, curve-based fractals, and space filling fractals. We develop a set of methods to generate fractals with a swarm of robots by using robots as vertices, and l...

Implicitizing rational surfaces is a fundamental computational task in Algorithmic Algebraic Geometry. Although the resultant of a μ-basis for a rational surface is guaranteed to contain the implicit equation of the surface as a factor, this resultant may also contain extraneous factors. Moreover, μ-bases for rational surfaces are, in general, noto...

We describe an algorithm to generate a normative infant cranial model from the input of 3D meshes that are extracted from CT scans of normal infant skulls. We generate a correspondence map between meshes based on a registration algorithm. Then we apply our averaging algorithm to construct the normative model. The goal of this normal model is to ass...

Given an implicit polynomial equation or a rational parametrization, we develop algorithms to determine whether the set of real and complex points defined by the equation, i.e. the surface defined by the equation, in the sense of Algebraic Geometry, is a cylindrical surface, a conical surface, or a surface of revolution. The algorithms are directly...

Scissor shears are affine transformations in 3-space that, in analogy with the usual rotations, can be understood as hyperbolic rotations about a fixed line, in a fixed coordinate frame. We study algebraic surfaces invariant under scissor shears, and investigate their similarities and differences with the algebraic surfaces invariant under the usua...

We introduce the G-blossom of a polynomial by altering the diagonal property of the classical blossom, replacing the identity function by arbitrary linear functions G=G(t). By invoking the G-blossom, we construct G-Bernstein bases and G-Bézier curves and study their algebraic and geometric properties. We show that the G-blossom provides the dual fu...

This paper shows that generic 2D-Free-Form Deformations of degree can be made birational by a suitable assignment of weights to the Bézier or B-spline control points. An FFD that is birational facilitates operations such as backward mapping for image warping, and Extended Free-Form Deformation. While birational maps have been studied extensively in...

Generalized quantum splines are piecewise polynomials whose generalized quantum derivatives agree up to some order at the joins. Just like classical and quantum splines, generalized quantum splines admit a canonical basis with compact support: the generalized quantum B-splines. Here we study generalized quantum B-spline bases and generalized quantu...

We develop a general, unified theory of splines for a wide collection of spline spaces, including trigonometric splines, hyperbolic splines, and special Müntz spaces of splines by invoking a novel variant of the homogeneous polar form where we alter the diagonal property. Using this polar form, we derive de Boor type recursive algorithms for evalua...

Real μ-bases for non-ruled real quadric surfaces have two potential drawbacks. First, the resultant of the three moving planes corresponding to a real μ-basis represents the implicit equation of the quadric surface, but in some cases contains a linear extraneous factor. Second, even when this resultant contains no extraneous factor, this resultant...

We present an algorithm for extracting the axis of revolution from the implicit equation of an algebraic surface of revolution based on three distinct computational methods: factoring the highest order form into quadrics, contracting the tensor of the highest order form, and using univariate resultants and gcds. We compare and contrast the advantag...

We show that a weighted least squares approximation of q-Bézier coefficients provides the best polynomial degree reduction in the q-L2-norm. We also provide a finite analogue of this result with respect to finite q-lattices and we present applications of these results to q-orthogonal polynomials.

We derive a collection of fundamental formulas for quantum B-splines analogous to known fundamental formulas for classical B-splines. Starting from known recursive formulas for evaluation and quantum differentiation along with quantum analogues of the Marsden identity, we derive quantum analogues of the de Boor–Fix formula for the dual functionals,...

We provide a simple, efficient technique for computing μ-bases for quadric surfaces from their rational quadratic parametrizations. Our major innovation is to simplify the computations by using complex parameters, even though all the surfaces we treat have only real coefficients in both their implicit and parametric representations. In addition to...

The -Bernstein-Bézier curves are generalizations of both the h-Bernstein-Bézier curves and the q-Bernstein-Bézier curves. We investigate two essential features of -Bernstein bases and -Bézier curves: the variation diminishing property and the degree elevation algorithm. We show that the -Bernstein bases for a non-empty interval satisfy Descartes' l...

We construct polar forms for diverse types of spaces, including trigonometric polynomials, hyperbolic polynomials and special Müntz spaces, by altering the diagonal property of the polar form for homogeneous polynomials. We use this polar form to develop recursive evaluation algorithms and subdivision procedures for the corresponding Bernstein Bézi...

We derive explicit formulas for the generating functions of the q-Bernstein basis functions in terms of q-exponential functions. Using these explicit formulas, we derive a collection of functional equations for these generating functions which we apply to prove a variety of identities, some old and some new, for the q-Bernstein bases.

We show how to compute perspective projections in 3-dimensions using rotations and spherical inversions in the homogeneous and conformal models of Clifford Algebra. One achievement of our paper is to show that although perspective is a purely projective operation, while a Clifford algebra by its very definition is a metric tool, nevertheless and su...

A software tool Synopsys' Educational Generic Memory Compiler (GMC) that enables automatic generation of static RAM cells (SRAMs) based on the parameters supplied by the user is presented. The software and the generated SRAMs are made to be free from intellectual property restrictions and can be easily integrated into educational designs. GMC deplo...

Due to the rapid development times, device complexity, etc. of the semiconductor industry it is challenging for universities to teach modern Integrated Circuit (IC) design. In particular, universities lack access to the necessary semiconductor technology data required to implement educational projects, as this information is normally proprietary. A...

We investigate the efficacy of the Clifford algebra R(4, 4) as a computational framework for contemporary 3-dimensional computer graphics. We give explicit rotors in R(4, 4) for all the standard affine and projective transformations in the graphics pipeline, including translation, rotation, reflection, uniform and nonuniform scaling, classical and...

We derive a closed formula for the generating functions of the uniform B-splines. We begin by constructing a PDE for these generating functions starting from the de Boor recurrence. By solving this PDE, we find that we can express these generating functions explicitly as sums of polynomials times exponentials. Using these generating functions, we d...

We derive explicit formulas for the generating functions of B-splines with knots in either geometric or affine progression. To find generating functions for B-splines whose knots have geometric or affine ratio $q$ , we construct a PDE for these generating functions in which classical derivatives are replaced by $q$ -derivatives. We then solve...

We provide a new technique to detect the singularities of rational space curves. Given a rational parametrization of a space curve, we first compute a μ-basis for the parametrization. From this μ-basis we generate three planar algebraic curves of different bidegrees whose intersection points correspond to the parameters of the singularities. To fin...

Quantum splines are piecewise polynomials whose quantum derivatives (i.e. certain discrete derivatives or equivalently certain divided differences) agree up to some order at the joins. Just like classical splines, quantum splines admit a canonical basis with compact support: the quantum B-splines. These quantum B-splines are the q-analogues of clas...

Unified Power Format (UPF) is an industry wide power format specification to implement low power techniques in a design flow. UPF is designed to reflect the power intent of a design at a relatively high level. UPF scripts help describe power intent such as: which power rails to be routed to individual blocks, when blocks are expected to be powered...

A rational surfaceS(s,t)=(a(t)a^@?(s),a(t)b^@?(s),b(t)c^@?(s),c(t)c^@?(s)) can be generated from two orthogonal rational planar directrices: P(t)=(a(t),b(t),c(t)) in the xz-plane and P^@?(s)=(a^@?(s),b^@?(s),c^@?(s)) in the xy-plane. Moving a scaled copy of the curve P^@?(s) up and down along the z-axis with the size controlled by the curve P(t), w...

We prove a result similar to the conjecture of Chen et al. (2008) concerning how to cal-culate the parameter values corresponding to all the singularities, including the infinitely near singularities, of rational planar curves from the Smith normal forms of certain Bezout resultant matrices derived from μ-bases. A great deal of mathematical lore is...

We provide a new technique for implicitizing rational surfaces of revolution using μ-bases. A degree n rational plane curve rotating around an axis generates a degree 2n rational surface. From a μ-basis p,qp,q of this directrix curve, where μ=deg(p)⩽deg(q)=n−μμ=deg(p)⩽deg(q)=n−μ, and a rational parametrization of the circle r(s)=(2s,1−s2,1+s2)r(s)=...

The aggressive scaling of CMOS technology toward nanometer lengths contributed to the surfacing of many effects that were not appreciable at the micrometer regime. Among them, Inverted Temperature Dependence (ITD) is certainly the most unusual. It manifests itself as a speed up of CMOS gates when the temperature increases, resulting in a reversal o...

The educational challenges to keep pace with today's microelectronics industry and Moore's Law result in universities being unable to provide graduates with the required qualifications to make them ready for employment immediately upon graduation. The most effective way of addressing this issue is the development and function of new educational mod...

We introduce a new variant of the blossom, the q-blossom, by altering the diagonal property of the standard blossom. This q-blossom is specifically adapted to developing identities and algorithms forq-Bernstein bases and q-Bézier curves over arbitrary intervals. By applying the q-blossom, we generate several new identities including an explicit for...

A new variant of the blossom, the h-blossom, is introduced by altering the diagonal property of the standard blossom. The significance of the h-blossom is that the h-blossom satisfies a dual functional property for h-Bézier curves over arbitrary intervals. Using the h-blossom, several new identities involving the h-Bernstein bases are developed inc...

New principles of verification system construction, with consideration of the impact of various internal and external destabilizing factors are presented. It is shown that the proposed principles allow keeping the main advantages of the traditional digital circuit logic simulation, while eliminating key limitations. Verification system is based on...

We extend some well known algorithms for planar Bezier and B-spline curves, including the de Casteljau subdivision algorithm for Bezier curves and several standard knot insertion procedures (Boehm's algorithm, the Oslo algorithm, and Schaefer's algorithm) for B-splines, from the real numbers to the complex domain. We then show how to apply these po...

Quaternion multiplication can be applied to rotate vectors in 3-dimensions. Therefore in Computer Graphics, quaternions are sometimes used in place of matrices to represent rotations in 3-dimensions. Yet while the formal algebra of quaternions is well-known in the Graphics community, the derivations of the formulas for this algebra and the geometri...

We investigate the Lane–Riesenfeld subdivision algorithm for uniform B-splines, when the arithmetic mean in the various steps of the algorithm is replaced by nonlinear, symmetric, binary averaging rules. The averaging rules may be different in different steps of the algorithm. We review the notion of a symmetric binary averaging rule, and we derive...

We construct a homogeneous model for Computer Graphics using the Clifford Algebra for ℝ 3 . To incorporate points as well as vectors within this model, we employ the odd-dimensional elements of this graded eight-dimensional algebra to represent mass-points by exploiting the pseudoscalars to represent mass. The even-dimensional elements of this Clif...

We show how to calculate three low degree set-theoretic generators (i.e., algebraic surfaces) for all rational space curves of low degree (degree ≤6) as well as for all higher degree rational space curves where at least one element of their μ-basis has degree 1 from a μ-basis of the parametrization. In addition to having low degree, at least two of...

We provide an algorithm to find a minimal set of generators for the Rees algebra associated to rational space curves of type (1, 1, d − 2) in projective 3-space based solely on a µ-basis of the curve. We also illustrate the geometry behind the generators via a case study of rational quartic space curves.

We provide a technique to detect the singularities of rational planar curves and to compute the correct order of each singularity including the infinitely near singularities without resorting to blow ups. Our approach employs the given parametrization of the curve and uses a μ-basis for the parametrization to construct two planar algebraic curves w...

We have developed a full-custom IC design flow based on Synopsys custom design tools and the recently released Synopsys 90 nm generic library. The developed design flow can be used for teaching VLSI and digital IC design courses. We have also developed a full-custom design project that was used as a course project in teaching ldquoDigital VLSI Desi...

An open Educational Design Kit (EDK) which supports a 90 nm design flow is described which includes all the necessary design rules, models, technology files, verification and extraction command decks, scripts, symbol libraries, and PCells. It also includes a Digital Standard Cell Library (DSCL) which supports all contemporary low power design techn...

Four unsolved problems that originate from research in Computer Graphics and Geometric Modeling will be presented. The first problem involves understanding the notion oscillation for Bezier surfaces, the freeform polynomial surfaces most common in Computer Graphics and Geometric Modeling. The second problem concerns generating smooth (C2) surfaces...

Relationships between the singularities of rational space curves and the moving planes that follow these curves are investigated. Given a space curve C with a generic 1–1 rational parametrization F(s,t) of homogeneous degree d, we show that if P and Q are two singular points of orders k and k′ on the space curve C, then there is a moving plane of d...

Our main result is that two point interpolatory subdivision schemes using Ck nonlinear averaging rules on pairs of real numbers generate real-valued functions that are also Ck. The signiflcance of this result is the following consequence: Suppose that S is a subdivision algorithm operating on sequences of real numbers using linear binary averaging...

The notion of a μ-basis for an arbitrary number of polynomials in one variable is defined. The basic properties of these μ-bases are derived, and an algorithm is presented based on Gaussian Elimination to calculate a μ-basis for any collection of univariate polynomials. These μ-bases are then applied to solve implicitization, inversion and intersec...

We present an efficient algorithm for subdividing non-uniform B-splines of arbitrary degree in a manner similar to the Lane–Riesenfeld subdivision algorithm for uniform B-splines of arbitrary degree. Our algorithm consists of doubling the control points followed by d rounds of non-uniform averaging similar to the d rounds of uniform averaging in th...

Rapid changes in integrated circuits (IC) technology and constantly shrinking process geometries demand a new curriculum that meets the contemporary requirements for IC design. This is especially important for 90nm and below technologies and the use of state-of-the-art EDA design tools and advanced IC design techniques. The creation of new curricul...

Taking a novel, more appealing approach than current texts, An Integrated Introduction to Computer Graphics and Geometric Modeling focuses on graphics, modeling, and mathematical methods, including ray tracing, polygon shading, radiosity, fractals, freeform curves and surfaces, vector methods, and transformation techniques. The author begins with f...

Increasing attention is being paid to complete machining, i.e., machining of the whole part in a single machine tool, in the metal working industry. For this purpose, complex machine tools equipped with machining components, such as multiple spindles ...

We investigate a general class of nonlinear subdivision algorithms for functions of a real or complex variable built from linear subdivision algorithms by replacing binary linear averages such as the arithmetic mean by binary nonlinear averages such as the geometric mean. Using our method, we can easily create stationary subdivision schemes for Gau...

Producing well-trained engineers who will work in the semiconductor industry poses unique challenges to existing educational systems worldwide. The rate of change is incongruent between industry and academia. It is necessary for students to combine theoretical knowledge with practical skills. Computer hardware and software for university engineerin...

A natural one to one correspondence is derived between the singular points of rational planar curves and the axial moving lines that follow these curves. This correspondence is applied to compute and to analyze all the singular points of low degree rational planar curves using μ-bases.

The standard proof of the Lane-Riesenfeld algorithm for inserting knots into uniform B-spline curves is based on the continuous convolution formula for the uniform B-spline basis functions. Here we provide two new, elementary, blossoming proofs of the Lane-Riesenfeld algorithm for uniform B-spline curves of arbitrary degree.

Recent trends in algebraic geometry emphasize effective computation over transcendent theory. The theme of this paper is that from the perspective of geometric modeling this trend is largely misguided – that for the purpose of geometric modeling the true role of algebraic geometry is insight not computation.

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...

Subdivision schemes generate self-similar curves and surfaces. There fore there is a close connection between curves and surfaces generated by subdivision algorithms and self-similar fractals generated by Iterated Function Systems (IFS). We demonstrate that this connection between subdivision s chemes and fractals is even deeper by showing that cur...

We provide a formal proof of equivalence between the class of fractals created by recursive-turtle programs (RTP) and iterated affine transformations (IAT). We begin by reviewing RTP (a geometric interpretation of non-bracketed L-systems with a single production rule) and IAT (iterated function systems restricted to affine transformations). Next, w...

We provide a formal proof of equivalence between the class of fractals created by Recursive Turtle Programs (RTP) and Iterated A#ne Transformations (IAT). We begin by reviewing RTP ( a geometric interpretation of non-bracketed L-systems with a single production rule) and IAT (Iterated Function Systems restricted to a#ne transformations). Next, we p...

LOGO is a programming language incorporating turtle graphics, originally devised for teaching computing to young children in elementary and middle schools. Here we advocate the use of LOGO to help introduce some of the basic concepts of computer graphics and computer aided design to undergraduate and graduate students in colleges and universities....

A survey is presented examining a variety of different techniques, some old and some new, for extending the construction of standard three sided and four sided Bézier patches to n-sided surface patches.Standard triangular and rectangular Bézier patches can be defined either explicitly using Bernstein blending functions or recursively using de Caste...

Fractals are attractors - fixed points of iterated function systems. Bezier curves are polynomials - linear combinations of Bernstein basis functions. The de Casteljau subdivision algorithm is used here to show that Bezier curves are also attractors. Thus, somewhat surprisingly, Bezier curves are fractals. This fractal nature of Bezier curves is ex...