About

Ronald N. Goldman is a Professor of Computer Science at Rice University in Houston, Texas. He received his B.S. in Mathematics from the Massachusetts Institute of Technology in 1968 and his M.A. and Ph.D. in Mathematics from Johns Hopkins University in 1973. His current research interests focus on the mathematical representation, manipulation, analysis, and reconstruction of shape using computers. His work includes research in computer aided geometric design, solid modeling, computer graphics, polynomials, and splines. He is particularly interested in algorithms for polynomial and piecewise polynomial curves and surfaces, as well as applications of algebraic and differential geometry to geometric modeling. Recently, he has concentrated on the uses of quaternions and Clifford algebras in computer graphics.

Research topics

• Mathematics
• Mathematical analysis
• Combinatorics
• Discrete mathematics
• Geometry
• Applied mathematics
• Pure mathematics
• Statistics

Selected publications

• NEGATIVE DEGREE q-BERNSTEIN BASES AND THE MULTIRATIONAL q-BLOSSOM

  Rocky Mountain Journal of Mathematics · 2025-08-01

  articleSenior author

  We investigate algebraic properties of the negative degree q-Bernstein bases. Our fundamental tool in this investigation is a recently introduced variant of the blossom, the multirational q-blossom, which provides the dual functionals for the negative degree q-Bernstein basis functions. By applying the dual functional property of the multirational q-blossom, we are readily able to generate several fundamental identities involving the negative degree q-Bernstein bases, including a new variant of Marsden’s identity, a partition of unity property, a reparametrization formula, and a formula for representing monomials. We also show how to use the homogeneous variant of the multirational q-blossom to convert between the q-Taylor bases and the negative degree q-Bernstein bases.

  PublisherDOI

• Proofs of two conjectures involving sums of normalized Narayana numbers

  Demonstratio Mathematica · 2025-01-01

  articleOpen access1st authorCorresponding

  Abstract Narayana numbers are well-known and have many applications in the field of combinatorics. The Narayana numbers form a triangular array, where the sum of the n th row is the n th Catalan number. Here we normalize the Narayana numbers by dividing each entry in the n th row by the n th Catalan number. Now each row of these normalized Narayana numbers defines a discrete probability distribution. We investigate two new properties of these normalized Narayana numbers: instead of summing along the rows, we derive the limit of the sums along the columns and the limit of the sums along the short diagonals.

  PublisherOA PDFDOI

• A universal approach to blossoming with applications to Bernstein bases and Bézier curves for arbitrary function spaces

  CALCOLO · 2025-05-08

  articleOpen access1st authorCorresponding

  Abstract We present a general constructive approach to blossoming for arbitrary finite dimensional univariate function spaces by altering the diagonal property of the classical blossom. We apply this approach to blossoming to develop the corresponding theory of Bernstein bases and Bézier curves including de Casteljau type algorithms for evaluation and subdivision as well as a Marsden identity for arbitrary finite dimensional univariate function spaces.

  PublisherOA PDFDOI

• Basic hypergeometric formulas and identities for negative degree q-Bernstein bases

  Filomat · 2024-01-01 · 1 citations

  articleOpen accessSenior author

  Weutilize formulas for basic hypergeometric series to derive identities and formulas for negative degree q-Bernstein bases, including the Marsden identity, the partition of unity property, the monomial representation formula, the reparametrization formula, and the degree reduction formula. We show that all these identities are just special forms of the q-analogue of Gauss? formula. We also provide a new proof for the q-analogue of Gauss? formula by using the Marsden identity for negative degree q-Bernstein bases together with the identity theorem for analytic functions.

  PublisherOA PDFDOI

• Subdivision algorithms with modular arithmetic

  Computer Aided Geometric Design · 2024-01-09 · 1 citations

  article1st authorCorresponding
  PublisherDOI

• Rational Askey–Wilson Bernstein bases and a multirational Askey–Wilson blossom

  Journal of Computational Algebra · 2024-10-21

  articleOpen accessSenior author
  PublisherDOI

• On the uniqueness of the multirational blossom

  Computer Aided Geometric Design · 2023 · 4 citations

  Senior authorCorresponding
  • Mathematics
  • Pure mathematics
  • Combinatorics
  PublisherDOI

• Birational Quadratic Planar Maps with Generalized Complex Rational Representations

  Mathematics · 2023-08-21 · 1 citations

  articleOpen accessSenior author

  Complex rational maps have been used to construct birational quadratic maps based on two special syzygies of degree one. Similar to complex rational curves, rational curves over generalized complex numbers have also been constructed by substituting the imaginary unit with a new independent quantity. We first establish the relationship between degree one, generalized, complex rational Bézier curves and quadratic rational Bézier curves. Then we provide conditions to determine when a quadratic rational planar map has a generalized complex rational representation. Thus, a rational quadratic planar map can be made birational by suitably choosing the middle Bézier control points and their corresponding weights. In contrast to the edges of complex rational maps of degree one, which are circular arcs, the edges of the planar maps can be generalized to hyperbolic and parabolic arcs by invoking the hyperbolic and parabolic numbers.

  PublisherOA PDFDOI

• An Upside Down Approach to the Koch Curve

  SSRN Electronic Journal · 2022-01-01

  articleOpen accessSenior author
  PublisherDOI

• Quantum Lorentz degrees of polynomials and a Pólya theorem for polynomials positive on q-lattices

  Applied Numerical Mathematics · 2021-03-19

  articleOpen access
  PublisherOA PDFDOI

Recent grants

• US-France (INRIA) Cooperative Research: Symbolic and Numerical Methods for Geometric Modeling

  NSF · $30k · 2004–2010

• Systematic Construction of Single Determinants Representing Sparse Resultants

  NSF · $287k · 2002–2009

Frequent coauthors

• Plamen Simeonov

  Medical University Plovdiv

  14 shared

• Xuhui Wang

  China Automotive Engineering Research Institute

  10 shared

• Xiaohong Jia

  10 shared

• Phillip J. Barry

  L3Harris (United States)

  9 shared

• Li‐Yong Shen

  University of Chinese Academy of Sciences

  8 shared

• Haohao Wang

  Northwest A&F University

  7 shared

• Thomas W. Sederberg

  Brigham Young University

  7 shared

• Géraldine Morin

  7 shared