Publications

Research monograph

Pythagorean-hodograph curves: algebra and geometry inseparable

Geometry and Computing Vol. 1, Springer, Berlin (728 p. + 204 illus.) ISBN 978-3-540-73397-3 (2008)

Springer website  table of contents  errata  MAA review  Amazon.com review  MathSciNet review

cdaBy virtue of their special algebraic structures, Pythagorean-hodograph (PH) curves offer unique advantages for computer-aided design and manufacturing, robotics, motion control, path planning, computer graphics, animation, and related fields. This book offers a comprehensive and self-contained treatment of the mathematical theory of PH curves, including algorithms for their construction and examples of their practical applications. Special features include an emphasis on the interplay of ideas from algebra and geometry and their historical origins, detailed algorithm descriptions, and many figures and worked examples. The book will appeal, in whole or in part, to mathematicians, computer scientists, and engineers.


Citations

Google scholar citations  Web of Science citations


Research papers

2019 2018 2017 2016 2015 2014 2013 2012 2011 2010
2009 2008 2007 2006 2005 2004 2003 2002 2001 2000
1999 1998 1997 1996 1995 1994 1993 1992 1991 1990
1989 1988 1987 1986 1985 1984 1983 1982 1981 1980

2019

  • R. T. Farouki and T. Sakkalis, Construction of rational curves with rational arc lengths by direct integration, preprint (2019)
  • B. Zhu and R. T. Farouki, A general framework for solving inverse dynamics problems in multi-axis motion control, ISA Transactions, to appear (2019)
  • R. T. Farouki, F. Pelosi, and M. L. Sampoli, Optimization of corner blending curves, preprint (2019)
  • R. T. Farouki and J. A. Strom, Computing the roots of sparse high-degree polynomials that arise from the study of random simplicial complexes, Numerical Algorithms, to appear (2019)
  • R. T. Farouki, C. Giannelli, and A. Sestini, New developments in theory, algorithms, and applications for Pythagorean-hodograph curves, Advanced Methods for Geometric Modeling and Numerical Simulation, Springer, to appear (2019)
  • R. T. Farouki, G. Gentili, H. P. Moon, and C. Stoppato, Minkowski products of unit quaternion sets, Advances in Computational Mathematics 45, 1607-1629 (2019)
  • R. T. Farouki, Existence of Pythagorean-hodograph quintic interpolants to spatial G^1 Hermite data with prescribed arc lengths, Journal of Symbolic Computation 95, 202-216 (2019)
  • R. T. Farouki, C. Giannelli, and A. Sestini, Rational minimal-twist motions on curves with rotation-minimizing Euler-Rodrigues frames, Journal of Computational and Applied Mathematics 352, 240-254 (2019)

2018

2017

2016

2015

2014

2013

2012

2011

2010

2009

2008

2007

2006

  • R. T. Farouki and C. Y. Han, Algorithms for spatial Pythagorean-hodograph curves, in Geometric Properties for Incomplete Data (R. Klette et al., eds.), Springer, 43-58 (2006)

2005

2004

2003

2002

2001

2000

1999

1998

1997

  • S. Hamaguchi and R. T. Farouki, Numerical simulation of phase transitions in dense dusty plasmas, in Advances in Dusty Plasmas (P. K. Shukla, D. A. Mendis, and T. Desai, eds.), Proceedings, International Conference on the Physics of Dusty Plasmas, Goa, India, World Scientific Press, 87-95, October 21-25 (1997)
  • S. Hamaguchi, R. T. Farouki, and D. H. E. Dubin, Triple point of Yukawa systems, Physical Review E 56, 4671-4682 (1997)
  • R. T. Farouki, Pythagorean-hodograph quintic transition curves of monotone curvature, Computer Aided Design 29, 601-606 (1997)
  • R. T. Farouki, Optimal parameterizations, Computer Aided Geometric Design 14, 153-168 (1997)
  • R. T. Farouki, Conic approximation of conic offsets, Journal of Symbolic Computation 23, 301-313 (1997)

1996

  • S. Hamaguchi, R. T. Farouki, and D. H. E. Dubin, Phase diagram of Yukawa systems near the one-component-plasma limit revisited, Journal of Chemical Physics 105, 7641-7647 (1996)
  • G. Albrecht and R. T. Farouki, Construction of C^2 Pythagorean-hodograph interpolating splines by the homotopy method, Advances in Computational Mathematics 5, 417-442 (1996)
  • R. T. Farouki and T. N. T. Goodman, On the optimal stability of the Bernstein basis, Mathematics of Computation 65, 1553-1566 (1996)
  • R. T. Farouki and S. Shah, Real-time CNC interpolators for Pythagorean-hodograph curves, Computer Aided Geometric Design 13, 583-600 (1996)
  • R. T. Farouki and R. Sverrisson, Approximation of rolling-ball blends for free-form parametric surfaces, Computer Aided Design 28, 871-878 (1996)
  • R. T. Farouki and T. Koenig, Computational methods for rapid prototyping of analytic solid models, Rapid Prototyping Journal 2 (3), 41-48 (1996)
  • R. T. Farouki and J. Peters, Smooth curve design with double-Tschirnhausen cubics, Annals of Numerical Mathematics 3, 63-82 (1996)
  • R. T. Farouki and H. Pottmann, Polynomial and rational Pythagorean-hodograph curves reconciled, in The Mathematics of Surfaces VI (G. Mullineux, ed.), Oxford University Press, 355-378 (1996)
  • R. T. Farouki, The elastic bending energy of Pythagorean-hodograph curves, Computer Aided Geometric Design 13, 227-241 (1996)

1995

  • A. Kaul and R. T. Farouki, Computing Minkowski sums of plane curves, International Journal of Computational Geometry and Applications 5, 413-432 (1995)
  • R. T. Farouki and C. A. Neff, Hermite interpolation by Pythagorean-hodograph quintics, Mathematics of Computation 64, 1589-1609 (1995)
  • R. T. Farouki, T. Koenig, K. Tarabanis, J. U. Korein, and J. S. Batchelder, Path planning with offset curves for layered manufacturing processes, Journal of Manufacturing Systems 14, 355-368 (1995)
  • R. T. Farouki and T. W. Sederberg, Analysis of the offset to a parabola, Computer Aided Geometric Design 12, 639-645 (1995)

1994

  • R. T. Farouki and S. Hamaguchi, Thermodynamics of strongly-coupled Yukawa systems near the one-component-plasma limit II. Molecular dynamics simulations, Journal of Chemical Physics 101, 9885-9893 (1994)
  • S. Hamaguchi and R. T. Farouki, Thermodynamics of strongly-coupled Yukawa systems near the one-component-plasma limit I. Derivation of the excess energy, Journal of Chemical Physics 101, 9876-9884 (1994)
  • R. T. Farouki and S. Hamaguchi, Spline approximation of “effective” potentials under periodic boundary conditions, Journal of Computational Physics 115, 276-287 (1994)
  • R. T. Farouki, K. Tarabanis, J. U. Korein, J. S. Batchelder, and S. R. Abrams, Offset curves in layered manufacturing, in Manufacturing Science and Engineering 1994 (Proceedings, 1994 International Mechanical Engineering Congress and Exposition), ASME PED Vol. 68-2, 557-568 (1994)
  • R. T. Farouki and J. K. Johnstone, Computing point/curve and curve/curve bisectors, in Design and Application of Curves and Surfaces, The Mathematics of Surfaces V (R. B. Fisher, ed.), Oxford University Press, 327-354 (1994)
  • R. T. Farouki, The conformal map z -> z^2 of the hodograph plane, Computer Aided Geometric Design 11, 363-390 (1994)
  • S. Hamaguchi and R. T. Farouki, Plasma-particulate interactions in non-uniform plasmas with finite flows, Physics of Plasmas 1, 2110-2118 (1994)
  • S. Hamaguchi, R. T. Farouki, and M. Dalvie, The Boltzmann-Poisson system in weakly collisional sheaths, in Semiconductors, Part I (W. M. Coughran, Jr. et al., eds.), IMA Volumes in Mathematics and its Applications, Vol. 58, 17-31, Springer (1994)
  • S. Hamaguchi and R. T. Farouki, Polarization force on a charged particulate in a non-uniform plasma, Physical Review E 49, 4430-4441 (1994)
  • R. T. Farouki and E. E. Salpeter, Mass segregation, relaxation, and the Coulomb logarithm in N-body systems (again), Astrophysical Journal 427, 676-683 (1994)
  • R. T. Farouki and J. K. Johnstone, The bisector of a point and a plane parametric curve, Computer Aided Geometric Design 11, 117-151 (1994)
  • R. T. Farouki and T. Sakkalis, Pythagorean-hodograph space curves, Advances in Computational Mathematics 2, 41-66 (1994)
  • R. T. Farouki, Watch your (parametric) speed!, in Computer-Aided Surface Geometry and Design, The Mathematics of Surfaces IV (A. Bowyer, ed.), Oxford University Press, 69-95 (1994)

1993

  • T. Sakkalis and R. T. Farouki, Algebraically rectifiable parametric curves, Computer Aided Geometric Design 10, 551-569 (1993)
  • S. Hamaguchi, M. Dalvie, R. T. Farouki, and S. Sethuraman, A shock-tracking algorithm for surface evolution under reactive-ion etching, Journal of Applied Physics 74, 5172-5184 (1993)
  • R. T. Farouki and S. Hamaguchi, Thermal energy of the crystalline one-component plasma from dynamical simulations, Physical Review E 47, 4330-4336 (1993)

1992

  • R. T. Farouki and S. Hamaguchi, Phase transitions of dense systems of charged “dust” grains in plasmas, Applied Physics Letters 61, 2973-2975 (1992)
  • R. T. Farouki, S. Hamaguchi, M. Surendra, and M. Dalvie, Dynamical N-body simulations of Coulomb scattering in plasma sheaths, Physical Review A 46, 7815-7829 (1992)
  • M. Dalvie, R. T. Farouki, S. Hamaguchi, and M. Surendra, Hydrodynamic analysis of electron motion in the cathode fall using a Monte Carlo simulation, Journal of Applied Physics 72, 2620-2631 (1992)
  • R. T. Farouki and J.-C. A. Chastang, Exact equations of “simple” wavefronts, Optik 91, 109-121 (1992)
  • T. W. Sederberg and R. T. Farouki, Approximation by interval Bezier curves, IEEE Computer Graphics and Applications 12 (5), 87-95 (1992)
  • M. Dalvie, S. Hamaguchi, and R. T. Farouki, Self-consistent Monte Carlo simulation of the cathode fall including treatment of negative glow electrons, Physical Review A 46, 1066-1077 (1992)
  • S. Hamaguchi, R. T. Farouki, and M. Dalvie, Ion energetics in collisionless sheaths of rf process plasmas, Physics of Fluids B 4, 2362-2367 (1992)
  • R. T. Farouki and J.-C. A. Chastang, Curves and surfaces in geometrical optics, in Mathematical Methods in Computer Aided Geometric Design II (T. Lyche and L. L. Schumaker, eds.), Academic Press, 239-260 (1992)
  • R. T. Farouki, S. Hamaguchi, and M. Dalvie, Analysis of a kinematic model for ion transport in rf plasma sheaths, Physical Review A 45, 5913-5928 (1992)
  • M. Dalvie, R. T. Farouki, and S. Hamaguchi, Flux considerations in the coupling of Monte Carlo plasma sheath simulations with feature evolution models, IEEE Transactions on Electron Devices 39, 1090-1099 (1992)
  • R. T. Farouki, Pythagorean-hodograph curves in practical use, in Geometry Processing for Design and Manufacturing (R. E. Barnhill, ed.), SIAM, Philadelphia, 3-33 (1992)
  • S. Hamaguchi, R. T. Farouki, and M. Dalvie, Ponderomotive force and ion energy distributions in an rf sheath, Physical Review Letters 68, 44-47 (1992)
  • J.-C. A. Chastang and R. T. Farouki, The mathematical evolution of wavefronts, Optics and Photonics News 3, 20-23 (1992)

1991

  • R. T. Farouki, Computing with barycentric polynomials, The Mathematical Intelligencer 13, 61-69 (1991)
  • S. Hamaguchi, R. T. Farouki, and M. Dalvie, Ion distribution function in a weakly collisional sheath, Physical Review A 44, 3804-3821 (1991)
  • R. T. Farouki, S. Hamaguchi, and M. Dalvie, Monte Carlo simulations of space-charge-limited ion transport through collisional plasma sheaths, Physical Review A 44, 2664-2681 (1991)
  • R. T. Farouki and T. Sakkalis, Real rational curves are not “unit speed,” Computer Aided Geometric Design 8, 151-157 (1991)
  • R. T. Farouki, On the stability of transformations between power and Bernstein polynomial forms, Computer Aided Geometric Design 8, 29-36 (1991)
  • R. T. Farouki, M. Dalvie, and L. F. Pavarino, Boundary-condition refinement of the Child-Langmuir law for collisionless dc plasma sheaths, Journal of Applied Physics 68, 6106-6116 (1991)

1990

  • R. T. Farouki and T. Sakkalis, Pythagorean hodographs, IBM Journal of Research and Development 34, 736-752 (1990)
  • R. T. Farouki and C. A. Neff, On the numerical condition of Bernstein-Bezier subdivision processes, Mathematics of Computation 55, 637-647 (1990)
  • R. T. Farouki and C. A. Neff, Algebraic properties of plane offset curves, Computer Aided Geometric Design 7, 101-127 (1990)
  • R. T. Farouki and C. A. Neff, Analytic properties of plane offset curves, Computer Aided Geometric Design 7, 83-99 (1990)
  • T. Sakkalis and R. T. Farouki, Singular points of algebraic curves, Journal of Symbolic Computation 9, 405-421 (1990)

1989

  • R. T. Farouki, Numerical stability in geometric algorithms and representations, in The Mathematics of Surfaces III (D. C. Handscomb, ed.), Oxford University Press, 83-113 (1989)
  • R. T. Farouki, C. A. Neff, and M. A. O’Connor, Automatic parsing of degenerate quadric surface intersections, ACM Transactions on Graphics 8, 174-203 (1989)
  • R. T. Farouki, Hierarchical segmentations of algebraic curves and some applications, in Mathematical Methods in Computer Aided Geometric Design (T. Lyche and L. L. Schumaker, eds.), Academic Press, 239-248 (1989)

1988

  • R. T. Farouki and V. T. Rajan, On the numerical condition of algebraic curves and surfaces I. Implicit equations, Computer Aided Geometric Design 5, 215-252 (1988)
  • R. T. Farouki and V. T. Rajan, Algorithms for polynomials in Bernstein form, Computer Aided Geometric Design 5, 1-26 (1988)
  • R. T. Farouki, Computational issues in solid boundary evaluation, Information Sciences 44, 87-171 (1988)

1987

  • R. T. Farouki, Graphical methods for surface differential geometry, in The Mathematics of Surfaces II (R. R. Martin, ed.), Oxford University Press, 363-385 (1987)
  • R. T. Farouki and V. T. Rajan, On the numerical condition of polynomials in Bernstein form, Computer Aided Geometric Design 4, 191-216 (1987)
  • R. T. Farouki, Direct surface section evaluation, in Geometric Modeling: Algorithms and New Trends (G. Farin, ed.), SIAM, 319-334 (1987)
  • R. T. Farouki, Trimmed surface algorithms for the evaluation and interrogation of solid boundary representations, IBM Journal of Research and Development 31, 314-334 (1987)

1986

  • J. M. Beck, R. T. Farouki, and J. K. Hinds, Surface analysis methods, IEEE Computer Graphics and Applications 6 (12), 18-36 (1986)
  • R. T. Farouki, The characterization of parametric surface sections, Computer Vision, Graphics, and Image Processing 33, 209-236 (1986)

1985

  • R. T. Farouki, Exact offset procedures for simple solids, Computer Aided Geometric Design 2, 257-279 (1985)
  • R. T. Farouki and J. K. Hinds, A hierarchy of geometric forms, IEEE Computer Graphics and Applications 5 (5), 51-78 (1985)
  • I. Wasserman, S. L. Shapiro, and R. T. Farouki, Do cosmic magnetic monopoles cause rapid decay of the galactic magnetic field?, Comments on Astrophysics 10, 257-265 (1985)

1984

  • R. T. Farouki, S. L. Shapiro, and I. Wasserman, Numerical simulation of the plasma and gravitational dynamics of a galactic magnetic monopole halo, Astrophysical Journal 284, 282-298 (1984)

1983

  • R. T. Farouki, G. L. Hoffman, and E. E. Salpeter, The collapse and violent relaxation of N-body systems: mass segregation and the secondary maximum, Astrophysical Journal 271, 11-21 (1983)
  • M. J. Duncan, R. T. Farouki, and S. L. Shapiro, Simulations of galaxy mergers: cannibalism and dynamical friction, Astrophysical Journal 271, 22-31 (1983)
  • R. T. Farouki, S. L. Shapiro, and S. A. Teukolsky, Computational astrophysics on the array professor, IEEE Computer 16 (6), 73-83 (1983)
  • R. T. Farouki, S. L. Shapiro, and M. J. Duncan, Hierarchical merging and the structure of elliptical galaxies, Astrophysical Journal 265, 597-605 (1983)

1982

  • R. T. Farouki and S. L. Shapiro, Simulations of merging disk galaxies, Astrophysical Journal 259, 103-115 (1982)
  • R. T. Farouki and E. E. Salpeter, Mass segregation, relaxation, and the Coulomb logarithm in N-body systems, Astrophysical Journal 253, 512-519 (1982)

1981

  • R. T. Farouki and S. L. Shapiro, Computer simulations of environmental influences on galaxy evolution in dense clusters II. Rapid tidal encounters, Astrophysical Journal 243, 32-41 (1981)

1980

  • R. T. Farouki and S. L. Shapiro, Computer simulations of environmental influences on galaxy evolution in dense clusters I. Ram-pressure stripping, Astrophysical Journal 241, 928-945 (1980)

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