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

By 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

Research papers

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2009 2008 2007 2006 2005 2004 2003 2002 2001 2000
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2025

F. Pelosi, M. L. Sampoli, and R. T. Farouki, Control point modifications that preserve the Pythagorean-hodograph nature of planar quintic curves,
Journal of Computational and Applied Mathematics 457, article 116301 (2025)
R. T. Farouki, M. Knez, V. Vitrih, and E. Zagar, Application of a metric for complex polynomials to bounded modification of planar Pythagorean-hodograph curves, Journal of Computational and Applied Mathematics 456, article 116235 (2025)

2024

R. T. Farouki, M. Knez, V. Vitrih, and E. Zagar, Least-squares approximation with planar Pythagorean-hodograph curves, preprint (2024)

2023

R. T. Farouki, Partition of the space of planar quintic Pythagorean-hodograph curves, Computer Aided Geometric Design, 106, article 102242 (2023)
R. T. Farouki, F. Pelosi, and M. L. Sampoli, Construction of planar quintic Pythagorean-hodograph curves by control-polygon constraints, Computer Aided Geometric Design 103, article 102192 (2023)

2022

R. T. Farouki, M. Knez, V. Vitrih, and E. Zagar, On the construction of polynomial minimal surfaces with Pythagorean normals, Applied Mathematics and Computation 435, article 127439 (2022)
R. T. Farouki, Identifying Pythagorean-hodograph curves closest to prescribed planar Bezier curves, Computer Aided Design 149, article 103266 (2022)
R. T. Farouki and J. R. Swett, Real-time compensation of backlash positional errors in CNC machines by localized feedrate modulation, International Journal of Advanced Manufacturing Technology 119, 5763-5776 (2022)
R. T. Farouki, The Cartesian ovals, The Mathematical Intelligencer 44, 343-353 (2022)

2021

R. T. Farouki, M. Knez, V. Vitrih, and E. Zagar, Planar projections of spatial Pythagorean-hodograph curves, Computer Aided Geometric Design 91, article 102049 (2021)
R. T. Farouki, J. R. Swett, R. W. Rohlen, and D. B. Smith, Real-time needle guidance for venipuncture based on optical coherence tomography, Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization 9, 643-650 (2021)
R. T. Farouki, Accurate real-time CNC interpolators based upon Richardson extrapolation, Computer Aided Design 135, article 103005 (2021)
R. T. Farouki, M. Knez, V. Vitrih, and E. Zagar, Spatial C2 closed loops of prescribed arc length defined by Pythagorean-hodograph curves, Applied Mathematics and Computation 391, article 125653 (2021)
R. T. Farouki, F. Pelosi, and M. L. Sampoli, Approximation of monotone clothoid segments by degree 7 Pythagorean-hodograph curves, Journal of Computational and Applied Mathematics 382, article 113110 (2021)
D. R. Garcia, B. S. Linke, and R. T. Farouki, Optimized routine of machining distortion characterization based on Gaussian surface curvature, Proceedings, 2nd IRTG 2057 Conference – Physical Modeling for Virtual Manufacturing, article 5, Springer (2021)

2020

R. T. Farouki, K. Hormann, and F. Nudo, Singular cases of planar and spatial Hermite interpolation problems based on Pythagorean-hodograph curves, Computer Aided Geometric Design 82, article 101930 (2020)
B. W. L. Margolis and R. T. Farouki, Inverse dynamics toolpath compensation for CNC machines based on model predictive control, International Journal of Advanced Manufacturing Technology 109, 2155-2172 (2020)
R. T. Farouki, S. H. Kim, and H. P. Moon, Construction of periodic adapted orthonormal frames on closed space curves, Computer Aided Geometric Design 76, article 101802 (2020)
R. T. Farouki and Z. Sir, Mapping rational rotation-minimizing frames from polynomial curves on to rational curves, Computer Aided Geometric Design 78, article 101833 (2020)
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 83, 1653-1670 (2020)

2019

S. Deshmukh, A. Alghanemi, and R. T. Farouki, Space curves defined by curvature-torsion relations and associated helices, Filomat 33, 4951-4966 (2019)
R. T. Farouki and T. Sakkalis, Construction of rational curves with rational arc lengths by direct integration, Computer Aided Geometric Design 74, article 101773 (2019)
B. Zhu and R. T. Farouki, A general framework for solving inverse dynamics problems in multi-axis motion control, ISA Transactions 95, 130-143 (2019)
R. T. Farouki and K. Hormann, Geometric design: new trends and challenges, SIAM News 52 (3), 8 (2019)
R. T. Farouki, F. Pelosi, and M. L. Sampoli, Optimization of corner blending curves, Computer Aided Design 117, article 102739 (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 (C. Giannelli and H. Speleers, eds.), Springer ISBN 978-3-030-27331-6, 127-177 (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

R. T. Farouki and J. Srinathu, Suppression of chip load variations by real-time spindle speed modulation, International Journal of Advanced Manufacturing Technology 99, 2005-2014 (2018)
R. T. Farouki and J. Srinathu, Feedrate modulation for accurate traversal of trimmed planar offset paths, International Journal of Advanced Manufacturing Technology 97, 3325-3337 (2018)
H. P. Moon and R. T. Farouki, C^1 and C^2 interpolation of orientation data along spatial Pythagorean-hodograph curves using rational adapted spline frames, Computer Aided Geometric Design 66, 1-15 (2018)
R. T. Farouki and H. P. Moon, Rational frames of minimal twist along space curves under specified boundary conditions, Advances in Computational Mathematics 44, 1627-1650 (2018)
R. T. Farouki, C. Giannelli, D. Mugnaini, and A. Sestini, Path planning with Pythagorean-hodograph curves for unmanned or autonomous vehicles, Journal of Aerospace Engineering, 232, 1361-1372 (2018)
R. T. Farouki, Reduced difference polynomials and self-intersection computations, Applied Mathematics and Computation 324, 174-190 (2018)

2017

R. T. Farouki, Helical polynomial curves interpolating G^1 data with prescribed axes and pitch angles, Computer Aided Geometric Design 56, 4-15 (2017)
R. T. Farouki and J. Srinathu, A real-time CNC interpolator algorithm for trimming and filling planar offset curves, Computer Aided Design 86, 1-11 (2017)
B. Fu, R. T. Farouki, and F. O. Eke, Equilibrium configuration of a bounded inextensible membrane subject to solar radiation pressure, Aerospace Science and Technology 68, 552-560 (2017)
K. M. Nittler and R. T. Farouki, Efficient high-speed cornering motions based on continuously-variable feedrates. II. Implementation and performance analysis, International Journal of Advanced Manufacturing Technology 88, 159-174 (2017)
R. T. Farouki, G. Gentili, C. Giannelli, A. Sestini, and C. Stoppato, A comprehensive characterization of the set of polynomial curves with rational rotation-minimizing frames, Advances in Computational Mathematics 43, 1-24 (2017)

2016

R. T. Farouki and B. S. Linke, Shigley Hauler – a competitive project illustrating basic machine design principles, International Journal of Mechanical Engineering Education 44, 284-301 (2016)
R. T. Farouki, Construction of G^1 planar Hermite interpolants with prescribed arc lengths, Computer Aided Geometric Design 46, 64-75 (2016)
R. T. Farouki and K. M. Nittler, Efficient high-speed cornering motions based on continuously-variable feedrates I. Real-time interpolator algorithm, International Journal of Advanced Manufacturing Technology 87, 3557-3568 (2016)
K. M. Nittler and R. T. Farouki, A real-time surface interpolator methodology for precision CNC machining of swept surfaces, International Journal of Advanced Manufacturing Technology 83, 561-574 (2016)
R. T. Farouki, C. Giannelli, and A. Sestini, Local modification of Pythagorean-hodograph quintic spline curves using the B-spline form, Advances in Computational Mathematics 42, 199-225 (2016)
R. T. Farouki, G. Gentili, C. Giannelli, A. Sestini, and C. Stoppato, Solution of a quadratic quaternion equation with mixed coefficients, Journal of Symbolic Computation 74, 140-151 (2016)
B. Fu, R. T. Farouki, and F. O. Eke, Mechanics of bounded inextensible membranes subject to solar radiation pressure, Paper AAS 16-547, Proceedings 26th AAS/AIAA Spaceflight Mechanics Meeting, Napa, CA, February 14-18 (2016)
R. T. Farouki, Rational rotation-minimizing frames – recent advances and open problems, Applied Mathematics and Computation 272 (1), 80-91 (2016)
R. T. Farouki and R. Gutierrez, Geometry of the ringed surfaces in R^4 that generate spatial Pythagorean hodographs, Journal of Symbolic Computation 37, 87-103 (2016)

2015

R. T. Farouki, F. Pelosi, M. L. Sampoli, and A. Sestini, Tensor-product surface patches with Pythagorean-hodograph isoparametric curves, IMA Journal of Numerical Analysis 36, 1389-1409 (2015)
R. T. Farouki, C. Giannelli, and A. Sestini, Identification and “reverse engineering” of Pythagorean-hodograph curves, Computer Aided Geometric Design 34, 21-36 (2015)
B. Dong and R. T. Farouki, Algorithm 952: PHquintic: A library of basic functions for the construction and analysis of planar quintic Pythagorean-hodograph curves, ACM Transactions on Mathematical Software 41 (4), Article 28 (2015)
R. T. Farouki, Arc lengths of rational Pythagorean-hodograph curves, Computer Aided Geometric Design 34, 1-4 (2015)
R. T. Farouki and K. M. Nittler, Rational swept surface constructions based on differential and integral sweep curve properties, Computer Aided Geometric Design 33, 1-16 (2015)
R. T. Farouki, C. Manni, M. L. Sampoli, and A. Sestini, Shape-preserving interpolation of spatial data by Pythagorean-hodograph quintic spline curves, IMA Journal of Numerical Analysis 35, 478-498 (2015)

2014

T. F. Schraeder and R. T. Farouki, Experimental performance analysis of an inverse dynamics CNC compensation scheme for high-speed execution of curved toolpaths, International Journal of Advanced Manufacturing Technology 73, 195-208 (2014)
R. T. Farouki, Construction of G^2 rounded corners with Pythagorean-hodograph curves, Computer Aided Geometric Design 31, 127-139 (2014)
R. T. Farouki, C. Giannelli, M. L. Sampoli, and A. Sestini, Rotation-minimizing osculating frames, Computer Aided Geometric Design 31, 27-42 (2014)
R. T. Farouki, C. Y. Han, and S. Li, Inverse kinematics for optimal tool orientation control in 5-axis machining, Computer Aided Geometric Design 31, 13-26 (2014)
M. Huard, R. T. Farouki, N. Sprynski, and L. Biard, C^2 interpolation of spatial data subject to arc-length constraints using Pythagorean-hodograph quintic splines, Graphical Models 76, 30-42 (2014)

2013

R. T. Farouki and T. Sakkalis, Corrigendum to “Rational rotation-minimizing frames on polynomial space curves of arbitrary degree” [J. Symbolic Comput. 45 (8) (2010) 844-856], Journal of Symbolic Computation 58, 99-102 (2013)
R. T. Farouki, P. Dospra, and T. Sakkalis, Scalar-vector algorithm for the roots of quadratic quaternion polynomials, and the characterization of quintic rational rotation-minimizing frame curves, Journal of Symbolic Computation 58, 1-17 (2013)
R. T. Farouki, C. Y. Han, P. Dospra, and T. Sakkalis, Rotation-minimizing Euler-Rodrigues rigid-body motion interpolants, Computer Aided Geometric Design 30, 653-671 (2013)
R. T. Farouki and S. Li, Optimal tool orientation control for 5-axis CNC milling with ball-end cutters, Computer Aided Geometric Design 30, 226-239 (2013)
J. R. Conway, A. L. Darling, C. A. Ernesto, R. T. Farouki, and C. A. Palomares, Experimental study of contouring accuracy for CNC machines executing curved paths with constant and curvature-dependent feedrates, Robotics and Computer Integrated Manufacturing 29, 357-369 (2013)
R. T. Farouki, C. Giannelli, and A. Sestini, An interpolation scheme for designing rational rotation-minimizing camera motions, Advances in Computational Mathematics 38, 63-82 (2013)

2012

T. Sakkalis and R. T. Farouki, Pythagorean-hodograph curves in Euclidean spaces of dimension greater than 3, Journal of Computational and Applied Mathematics 236, 4375-4382 (2012)
D. A. Reiter, F. Fathallah, R. T. Farouki, and J. H. Walton, Noninvasive high resolution mechanical strain maps of the spine intervertebral disc using nonrigid registration of magnetic resonance images, Journal of Biomechanics 45, 1534-1539 (2012)
R. T. Farouki, The Bernstein polynomial basis: a centennial retrospective, Computer Aided Geometric Design 29, 379-419 (2012)
R. T. Farouki and T. Sakkalis, A complete classification of quintic space curves with rational rotation-minimizing frames, Journal of Symbolic Computation 47, 214-226 (2012)
J. R. Conway, C. A. Ernesto, R. T. Farouki, and M. Zhang, Performance analysis of cross-coupled controllers for CNC machines based upon precise real-time contour error measurement, International Journal of Machine Tools and Manufacture 52, 30-39 (2012)
C. A. Ernesto and R. T. Farouki, High-speed cornering by CNC machines under prescribed bounds on axis accelerations and toolpath contour error, International Journal of Advanced Manufacturing Technology 58, 327-338 (2012)
R. T. Farouki, C. Manni, F. Pelosi, and M. L. Sampoli, Design of C^2 spatial Pythagorean-hodograph quintic splines by control polygons, in J. D. Boissonnat et al. (eds.), Lecture Notes in Computer Science 6920, 253-269, Springer (2012)
R. T. Farouki, C. Giannelli, C. Manni, and A. Sestini, Design of rational rotation-minimizing rigid body motions by Hermite interpolation, Mathematics of Computation 81, 879-903 (2012)

2011

R. T. Farouki and T. Sakkalis, Equivalence of distinct characterizations for rational rotation-minimizing frames on quintic space curves, Computer Aided Geometric Design 28, 436-445 (2011)
R. T. Farouki and Z. Sir, Rational Pythagorean-hodograph space curves, Computer Aided Geometric Design 28, 75-88 (2011)

2010

R. T. Farouki, Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves, Advances in Computational Mathematics 33, 331-348 (2010)
C. A. Ernesto and R. T. Farouki, Solution of inverse dynamics problems for contour error minimization in CNC machines, International Journal of Advanced Manufacturing Technology 49, 589-604 (2010)
R. T. Farouki and T. Sakkalis, Rational rotation-minimizing frames on polynomial space curves of arbitrary degree, Journal of Symbolic Computation 45, 844-856 (2010)
L. Biard, R. T. Farouki, and N. Szafran, Construction of rational surface patches bounded by lines of curvature, Computer Aided Geometric Design 27, 359-371 (2010)
R. T. Farouki, N. Szafran, and L. Biard, Construction and smoothing of triangular Coons patches with geodesic boundary curves, Computer Aided Geometric Design 27, 301-312 (2010)
R. T. Farouki, C. Giannelli, and A. Sestini, Geometric design using space curves with rational rotation-minimizing frames, in M. Daehlen et al. (eds.), Lecture Notes in Computer Science 5862, 194-208, Springer (2010)

2009

R. T. Farouki, N. Szafran, and L. Biard, Construction of Bezier surface patches with Bezier curves as geodesic boundaries, Computer Aided Design 41, 772-781 (2009)
R. T. Farouki and C. Giannelli, Spatial camera orientation control by rotation-minimizing directed frames, Computer Animation and Virtual Worlds 20, 457-472 (2009)
R. T. Farouki, N. Szafran, and L. Biard, Existence conditions for Coons patches interpolating geodesic boundary curves, Computer Aided Geometric Design 26, 599-614 (2009)
R. T. Farouki, C. Giannelli, C. Manni, and A. Sestini, Quintic space curves with rational rotation-minimizing frames, Computer Aided Geometric Design 26, 580-592 (2009)
T. Sakkalis, R. T. Farouki, and L. Vaserstein, Non-existence of rational arc length parameterizations for curves in R^n, Journal of Computational and Applied Mathematics 228, 494-497 (2009)
R. T. Farouki, C. Giannelli, and A. Sestini. Helical polynomial curves and double Pythagorean hodographs II. Enumeration of low-degree curves, Journal of Symbolic Computation 44, 307-322 (2009)
R. T. Farouki, C. Giannelli, and A. Sestini. Helical polynomial curves and double Pythagorean hodographs I. Quaternion and Hopf map representations, Journal of Symbolic Computation 44, 161-179 (2009)

2008

H. I. Choi, R. T. Farouki, C. Y. Han, and H. P. Moon, Computing the Minkowski value of the exponential function over a complex disk, in Computer Mathematics (D. Kapur, ed.), Lecture Notes in Artificial Intelligence 5081, 1-21, Springer (2008)
H. I. Choi, R. T. Farouki, S. H. Kwon, and H. P. Moon, Topological criterion for selection of quintic Pythagorean-hodograph Hermite interpolants, Computer Aided Geometric Design 25, 411-433 (2008)
R. T. Farouki, C. Giannelli, C. Manni, and A. Sestini, Identification of spatial PH quintic Hermite interpolants with near-optimal shape measures, Computer Aided Geometric Design 25, 274-297 (2008)
R. T. Farouki, B. Juettler, and C. Manni, Preface: Pythagorean-hodograph curves and related topics, Computer Aided Geometric Design 25, 203-204 (2008)

2007

R. T. Farouki and J. Hass, Evaluating the boundary and covering degree of planar Minkowski sums and other geometrical convolutions, Journal of Computational and Applied Mathematics 209, 246-266 (2007)
S. D. Timar, R. T. Farouki, and C. L. Boyadjieff, Time-optimal feedrates along curved paths for Cartesian CNC machines with prescribed bounds on axis velocities and accelerations, International Journal of Robotics and Automation 22, 112-125 (2007)
J. Hass, R. T. Farouki, C. Y. Han, X. Song, and T. W. Sederberg, Guaranteed consistency of surface intersections and trimmed surfaces using a coupled topology resolution and domain decomposition scheme, Advances in Computational Mathematics 27, 1-26 (2007)
S. D. Timar and R. T. Farouki, Time-optimal traversal of curved paths by Cartesian CNC machines under both constant and speed-dependent axis acceleration bounds, Robotics and Computer Integrated Manufacturing 23, 563-579 (2007)
R. T. Farouki and T. Sakkalis, Rational space curves are not “unit speed,” Computer Aided Geometric Design 24, 238-240 (2007)
C. B. U. Perwass, R. T. Farouki, and L. Noakes, A geometric product formulation for spatial Pythagorean-hodograph curves with applications to Hermite interpolation, Computer Aided Geometric Design 24, 220-237 (2007)
R. T. Farouki and C. Y. Han, Root neighborhoods, generalized lemniscates, and robust stability of dynamic systems, Applicable Algebra in Engineering, Communication, and Computing 18, 169-189 (2007)
F. Pelosi, M. L. Sampoli, R. T. Farouki, and C. Manni, A control polygon scheme for design of planar C^2 PH quintic spline curves, Computer Aided Geometric Design 24, 28-52 (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

C. L. Boyadjieff, R. T. Farouki, and S. D. Timar, Smoothing of time-optimal feedrates for Cartesian CNC machines, in Mathematics of Surfaces XI (R. Martin, H. Bez, and M. Sabin, eds.), Springer, 84-101 (2005)
R. T. Farouki, C. Y. Han, and J. Hass, Boundary evaluation algorithms for Minkowski combinations of complex sets using topological analysis of implicit curves, Numerical Algorithms 40, 251-283 (2005)
F. Pelosi, R. T. Farouki, C. Manni, and A. Sestini, Geometric Hermite interpolation by spatial Pythagorean-hodograph cubics, Advances in Computational Mathematics 22, 325-352 (2005)
R. T. Farouki and C. Y. Han, Solution of elementary equations in the Minkowski geometric algebra of complex sets, Advances in Computational Mathematics 22, 301-323 (2005)
S. D. Timar, R. T. Farouki, T. S. Smith, and C. L. Boyadjieff, Algorithms for time-optimal control of CNC machines along curved tool paths, Robotics and Computer Integrated Manufacturing 21, 37-53 (2005)

2004

R. T. Farouki and C. Y. Han, Robust plotting of generalized lemniscates, Applied Numerical Mathematics 51, 257-272 (2004)
R. T. Farouki, C. Y. Han, J. Hass, and T. W. Sederberg, Topologically consistent trimmed surface approximations based on triangular patches, Computer Aided Geometric Design 21, 459-478 (2004)
R. T. Farouki and C. Y. Han, Computation of Minkowski values of polynomials over complex sets, Numerical Algorithms 36, 13-29 (2004)
X. W. Song, T. W. Sederberg, J. Zheng, R. T. Farouki, and J. Hass, Linear perturbation methods for topologically consistent representations of free-form surface intersections, Computer Aided Geometric Design 21, 303-319 (2004)
R. T. Farouki, C. Y. Han, C. Manni, and A. Sestini, Characterization and construction of helical polynomial space curves, Journal of Computational and Applied Mathematics 162, 365-392 (2004)

2003

R. T. Farouki and C. Y. Han, Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves, Computer Aided Geometric Design 20, 435-454 (2003)
R. T. Farouki, C. Manni, and A. Sestini, Spatial C^2 PH quintic splines, Curve and Surface Design: Saint-Malo 2002 (T. Lyche, M-L. Mazure, and L. L. Schumaker, eds.), Nashboro Press, 147-156 (2003)
R. T. Farouki, Minkowski combinations of complex sets – geometry, algorithms, and applications, Curve and Surface Design: Saint-Malo 2002 (T. Lyche, M-L. Mazure, and L. L. Schumaker, eds.), Nashboro Press, 123-146 (2003)
R. T. Farouki, T. N. T. Goodman, and T. Sauer, Construction of orthogonal bases for polynomials in Bernstein form on triangular and simplex domains, Computer Aided Geometric Design 20, 209-230 (2003)
R. T. Farouki, C. Manni, and A. Sestini, Shape-preserving interpolation by G^1 and G^2 PH quintic splines, IMA Journal of Numerical Analysis 23, 175-195 (2003)
R. T. Farouki and H. P. Moon, Minkowski geometric algebra and the stability of characteristic polynomials, in Visualization and Mathematics III (H-C. Hege and K. Polthier, eds.), Springer, 163-188 (2003)
R. T. Farouki, Exact rotation-minimizing frames for spatial Pythagorean-hodograph curves, Graphical Models 64, 382-395 (2003)

2002

R. T. Farouki, M. al-Kandari, and T. Sakkalis, Hermite interpolation by rotation-invariant spatial Pythagorean-hodograph curves, Advances in Computational Mathematics 17, 369-383 (2002)
R. T. Farouki, M. al-Kandari, and T. Sakkalis, Structural invariance of spatial Pythagorean hodographs, Computer Aided Geometric Design 19, 395-407 (2002)
R. T. Farouki, Preface: Advances in geometrical algorithms and representations, Advances in Computational Mathematics 17, 1-3 (2002)
T. S. Smith, R. T. Farouki, M. al-Kandari, and H. Pottmann, Optimal slicing of free-form surfaces, Computer Aided Geometric Design 19, 43-64 (2002)
R. T. Farouki, Pythagorean-hodograph curves, in Handbook of Computer Aided Geometric Design (G. Farin, J. Hoschek, and M-S. Kim, eds.). Elsevier, 405-427 (2002)
R. T. Farouki and H. Pottmann, Exact Minkowski products of N complex disks, Reliable Computing 8, 43-66 (2002)

2001

P. Costantini, R. T. Farouki, C. Manni, and A. Sestini, Computation of optimal composite parameterizations, Computer Aided Geometric Design 18, 875-897 (2001)
T. S. Smith and R. T. Farouki, Gauss map computation for free-form surfaces, Computer Aided Geometric Design 18, 831-850 (2001)
R. T. Farouki, B. K. Kuspa, C. Manni, and A. Sestini, Efficient solution of the complex quadratic tridiagonal system for C^2 PH quintic splines, Numerical Algorithms 27, 35-60 (2001)
Y-F. Tsai and R. T. Farouki, Algorithm 812: BPOLY: An object-oriented library of numerical algorithms for polynomials in Bernstein form, ACM Transactions on Mathematical Software 27, 267-296 (2001)
R. T. Farouki, C. Manni, and A. Sestini, Real-time CNC interpolators for Bezier conics, Computer Aided Geometric Design 18, 639-655 (2001)
Y-F. Tsai, R. T. Farouki, and B. Feldman, Performance analysis of CNC interpolators for time-dependent feedrates along PH curves, Computer Aided Geometric Design 18, 245-265 (2001)
R. T. Farouki, H. P. Moon, and B. Ravani, Minkowski geometric algebra of complex sets, Geometriae Dedicata 85, 283-315 (2001)
A. S. al-Yahmadi, T. C. Hsia, and R. T. Farouki, Sliding mode control of two arms manipulating a flexible beam, Proceedings, 32nd International Symposium on Robotics, Seoul, Korea, April 19-21 (2001)
H. P. Moon, R. T. Farouki, and H. I. Choi, Construction and shape analysis of PH quintic Hermite interpolants, Computer Aided Geometric Design 18, 93-115 (2001)
R. T. Farouki and Y-F. Tsai, Exact Taylor series coefficients for variable-feedrate CNC curve interpolators, Computer Aided Design 33, 155-165 (2001)

2000

R. T. Farouki and H. P. Moon, Bipolar and multipolar coordinates, in The Mathematics of Surfaces IX (R. Cippola and R. Martin, eds.), Springer, 348-371 (2000)
R. T. Farouki, W. Gu, and H. P. Moon, Minkowski roots of complex sets, in Geometric Modeling and Processing 2000, IEEE Computer Society Press, 287-300 (2000)
R. T. Farouki, H. P. Moon, and B. Ravani, Algorithms for Minkowski products and implicitly-defined complex sets, Advances in Computational Mathematics 13, 199-229 (2000)
R. T. Farouki, Curves from motion, motion from curves, in Curve and Surface Design: Saint-Malo 1999 (P-J. Laurent, P. Sablonniere, and L. L. Schumaker, eds.), Vanderbilt University Press, 63-90 (2000)
R. T. Farouki, Y-F. Tsai, and C. Wilson, Physical constraints on feedrates and feed accelerations along curved tool paths, Computer Aided Geometric Design 17, 337-359 (2000)
R. T. Farouki, Legendre-Bernstein basis transformations, Journal of Computational and Applied Mathematics 119, 145-160 (2000)
R. T. Farouki, Convergent inversion approximations for polynomials in Bernstein form, Computer Aided Geometric Design 17, 179-196 (2000)

1999

R. T. Farouki, Closing the gap between CAD model and downstream application, SIAM News 32 (5), 1-3 (1999)
R. Ramamurthy and R. T. Farouki, Voronoi diagram and medial axis algorithm for planar domains with curved boundaries II. Detailed algorithm description, Journal of Computational and Applied Mathematics 102, 253-277 (1999)
R. Ramamurthy and R. T. Farouki, Voronoi diagram and medial axis algorithm for planar domains with curved boundaries I. Theoretical foundations, Journal of Computational and Applied Mathematics 102, 119-141 (1999)
R. T. Farouki, Y-F. Tsai, and G-F. Yuan, Contour machining of free-form surfaces with real-time PH curve CNC interpolators, Computer Aided Geometric Design 16, 61-76 (1999)
R. T. Farouki, J. Manjunathaiah, and G-F. Yuan, G codes for the specification of Pythagorean-hodograph tool paths and associated feedrate functions on open-architecture CNC machines, International Journal of Machine Tools and Manufacture 39, 123-142 (1999)

1998

R. T. Farouki, K. Saitou, and Y-F. Tsai, Least-squares path approximation with Pythagorean-hodograph curves for high-speed CNC machining, in The Mathematics of Surfaces VIII (R. Cripps, ed.), Information Geometers, 245-264 (1998)
R. Ramamurthy and R. T. Farouki, Topological and computational issues in Voronoi diagram and medial axis constructions for planar domains with curved boundaries, Proceedings, Workshop on Differential/Topological Techniques in Geometric Modeling and Processing, Pohang, Korea, Bookplus Press, 1-26, April 7-9 (1998)
R. T. Farouki and J. Rampersad, Cycles upon cycles: An anecdotal history of higher curves in science and engineering, in Mathematical Methods for Curves and Surfaces II (M. Daehlen, T. Lyche, and L. L. Schumaker, eds.), Vanderbilt University Press, 95-116 (1998)
R. T. Farouki and R. Ramamurthy, Degenerate point/curve and curve/curve bisectors arising in medial axis computations for planar domains with curved boundaries, Computer Aided Geometric Design 15, 615-635 (1998)
R. T. Farouki, J. Manjunathaiah, D. Nicholas, G-F. Yuan, and S. Jee, Variable-feedrate CNC interpolators for constant material removal rates along Pythagorean-hodograph curves, Computer Aided Design 30, 631-640 (1998)
R. T. Farouki, J. Manjunathaiah, and S. Jee, Design of rational cam profiles with Pythagorean-hodograph curves, Mechanism and Machine Theory 33, 669-682 (1998)
R. T. Farouki, On integrating lines of curvature, Computer Aided Geometric Design 15, 187-192 (1998)
R. T. Farouki and R. Ramamurthy, Specified-precision computation of curve/curve bisectors, International Journal of Computational Geometry and Applications 8, 599-617 (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)

1990

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 (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 approximation of non-degenerate offset surfaces, Computer Aided Geometric Design 3, 15-43 (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)