**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 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**

- 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) - 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 (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, 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, 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 intervetrebral 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)

**2002**

- R. T. Farouki, Exact rotation-minimizing frames for spatial Pythagorean-hodograph curves,
*Graphical Models***64**, 382-395 (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 Desig*n**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) - 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)