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Advanced Methods for Geometric Modeling and Numerical Simulation pp 127–177Cite as

New Developments in Theory, Algorithms, and Applications for Pythagorean–Hodograph Curves

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Part of the book series: Springer INdAM Series ((SINDAMS,volume 35))

Abstract

The past decade has witnessed sustained interest in elucidating the basic theory of Pythagorean–hodograph (PH) curves, developing construction algorithms, formulating generalizations, and investigating applications. The rapid pace of this activity, encompassing diverse lines of inquiry, makes it desirable at this point to take a broad perspective of these recent developments, and assess their relationships. In the present article, we aim to address this need by categorizing recent results into a number of broad themes—extensions and specializations of the basic polynomial PH curves; rational orthonormal frames along spatial PH curves; construction and analysis algorithms for PH curves; surface design based on PH curves; and the use of PH curves in practical applications.

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Notes

  1. 1.

    The use of the basis vector \(\mathbf{i}\) in the product (5) is only conventional—any other unit vector \(\mathbf{u}\) may be used instead, corresponding to a change of coordinates.

  2. 2.

    Since they have just a one–parameter family of tangent planes, the developable surfaces require a separate treatment.

  3. 3.

    For the rotation–minimizing motions with fixed initial/final orientations in [58, 71] the curve was not defined a priori, but was rather an outcome of the construction procedure.

  4. 4.

    The assumption that the object is static and situated at the origin is not essential, since only the relative position of the object and camera matter.

  5. 5.

    To exploit the advantageous features of PH splines, one must also store the complex or quaternion pre–image polynomials of each spline segment—see Sects. 5.4 and 7.3.

  6. 6.

    This is a consequence of the fact that, under the map (5), the pre–image of a point in \(\mathbb {R}^3\) is a circle in the quaternion space \(\mathbb {H}\)—see [68].

  7. 7.

    The system of lines of curvature is singular at umbilic points, where the principal directions are undefined [144].

  8. 8.

    The expression (36) assumes that these constants are the same for each machine axis, but this assumption is not essential.

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Acknowledgements

C. Giannelli and A. Sestini are members of the INdAM Research group GNCS. The INdAM support through GNCS and Finanziamenti Premiali SUNRISE is gratefully acknowledged.

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  1. Department of Mechanical and Aerospace Engineering, University of California, Davis, CA, USA

    Rida T. Farouki

  2. Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze, Florence, Italy

    Carlotta Giannelli & Alessandra Sestini

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  1. Rida T. Farouki
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    Carlotta Giannelli

  2. Department of Mathematics, University of Rome Tor Vergata, Rome, Italy

    Hendrik Speleers

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Farouki, R.T., Giannelli, C., Sestini, A. (2019). New Developments in Theory, Algorithms, and Applications for Pythagorean–Hodograph Curves. In: Giannelli, C., Speleers, H. (eds) Advanced Methods for Geometric Modeling and Numerical Simulation. Springer INdAM Series, vol 35. Springer, Cham. https://doi.org/10.1007/978-3-030-27331-6_7

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