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Low Degree Euclidean and Minkowski Pythagorean Hodograph Curves

  • Conference paper
Mathematical Methods for Curves and Surfaces (MMCS 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5862))

Abstract

In our contribution we study cubic and quintic Pythagorean Hodograph (PH) curves in the Euclidean and Minkowski planes. We analyze their control polygons and give necessary and sufficient conditions for cubic and quintic curves to be PH. In the case of Euclidean cubics the conditions are known and we provide a new proof. For the case of Minkowski cubics we formulate and prove a new simple geometrical condition. We also give conditions for the control polygons of quintics in both types of planes.

Moreover, we introduce the new notion of the preimage of a transformation, which is closely connected to the so-called preimage of a PH curve. We determine which transformations of the preimage curves produce similarities of PH curves in both Euclidean and Minkowski plane. Using these preimages of transformations we provide simple proofs of the known facts that up to similarities there exists only one Euclidean PH cubic (the so-called Tschirnhausen cubic) and two Minkowski PH cubics. Eventually, with the help of this novel approach we classify and describe the systems of Euclidean and Minkowski PH quintics.

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Author information

Authors and Affiliations

  1. Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, 186 75, Prague

    Zbyněk Šír

  2. Centre of Mathematics for Applications, Department of Informatics, University of Oslo, P.O. Box 1053, 0316, Blindern, Oslo, Norway

    Jiří Kosinka

Authors
  1. Zbyněk Šír
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  2. Jiří Kosinka
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Editor information

Editors and Affiliations

  1. Department of Informatics, University of Oslo, Norway

    Morten Dæhlen  & Tom Lyche  & 

  2. Department of Informatics and Centre of Mathematics for Applications, University of Oslo, Norway

    Michael Floater

  3. INSA de Rennes, Rennes, France

    Jean-Louis Merrien

  4. Department of Informatics and Centre of Mathematics for Applications, University of Oslo, Oslo, Norway

    Knut Mørken

  5. Department of Mathematics, Vanderbilt University, 37240, Nashville, TN, USA

    Larry L. Schumaker

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Šír, Z., Kosinka, J. (2010). Low Degree Euclidean and Minkowski Pythagorean Hodograph Curves. In: Dæhlen, M., Floater, M., Lyche, T., Merrien, JL., Mørken, K., Schumaker, L.L. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2008. Lecture Notes in Computer Science, vol 5862. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11620-9_26

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  • DOI: https://doi.org/10.1007/978-3-642-11620-9_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-11619-3

  • Online ISBN: 978-3-642-11620-9

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