IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v272y2016ip1p80-91.html
   My bibliography  Save this article

Rational rotation-minimizing frames—Recent advances and open problems

Author

Listed:
  • Farouki, Rida T.

Abstract

Recent developments in the basic theory, algorithms, and applications for curves with rational rotation-minimizing frames (RRMF curves) are reviewed, and placed in the context of the current state-of-the-art by highlighting the many significant open problems that remain. The simplest non-trivial RRMF curves are the quintics, characterized by a scalar condition on the angular velocity of the Euler–Rodrigues frame (ERF). Two different classes of RRMF quintics can be identified. The first class of curves may be characterized by quadratic constraints on the quaternion coefficients of the generating polynomials; by the root structure of those polynomials; or by a certain polynomial divisibility condition. The second class has a strictly algebraic characterization, less well-suited to geometrical construction algorithms. The degree 7 RRMF curves offer more shape freedoms than the quintics, but only one of the four possible classes of these curves has been satisfactorily described. Generalizations of the adapted rotation-minimizing frames, for which the angular velocity has no component along the tangent, to directed and osculating frames (with analogous properties relative to the polar and binormal vectors) are also discussed. Finally, a selection of applications for rotation-minimizing frames are briefly reviewed—including construction of swept surfaces, rigid-body motion planning, 5-axis CNC machining, and camera orientation control.

Suggested Citation

  • Farouki, Rida T., 2016. "Rational rotation-minimizing frames—Recent advances and open problems," Applied Mathematics and Computation, Elsevier, vol. 272(P1), pages 80-91.
  • Handle: RePEc:eee:apmaco:v:272:y:2016:i:p1:p:80-91
    DOI: 10.1016/j.amc.2015.04.122
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300315005937
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.amc.2015.04.122?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Krajnc, Marjeta & Vitrih, Vito, 2012. "Motion design with Euler–Rodrigues frames of quintic Pythagorean-hodograph curves," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(9), pages 1696-1711.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.

      Corrections

      All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:apmaco:v:272:y:2016:i:p1:p:80-91. See general information about how to correct material in RePEc.

      If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

      If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

      If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

      For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

      Please note that corrections may take a couple of weeks to filter through the various RePEc services.

      IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.