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Offsets, conchoids and pedal surfaces

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Abstract

We discuss three geometric constructions and their relations, namely the offset, the conchoid and the pedal construction. The offset surface F d of a given surface F is the set of points at fixed normal distance d of F. The conchoid surface G d of a given surface G is obtained by increasing the radius function by d with respect to a given reference point O. There is a nice relation between offsets and conchoids: The pedal surfaces of a family of offset surfaces are a family of conchoid surfaces. Since this relation is birational, a family of rational offset surfaces corresponds to a family of rational conchoid surfaces and vice versa. We present theoretical principles of this mapping and apply it to ruled surfaces and quadrics. Since these surfaces have rational offsets and conchoids, their pedal and inverse pedal surfaces are new classes of rational conchoid surfaces and rational offset surfaces.

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References

  1. Albano, A., Roggero, M.: Conchoidal transform of two plane curves. Appl. Algebra Eng. Commun. Comput. 21, 309–328 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arrondo, E., Sendra, J., Sendra, J.R.: Parametric generalized offsets to hypersurfaces. J. Symb. Comput. 23, 267–285 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. Dietz, R., Hoschek, J., Jüttler, B.: An algebraic approach to curves and surfaces on the sphere and other quadrics. Comput. Aided Geom. Design 10, 211–229 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  4. Farouki, R.T.: Pythagorean–Hodograph curves. In: Farin, G., Hoschek, J., Kim, M.-S. (eds.) Handbook of Computer Aided Geometric Design. Elsevier, Amsterdam (2002)

  5. Farouki, R.T.: Pythagorean–Hodograph Curves: Algebra and Geometry Inseparable. Springer, Berlin (2008)

  6. Gruber, D., Peternell, M.: Conchoid surfaces of quadrics. J. Symb. Comput. 59, 36–53 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hilton, H.: Plane Algebraic Curves. Clarendon Press, Oxford (1920)

  8. Kerrick, A.H.: The limacon of Pascal as a basis for computed and graphic methods of determining astronomic positions. J. Inst. Navig. 6, 310–316 (1959)

    Article  Google Scholar 

  9. Krasauskas, R., Peternell, M.: Rational offset surfaces and their modeling applications. In: Emiris, I.Z., Sottile, F., Theobald, Th. (eds.) IMA Volume 151: Nonlinear Computational Geometry, pp. 109–136 (2010)

  10. Lin, W., Yu, Z., Yuang, E., Luk, K.: Conchoid of nicomedes and limacon of pascal as electrode of static field and a waveguide of high frecuency wave. PIER 30, 273–284 (2001)

    Article  Google Scholar 

  11. Loria, G.: Spezielle Algebraische und Transzendente Ebene Kurven: Theorie und Geschichte. Teubner, Leipzig (1911)

  12. Peternell, M., Pottmann, H.: A Laguerre geometric approach to rational offsets. Comput. Aided Geom. Des. 15, 223–249 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  13. Peternell, M., Gruber, D., Sendra, J.: Conchoid surfaces of rational ruled surfaces. Comput. Aided Geom. Des. 28, 427–435 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Pottmann, H.: Rational curves and surfaces with rational offsets. Comput. Aided Geom. Des. 12, 175–192 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  15. Pottmann, H., Shi, L., Skopenkov, M.: Darboux cyclides and webs from circles. Comput. Aided Geom. Des. 29, 77–97 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Schicho, J.: Proper parametrization of real tubular surfaces. J. Symb. Comput. 30, 583–593 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Sendra, J.R., Sendra, J.: Algebraic analysis of offsets to hypersurfaces. Math. Z. 234, 697–719 (1999)

    Article  MathSciNet  Google Scholar 

  18. Sendra, J.R., Sendra, J.: An algebraic analysis of conchoids to algebraic curves. Appl. Algebra Eng. Commun. Comput. 19, 285–308 (2008)

    Article  MathSciNet  Google Scholar 

  19. Sendra, J., Sendra, J.R.: Rational parametrization of conchoids to algebraic curves. Appl. Algebra Eng. Commun. Comput. 21, 413–428 (2010)

    Article  MathSciNet  Google Scholar 

  20. Sendra, J.R., Winkler, F., Pérez-Diaz, S.: Rational Algebraic Curves: A Computer Algebra Approach. In: Series Algorithms and Computation in Mathematics, Vol. 22. Springer, Heidelberg (2007)

  21. Szmulowicz, F.: Conchoid of Nicomedes from reflections and refractions in a cone. Am. J. Phys. 64, 467–471 (1996)

    Article  Google Scholar 

  22. Vršek, J., Lávicka, M.: Exploring hypersurfaces with offset-like convolutions. Comput. Aided Geom. Des. 29, 676–690 (2012)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. University of Technology Vienna, 1040, Wien, Austria

    Martin Peternell & Lukas Gotthart

  2. Dpto. Matemática Aplicada a la I.T. de Telecomunicación CITSEM, Universidad Politécnica de Madrid, 28031, Madrid, Spain

    Juana Sendra

  3. Dpto. de Fisica y Matemáticas, Universidad de Alcalá, 28871, Alcalá de Henares, Spain

    J. Rafael Sendra

Authors
  1. Martin Peternell
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  2. Lukas Gotthart
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  3. Juana Sendra
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  4. J. Rafael Sendra
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Correspondence to Martin Peternell.

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Peternell, M., Gotthart, L., Sendra, J. et al. Offsets, conchoids and pedal surfaces. J. Geom. 106, 321–339 (2015). https://doi.org/10.1007/s00022-014-0251-1

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  • DOI: https://doi.org/10.1007/s00022-014-0251-1

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