Journal of Geometry

, Volume 106, Issue 2, pp 321–339 | Cite as

Offsets, conchoids and pedal surfaces

  • Martin Peternell
  • Lukas Gotthart
  • Juana Sendra
  • J. Rafael Sendra


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.


Offset surfaces conchoid surfaces pedal surfaces inverse pedal surfaces Darboux and Dupin cyclides 

Mathematics Subject Classification

51M 51N 53A 


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Copyright information

© Springer Basel AG 2014

Authors and Affiliations

  • Martin Peternell
    • 1
  • Lukas Gotthart
    • 1
  • Juana Sendra
    • 2
  • J. Rafael Sendra
    • 3
  1. 1.University of Technology ViennaWienAustria
  2. 2.Dpto. Matemática Aplicada a la I.T. de Telecomunicación CITSEMUniversidad Politécnica de MadridMadridSpain
  3. 3.Dpto. de Fisica y MatemáticasUniversidad de AlcaláAlcalá de HenaresSpain

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