Article

Helices through 3 or 4 pointsΦ

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Abstract

How many points in space are needed to define a circular helix? We show here that given 3 distinct points in space there exist continuous families of helices passing through these points. Given 4 generic distinct points there is no helix. However, a discrete family of helices through 3 points can be specified if an additional property of the helix is prescribed. In particular, the case where the helical radius is specified is studied in detail.

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... for t ∈ (0, 2π), where r is the helix radius and 2πc is the separation between helix loops. In this case, the number of points required for the random sampling is n = 3, but the model might also need to constraint the radius with a cylinder [29]. ...
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  • M J Keil
  • J Rodriguez
M. J. Keil and J. Rodriguez. Methods for generating compound spring element curves. J. Geometry and Graphics, 3:67-76, 1999.
  • N Chouaieb
  • A Goriely
  • J H Maddocks
N. Chouaieb, A. Goriely, and J. H. Maddocks. Helices. Proc. Nat. Acad. Sci. USA, 103(25):9398-9403, 2006.