Quaternion Function Reference

frenet

Frenet-Serret frames

Syntax

[T, N1, N2, N3, J, K, L, M] = frenet(q)

Description

frenet(q) computes the Frenet-Serret frames of a quaternion sequence or curve.

Given a vector of quaternions (full or pure), representing a curve in 3-space, 4-space or spacetime, this function computes the vectors of tangents, normals, binormals, and trinormals and the curvatures. The output parameters are as follows. The first four results are unit quaternions; the remaining four are scalars. T are the tangents; N1 are the normals; N2 are the binormals; N3 are the trinormals; J is the moduli of the tangents, so that non-unit tangents are J .* T; K is the curvature; again K .* N1 gives un-normalised normals; L is the second curvature or torsion; M is the third curvature or bitorsion.

Note that the first and last values of T are not necessarily meaningful because T is an approximation to the derivative, and therefore cannot be correctly computed for the first and last values in q. The same applies to N1, N2 and N3, but as these are computed using successive derivatives, additional values at the extremities are not necessarily meaningful.

Comments on references: the article by Bharathi and Nagaraj was probably the first to explain the Frenet-Serret frames in 4-space using quaternions, but the Wikipedia article makes clear that Camille Jordan worked out the n-dimensional case in 1874. The article by Kurt Nalty is of value for its clear tutorial explanation of the formulae.

References

  1. K. Bharathi and M. Nagaraj, 'Quaternion Valued Function of a Real Variable Serret-Frenet Formula', Indian J. Pure and Applied Mathematics, 18(6), 507-511, June 1987.
  2. 'Frenet-Serret formulas', subsection: 'Formulas in n dimensions', Wikipedia (English), accessed 27 April 2011. http://en.wikipedia.org/wiki/Frenet%E2%80%93Serret_formulas
  3. Kurt Nalty, 'Trajectories and curves in 3 and 4 dimensions', self-published article, 11pp, 19 January 2007, available at http://kurtnalty.com/.

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