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Spherical Excess and Reference Twist

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A Primer on the Kinematics of Discrete Elastic Rods

Part of the book series: SpringerBriefs in Applied Sciences and Technology ((BRIEFSTHERMAL))

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Abstract

The method by which a component of the rotation of the cross-section is computed in the discrete elastic rod formulation is exceptional and exploits a phenomenon in differential geometry known as a holonomy. In this chapter, relevant background from differential geometry and spherical geometry are presented so the reader can understand how the reference twist in the rod can be related to a solid angle enclosed by the trace of a unit tangent vector on a sphere. A derivation of the expression for the variation of the reference twist as a function of the variation of the tangent vector is also discussed in detail. Some of the developments in this chapter are illuminated with the help of a simple example of a rod with three vertices that was also considered in a previous chapter.

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Notes

  1. 1.

    As discussed in [42, 44, 48, 68], Kelvin and Tait’s result has been independently rediscovered several times since 1867. The most notable instance lies in a wonderful paper by Goodman and Robinson [15] where it is used to compute drift in navigation estimates.

  2. 2.

    It might be helpful to recall that examples of parallel propagation of Bishop frame vectors were shown earlier in Figs. 2.6 and 2.7 on Page 12 for the cases where c t was a great circle and a circle, respectively. Additional examples of parallel transport of vectors along a curve on a surface can be found in the textbooks on elementary differential geometry (see, e.g., [23, 53]).

  3. 3.

    The referential discrete (integrated) twist \(m^k_{\mathrm{ref}}\) in [3] and [29] is denoted by \( \underline {m}_k\) and \(\hat {\vartheta }^k\), respectively.

  4. 4.

    We have been unable to resolve this sign difference, but based on the example discussed in Sect. 7.6, we believe the sign difference is a typographical error.

  5. 5.

    In the notation of [4], the solid angle f j is denoted by ψ j .

  6. 6.

    Kirsch’s thesis [32, Sect. A.2, Appendix A] also contains a helpful discussion which illuminate remarks in [4, Sect. 6] on the similarities between Eq. (7.48) and a formula for the writhing of a curve proposed by de Vries in [66, Eq. (4)].

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Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, Los Angeles, CA, USA

    M. Khalid Jawed

  2. Department of Mechanical Engineering, University of California, Berkeley, Berkeley, CA, USA

    Alyssa Novelia & Oliver M. O’Reilly

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  1. M. Khalid Jawed
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  2. Alyssa Novelia
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  3. Oliver M. O’Reilly
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Jawed, M.K., Novelia, A., O’Reilly, O.M. (2018). Spherical Excess and Reference Twist. In: A Primer on the Kinematics of Discrete Elastic Rods. SpringerBriefs in Applied Sciences and Technology(). Springer, Cham. https://doi.org/10.1007/978-3-319-76965-3_7

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  • DOI: https://doi.org/10.1007/978-3-319-76965-3_7

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