Abstract
A tutorial on the modern definition and application of moving frames, with a variety of examples and exercises, is given. We show three types of invariants; differential, joint and integral, and the running example is the linear action of SL(2) on smooth surfaces, on sets of points in the plane, and path integrals over curves in the plane. We also give details of moving frames for the group of rotations and translations acting on smooth curves, and on discrete sets of points, in the conformal geometric algebra.
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Notes
- 1.
Editorial note: In this chapter only, the ‘⋅’ does not denote the dot product, but function composition; this can also be used for a function ‘acting on’ its argument.
- 2.
A right action satisfies gâ‹…(hâ‹…z)=(hg)â‹…z. The moving frame theory for right actions is entirely equivalent.
- 3.
Editorial note: To relate to a standard term in robotics, in Cartan’s examples moving frames simplify to ‘the group element that sends the frame of vectors at a point to a reference frame of vectors at the origin’.
- 4.
For any
, if one stacks the tangent vectors to 
at z and the tangent vectors of the orbit at z as columns in a matrix, then the matrix must have \(n=\dim M\) columns and have full rank. The tangent vectors to the orbits can be obtained by differentiating gâ‹…z with respect to the group parameters at g=e, the identity of the group. - 5.
The formulae are fully explained in terms of undergraduate multi-variable calculus in [21], while the Fels and Olver papers use (nontrivial) exterior calculus.
- 6.
We have \(I^{\alpha}_{KJ}=I^{\alpha}_{JK}\) since they are equal to the invariantisation of \(u^{\alpha}_{KJ}=u^{\alpha}_{JK}\) respectively.
, if one stacks the tangent vectors to 
at z and the tangent vectors of the orbit at z as columns in a matrix, then the matrix must have References
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Mansfield, E., Zhao, J. (2011). On the Modern Notion of a Moving Frame. In: Dorst, L., Lasenby, J. (eds) Guide to Geometric Algebra in Practice. Springer, London. https://doi.org/10.1007/978-0-85729-811-9_20
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