## Abstract

This note provides the details and proofs of the results announced by Levi 1993 Fields Insitute Communications vol 1 pp 133-8. The main result of this note is a geometrical representation of the reconstruction problem for SO(3) in terms of parallel transport. It is, of course, well known that the solution of a linear equation ẋ = Ω(t)x in R^{n} cannot in general be expressed by x(t) = e^{∫0tΩ(τ)dτ}x(0) (false) because the coefficient matrices Ω(t_{1}) and Ω(t_{2}) may fail to commute for t_{1} ≠ t_{2}. Nevertheless, when n = 3 and when Ω(t) is skew-symmetric, i.e. when it lies in the Lie algebra of the group of rigid rotations in R^{3}, the above false formula is almost correct, as we will show here. The main result of this note is a geometrical expression for the matrix solution X(t) of matrix equations on TSO(3) of the form Ẋ = Ω(t)X, Ω^{T} = -Ω (*) where X, Ω are 3 × 3 matrices with real coefficients. The argument relies on a theorem of Poinsot together with some observations on geodesic curvatures of moving curves.

Original language | English (US) |
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Pages (from-to) | 413-419 |

Number of pages | 7 |

Journal | Nonlinearity |

Volume | 9 |

Issue number | 2 |

DOIs | |

State | Published - Dec 1 1996 |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics
- General Physics and Astronomy
- Applied Mathematics