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By F. Monroy-Perez, B. Bonnard, J. P. Gauthier

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12 consider liftings with as few free parameters as possible. ,un-i(t) arbitrary parameters. Then the ui(t) is called first curvature, or simply the curvature, associated with a curve j(t) = g(t)e\, because —7(£) = ui{t). For curves in E 3 , the remaining parameter U2(t) is called the torsion of 7, and is usually denoted by r(i). There are several problems with Serret-Frenet frames. First of all, not all arc-length parameterized curves can be lifted to a Serret-Frenet system. It is easy to see that only curves for which £>7 7(t), ~dT' D n-l ' dtn- T7W are linearly independent at all points along the curve can be Serret-Frenet lifted, as a consequence of the Gramm-Schmidt procedure applied to the above vectors.

Again we draw from the calculations obtained in the Euler-Grifnths case. , Dvi —j— = at 1 ^ > MiVi fi\ *—' and Dvi 1 —j— = — M t v i , at n\ . ,n. Since the curvature is equal to 1 in the interval (to, £i), it follows that the normal vector n(t) is given by n(t) = ^ MM, or H\ti(t) = - J2 MM. MA i=2 j=2 After the calculations analogous to the previous problem, with n replaced by fJ-\, we get that (MAT)2 = YlPivl «=2 hence, (fj,\(t)) r(t) = constant. ~ PAAA = ~Is ' 35 We shall omit the calculations which show that the derivative —r- of the dt binormal vector is in the linear span of the triad t, n, b , since they are the same as before.

The algebraic surface S given by Mi = M 2 = • • • = M„_i = 0 is called the switching surface for this problem. We shall not go into any detailed analysis of the type of crossings of S, except to quote the following theorem 13 : A curve of minimal length for the problem of Dubins is necessarily one of the following: a. A smooth arc with constant curvature equal to 1. b. A geodesic arc. c. A continuously differentiable curve that is a concatenation of geodesic arcs and arcs with curvature equal to 1.

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