By Jean Laurent
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Extra resources for Approximation et optimisation
This lemma has numerous corollaries. Without enumerating them, we shall merely point out that they show the existence and continuity from the right of a right semi-tangent at each point of a geodesic. A similar statement holds for left semi-tangents. Let S be a point on the convex surface in elliptic space and let 1 be a geodesic extending from S. Let us take the point X close to S on T and join it to S by a line segment. Let s be the distance between the points Sand X along the geodesic, a the spatial distance between these points, and & the angle which is formed by the semi-tangent to y at S with the line segment SX.
Let us plot the osculating Euclidean space at the point P, as in para. 1. Let F be the surface and 1 be a curve on it, corresponding to F and y. Clearly, 1 at Pon F has a geodesic curvature equal to zero. Hence the principal normal of the curve -=[ at P coincides with the normal of the surface 18 I. ELLIPTIC SPACE (vxu) = 0, (;Xv) = 0. Substituting into this and noting that x = e 0 at P, we obtam (YXu) = 0, where v= (x" (vxv) = 0, + x) i is the unit vector of the principal normal of T at P. Since, moreover, xv = 0, the vectors v and ~ either coincide or are oppositely directed.
The vectors x, 't, v are mutually perpendicular. 4. SURFACES IN ELLIPTIC SPACE Let us assume we have the surface F in elliptic space given by the equation x = x(u, v). A tangent vector at the point P of this surface will mean a tangent vector of any curve on the surface originating from P. In particular, xu and Xo are tangent vectors. The plane given by the equation in parametric form ox= x 4- rzxu + ~Xti (parameteIS a, ~), is the tangent plane of the surface, which is easily proved by the argument given for the tangent curve in para.
Approximation et optimisation by Jean Laurent