| Popis: |
We prove that the vertices of a curve γ⊂R n are critical points of the radius of the osculating hypersphere. Using Sturm theory, we give a new proof of the (2k+2)-vertex theorem for convex curves in the Euclidean space R 2k . We obtain a very practical formula to calculate the vertices of a curve in R n . We apply our formula and Sturm theory to calculate the number of vertices of the generalized ellipses in R 2k . Moreover, we explain the relations between vertices of curves in Euclidean n-space, singularities of caustics and Sturm theory (for the fundamental systems of solutions of disconjugate homogeneous linear differential operators L:C ∞(S 1)→C ∞(S 1)). |
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