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The shortest way to the geodesics of spheres

Mauro Patrão

Abstract

In this paper, we prove, using only elementary geometric arguments and only assuming that the curves are continuous, that the geodesics on a sphere are the minor arcs of the great circles. Our result are valid for any sphere in any inner product space.

The shortest way to the geodesics of spheres

Abstract

In this paper, we prove, using only elementary geometric arguments and only assuming that the curves are continuous, that the geodesics on a sphere are the minor arcs of the great circles. Our result are valid for any sphere in any inner product space.

Paper Structure

This paper contains 3 theorems, 12 equations.

Key Result

Lemma 1

If there exist $a < s < t < b$ such that $\pi_p C(s) = \pi_p C(t)$ or that $\pi_q C(s) = \pi_q C(t)$, then $C$ is not a geodesic between $p$ and $q$.

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof