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In Euclidean geometry, all metric notions (arc length for curves, the firstfundamental form for surfaces, etc.) are derived from the Euclidean innerproduct on tangent vectors, and this inner product is preserved by the fullsymmetry group of Euclidean space (translations, rotations, and reflections).In equiaffine geometry, there is no invariant notion of inner product ontangent vectors that is preserved by the full equiaffine symmetry group.Nevertheless, it is possible to define an invariant notion of arc length fornondegenerate curves, and an invariant first fundamental form for nondegeneratesurfaces in equiaffine space. This leads to two possible notions of arc lengthfor a curve contained in a surface, and these two arc length functions do notnecessarily agree. In this paper we will derive necessary and sufficientconditions under which the two arc length functions do agree, and illustratewith examples.

A Tale of Two Arc Lengths: Metric notions for curves in surfaces in equiaffine space