Robotics research has found numerous important applications of Riemannian geometry. Despite that, the concept remain challenging to many roboticists because the background material is complex and strikingly foreign. Beyond {\em Riemannian} geometry, there are many natural generalizations in the mathematical literature -- areas such as Finsler geometry and spray geometry -- but those generalizations are largely inaccessible, and as a result there remain few applications within robotics. This paper presents a re-derivation of spray and Finsler geometries we found critical for the development of our recent work on a powerful behavioral design tool we call geometric fabrics. These derivations build from basic tools in advanced calculus and the calculus of variations making them more accessible to a robotics audience than standard presentations. We focus on the pragmatic and calculable results, avoiding the use of tensor notation to appeal to a broader audience, emphasizing geometric path consistency over ideas around connections and curvature. We hope that these derivations will contribute to an increased understanding of generalized nonlinear, and even classical Riemannian, geometry within the robotics community and inspire future research into new applications.
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