  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.
