Before revision:  This paper presents a re-derivation of spray and Finsler geometries , critical for the development of our recent work on geometric fabrics, which builds the ideas from familiar concepts in advanced calculus and the calculus of variations  .
After revision:  This paper presents a re-derivation of spray and Finsler geometries , 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 .