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 geometric fabrics, which builds the ideas from familiar concepts in advanced calculus and the calculus of variations making them more accessible to a robotics audience than standard presentations .