Robotics research has found numerous important applications of Riemannian geometry , despite the background mathematical material being strikingly challenging. This material is often remains inaccessible to roboticists, and despite many natural generalizations within the mathematical literature, there remain very few applications of modern highly-relevant geometric generalizations such as spray or Finsler geometry . This paper presents an entire re-derivation of generalized spray and Finsler geometries focusing on building the ideas from familiar concepts in advanced calculus and the calculus of variations. We focus on the pragmatic and calculable results and avoid the use of tensor notation to appeal to a broader audience . We have already used these derivations for important advances in our own work (for instance, geometric fabrics); it is our hope that they will contribute to an increased understanding of these ideas within the robotics community and inspire future research into applications of generalized nonlinear, Finsler, and even Riemannian geometries .
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 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 , 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. We focus on the pragmatic and calculable results , avoiding the use of tensor notation to appeal to a broader audience and emphasizing geometric path consistency over ideas around connections and curvature. It is our hope that they will contribute to an increased understanding generalized nonlinear, and even classical Riemannian, geometry within the robotics community and inspire future research into new applications .
R
, despite the background mathematical material being strikingly challenging. This material is often remains inaccessible to roboticists, and despite
. Despite that, the concept remain challenging to many roboticists because the background material is complex and strikingly foreign. Beyond Riemannian geometry, there are
R
within the mathematical literature, there remain very few applications of modern highly-relevant geometric generalizations such as spray or Finsler geometry
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
R
an entire
a
D
generalized
R
focusing on building
, critical for the development of our recent work on geometric fabrics, which builds
R
and avoid
, avoiding
R
. We have already used these derivations for important advances in our own work (for instance, geometric fabrics); it
and emphasizing geometric path consistency over ideas around connections and curvature. It
R
of these ideas
generalized nonlinear, and even classical Riemannian, geometry
R
applications of generalized nonlinear, Finsler, and even Riemannian geometries
new applications