# A car as parabolic geometry

## TL;DR

This paper demonstrates that a car, modeled as a nonholonomic system, exemplifies a flat parabolic geometry, linking it to various classical geometries and their interpretations through the car's kinematics.

## Contribution

It establishes a novel connection between car kinematics and advanced geometric structures, specifically flat parabolic geometries of a certain type.

## Key findings

- Car as an example of flat parabolic geometry
- Connections to circle geometry and conformal Minkowski spacetime
- Interpretation of classical geometries via car kinematics

## Abstract

We show that a car, viewed as a nonholonomic system, provides an example of a flat parabolic geometry of type $({\bf SO}(2,3),P_{12})$, where $P_{12}$ is a Borel parabolic subgroup in ${\bf SO}(2,3)$. We discuss the relations of this geometry of a car with the geometry of circles in the plane (a low dimensional Lie sphere geometry), the geometry of 3-dimensional conformal Minkowski spacetime, the geometry of 3-rd order ODEs, projective contact geometry in three dimensions, and the corresponding twistor fibrations. We indicate how all these classical geometries can be interpreted in terms of the nonholonomic kinematics of a car.