Vehicle Trajectory Modelling - ISISS M. Casagrande (Pieve di Soligo)

The text “The Geometry of Parallel Parking” explores a new geometric model inspired by the curved paths vehicles follow while maneuvering, particularly during parallel parking. Starting from the observation that a car’s trajectory can be represented by two quarter-circle arcs of opposite curvature, the authors build a non-Euclidean plane geometry where “distance” between points is defined as the length of such composite arcs rather than straight segments. They prove several theorems about these arcs—showing, for example, that aligned arcs behave additively while non-aligned ones satisfy a triangle-inequality-type relation—thus establishing a metric space with a modified value of π. Further sections define fundamental geometric concepts (segments, angles, triangles) within this framework and demonstrate that areas coincide with Euclidean ones, while perimeters are scaled by a constant factor. Finally, the paper analyzes how this geometry models the motion of a car in parallel parking, providing analytic formulas for trajectory angles and classifying possible “triangles” formed under the new rules.