The Flat Earth Error: Differential Geometry in Vehicle Dynamics
The widespread idealization of road surfaces as horizontal planes can introduce significant inaccuracies into vehicle dynamics simulations, a phenomenon termed the ``Flat Earth Error.'' This article provides an expository guide to the use of classical differential geometry to model vehicular motion on curved surfaces. The influence of road curvature is characterized using the metric tensor, the second fundamental form, the shape operator, and the Christoffel symbols. Reproducible MATLAB scripts using an elliptic cone as a benchmark example illustrate the construction of geodesic curves and the simulation of particle dynamics on curved surfaces. The resulting geometric structures are integrated into a single-track vehicle-and-track model within a pseudospectral optimal control framework. Trajectory optimization results over a high-density mobile LiDAR profile of Darlington Raceway are used to generate a high-fidelity road-surface model. This model is used within an hp-adaptive collocation framework to investigate minimum lap time optimal control vehicular trajectories. These computations capture the non-smooth traction saturation limits of the tyres alongside position-dependent variations in gravitational forcing. Integrating differential geometry, multibody mechanics, and optimal control is essential for high-fidelity driven and autonomous vehicle-dynamics simulations. The optimized velocity and tyre slip profiles show that real-world racing track geometries induce dynamically significant three-dimensional effects. These results are particularly relevant to performance-limited simulations on highly banked track surfaces such as NASCAR ovals.