The Simple Geometry Behind Road Construction

In my exploration of road geometry, I delve into the intricacies of creating smooth, procedurally generated roads using a profile-based approach. Much like Bezier splines, these profiles act as control points, allowing us to interpolate and construct the actual road path. The challenge lies in connecting two profiles with smooth, parallel arcs using only lines and circular arcs, avoiding the pitfalls of centerline Bezier splines commonly used in game development.

The geometric problem is simplified by considering two endpoints with direction vectors and finding a circular arc that connects them while being tangent at both points. However, a single arc often isn't sufficient, leading to the need for line extensions and a two-line fillet construction. This involves extending lines from the endpoints, finding intersection points, and using the Tangent–Radius Theorem to ensure smooth transitions.

For more complex transitions, such as when roads shift direction, I introduce the concept of intermediary profiles. By using cubic Hermite splines, we can determine the best position for these profiles, allowing us to split the problem into manageable parts. This approach is particularly useful for creating S-shaped paths, reminiscent of LEGO train tracks.

Special cases arise when continuation lines are parallel or when they don't intersect cleanly. In such scenarios, constraints are built into the road placement tool to simplify the setup, ensuring robust connections. The ultimate goal is to dynamically establish intersections and stitch these road segments into intricate networks, a topic for future exploration.