The Simple Geometry Behind Everything on the Road

This summarizes a geometric method that represents roads based on profiles, and connects between them using only lines and circular arcs.

A profile is information like a snapshot of a road cross-section, and the actual road shape is created by interpolating between these profiles. The core problem is connecting the endpoints of two profiles with different positions and directions into a smooth, parallel path.

Basic Construction

• Use two endpoints A, B and the direction vector at each endpoint.
• Draw a continuation line at each endpoint in the direction the road continues.
• Let the point where the two continuation lines meet be C.
• If one side is longer, place an auxiliary point M on the longer one, matching the length of the shorter side.
• Draw lines perpendicular to the continuation lines at A and M respectively, and find their intersection point O.
• By the Tangent-Radius Theorem, the circle centered at O is tangent to those lines.

With this construction, the final path becomes an arc from A to M + a straight line from M to B. This is effectively the same construction as what's called a two-line fillet in CAD.

Connecting Profiles

• If the profile lengths are equal and the conditions match, the same geometric construction can be applied at both ends to connect both boundaries simultaneously.
• This naturally keeps both edges of the road parallel.

The S-curve Transition Problem

Not all cases are solved with a single arc. If the continuation lines don't meet in the desired way, the road needs an S-curve that bends one way and then bends back the other way.

For this, an intermediary profile is placed in the middle.

• Candidate positions and directions are determined using a cubic Hermite spline.
• Ideally, it would be correct to find the inflection point where curvature changes, but in actual implementation there's no need to go as far as solving complex cubic equations.
• When the tangent magnitude relative to the distance between center points is roughly 1.0 to 1.5 times, using P(0.5) and P'(0.5) gives a sufficiently good approximation.
• P(0.5) gives the position of the intermediate profile, and P'(0.5) gives the tangent direction at that point.

Special Cases

• If the continuation lines are parallel and in the same direction, this can be handled with a semicircular arc + a straight line.
• Beyond this, there are cases that aren't cleanly resolved with a single intermediate profile.

Design Constraints

Rather than solving every impossible arrangement with a general solution, constraints are built into the tool itself.

• The infinite straight line of the source profile divides the plane in half.
• If a user tries to place the next profile on the half-plane behind it, the new profile's direction is forced to match the source profile.
• This transforms complex cases into simpler special cases that can be connected stably.

Finally, this approach creates the basic building blocks of a road network. The next step is to dynamically create intersections where roads meet, and assemble these blocks into more complex networks.