A path across a height field must be solved from the surface's own slope, not given a height of its own

Fact

Laying a route over a surface by choosing its (x, y) shape and its height separately gives you two independent functions of the same parameter. They agree only by accident, and where they disagree the route becomes a causeway on stilts or a trench — in one measured case 110 units above the ground on a 260-unit mountain.

Solve the direction instead. On a surface whose fall-line slope is g, a step that heads a fraction fall of the way into the fall line and spends the rest across the slope has a grade of exactly g · fall. So for a wanted grade G:

fall = min(1, G / g)          # radial/uphill component of the unit step
side = sqrt(1 - fall²)        # across-slope component

Step by a fixed arc length along that direction and read the height from the surface. The route now lies on the surface by construction, at the grade asked for, and its length falls out of the geometry rather than being guessed.

Why it matters

The stilts version looks fine from directly above and from far away, which is where a generated world usually gets checked. It shows up as a wall beside the path, as physics that cannot step onto it, or as a shadow with nothing casting it — none of which point at the route generator.

The solved version also makes the design controllable: grade becomes an input, and switchbacks come from flipping the across-slope sign while fall rises briefly to 1.

How to apply

• Never write the route's height. Read it from the field at the position you stepped to.
• fall is capped by 1, so a path can never be steeper than the surface's own fall line. Clamp it below that too, or a hairpin takes the fall line wherever the ground happens to be steepest and produces a pitch nothing can climb.
• Assert the result: sample the finished field along the route and check the measured grade against the one you asked for.