A raised-cosine contour is flat at its own rim, so contours sum into a height field with no creases

Fact

For a radial bump of amplitude A and radius R,

h(d) = A * 0.5 * (1 + cos(pi * d / R))     for d < R, else 0

is zero at d = R and has zero derivative there. So it drops into the surrounding ground with matching value and matching slope, and any number of these can be added together in any arrangement without a crease at any rim.

The obvious alternatives do not have that property. A linear cone A * (1 - d/R) matches in value but arrives at the rim with a finite slope, leaving a visible ring and — in a simulation — a step change in force. A plateau matches in neither and is a cliff.

Steepest slope is A * pi / (2R), at d = R/2. That one number is the thing to constrain, not the amplitude.

Why it matters

A height field is usually assembled from primitives, and "the pieces line up" is naturally checked on value alone, because that is what is visible in a still frame. Slope discontinuities show up later and elsewhere: as a shading seam that looks like a normals bug, and as an impulse that flicks a moving body sideways for no reason the code can be seen to contain.

The bigger payoff is structural. If the surface is smooth everywhere, a body can be glued to it — position on the plane, height read from the field — and there is no edge to leave. That removes airborne physics, landing detection and the whole class of bugs where something ends up under the terrain.

How to apply

• Gradient is radial, magnitude -A * 0.5 * (pi/R) * sin(pi * d / R); split it by (dx/d, dy/d), and use zero at d = 0 rather than dividing by it.
• With a body glued to the surface, downhill acceleration is -g * gradient. Assert g * steepest_slope < friction_deceleration, or a body can accelerate downhill for ever.
• Contours add, so the combined gradient can exceed every individual one. Measure the assembled field over a grid; do not trust the per-primitive number.
• Check that goal positions — a cup, a spawn, a resting place — sit where the gradient is near zero, or nothing will settle there.