Polynomial smooth-min functions preserve SDF shape outside the blend zone; exponential versions distort the whole field

Multiple smooth-minimum formulas exist and differ in a property IQ calls rigidity. Polynomial-style smins (the CD family, plus the Circular Geometrical form) preserve the shape of the two SDFs everywhere except in the blending region — outside the blend zone the result equals one of the original SDFs. This means the raymarcher’s step size stays correct everywhere except in the blend region, and the result remains a true (or near-true) SDF. Exponential smin lacks this rigidity: it distorts the field no matter how far apart the shapes are, never recovering the original SDF outside the blend zone. That makes exponential smin less suitable for raymarching when performance and accuracy outside blend regions matter, so IQ recommends the polynomial CD family for most use.

Examples

Quadratic smin: smin(a,b,k) = min(a,b) - h*h*k*0.25 where h = max(k-abs(a-b),0)/k. Outside the blend zone (|a-b|>k) it returns exact min(a,b); an exponential smin would still perturb the field there.

Assessment

Explain what ‘rigidity’ means for a smooth-minimum function, and state which family (polynomial vs. exponential) IQ prefers for raymarching and why.