Cone trace analytic solution

ConeTraceAnalytic.txt

Spherical cap cone analytic solution is a 1d problem, since the cone cap sphere slides along the ray. The intersection point to empty space sphere is always on the ray.

S : radius of cone cap sphere at t=1
r(d) : cone cap sphere radius at distance d

r(d) = d*S

p = distance of current SDF sample
SDF(p) = sdf function result at location p
x = distance after conservative step

The equation:
p + SDF(p) = x + r(x)

Simplify:
p + SDF(p) = x + x*S   ; substitute r(d) = d*S
x(1+S) = p + SDF(p)
x = (p + SDF(p)) / (1+S)

Substitute: C = 1 / (1+S)
x = (p + SDF(p)) * C   ; <--- this is the code executed in shader

C is stored to constant buffer and precalculated on CPU.

In the gist, p is used both as the point (center of SDF sample) and the distance from origin to p. The meaning however should be clear from the context.