I've been using this algorithm to improve the results of the DE step search.
bool yams(const Vector& a, const Vector& b, Real* x, Real* e) {
if (m(a)) {
return false; /* ray starts inside the Mandelbulb */
}
/* locate the surface with DE steps */
Real min = 0;
Real max = 0;
bool found = false;
for (int i = 0; i < 128; ++i) {
Real next = min + md(a.add(b.mul(min))) * 0.1;
if (m(a.add(b.mul(next)))) {
max = next;
found = true;
break;
}
min = next;
}
if (!found) { /* unable to locate surface */
*x = min;
*e = 0;
return true;
}
/* solution refinement */
for (int i = 0; i < 128; ++i) {
Real mid = (min + max) * 0.5;
if (m(a.add(b.mul(mid)))) {
max = mid;
} else {
min = mid;
}
}
*x = min;
*e = max - min;
return true;
}
After the surface is found using DE steps, it then refines the solution by bisection/binary search to "back up" to where the ray hits the surface.
It depends on having the boolean version of the Mandelbulb surface functioin,
m(), available.
The function
md() is the common
log|arg| * |arg| / |de| distance metric.
It returns false if the ray does not hit the surface, and true otherwise.
If the intersection is found, then the variable 'x' contains the distance along the line
a + b * x to the intersection, and e contains the error with the intention that
m(a + b * x) == falseand
m(a + b * (x+e)) == truewhere e is as small as possible.
Let me know what you think
And here are some graphs from my analysis:
The white line shows the distance function |arg| as the ray approaches the surface.
The green lines mark the areas of the x-axis that are contained inside the surface.
The blue line marks the solution.
November 28, 2008, 10:27:01 PM
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