Raytracing a cone along an arbitrary axis

StackOverflow https://stackoverflow.com/questions/21783235

  •  11-10-2022
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Question

I'm working on a RayTracer and I can't figure out what I'm doing wrong when I try to calculate an intersection with a cone. I have my ray vector and the position of the cone with its axis. I know that compute a cone along a simple axis is easy but I want to do it with an arbitrary axis.

I'm using this link http://mrl.nyu.edu/~dzorin/rend05/lecture2.pdf for the cone equation (page 7-8) and here is my code :

alpha = cone->angle * (PI / 180);
axe.x = 0;
axe.y = 1;
axe.z = 0;

delt_p = vectorize(cone->position, ray.origin);
tmp1.x = ray.vector.x - (dot_product(ray.vector, axe) * axe.x);
tmp1.y = ray.vector.y - (dot_product(ray.vector, axe) * axe.y);
tmp1.z = ray.vector.z - (dot_product(ray.vector, axe) * axe.z);
tmp2.x = (delt_p.x) - (dot_product(delt_p, axe) * axe.x);
tmp2.y = (delt_p.y) - (dot_product(delt_p, axe) * axe.y);
tmp2.z = (delt_p.z) - (dot_product(delt_p, axe) * axe.z);

a = (pow(cos(alpha), 2) * dot_product(tmp1, tmp1)) - (pow(sin(alpha), 2) * dot_product(ray.vector, axe));
b = 2 * ((pow(cos(alpha), 2) * dot_product(tmp1, tmp2)) - (pow(sin(alpha), 2) * dot_product(ray.vector, axe) * dot_product(delt_p, axe)));
c = (pow(cos(alpha), 2) * dot_product(tmp2, tmp2)) - (pow(sin(alpha), 2) * dot_product(delt_p, axe));
delta = pow(b, 2) - (4 * a * c);

if (delta >= 0)
{
    t1 = (((-1) * b) + sqrt(delta)) / (2 * a);
    t2 = (((-1) * b) - sqrt(delta)) / (2 * a);
    t = (t1 < t2 ? t1 : t2);
    return (t);
}

I initialised my axis with the y axis so I can rotate it. Here is what I get : http://i.imgur.com/l3kaavc.png Instead of a cone, I have that paraboloid red shape on the right, and I know that it's almost the same equation as a cone.

Was it helpful?

Solution

You probably need to implement arbitrary transformations on primitives using homogenous matrices, rather than support arbitrary orientation for each primitive.

For example, it's not uncommon for ray tracers to only support cones that have their base on the origin, and that point along the vertical axis. You would then use affine transformations to move the cone to the right place and orientation.

My own ray tracer (which thus far only supports planes, boxes and spheres) has the same problem, and implementation transformation matrices is my next task.

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