Ray-Sphere Intersection

Ray-Sphere Intersection

The heart of any ray tracer: does a ray hit an object? For a unit sphere centered at the origin, the math is elegant.

The Discriminant

Substituting the ray equation into the sphere equation x² + y² + z² = 1 gives a quadratic in t:

a*t² + b*t + c = 0

Where:

a = dot(direction, direction)
b = 2 * dot(direction, sphere_to_ray)
c = dot(sphere_to_ray, sphere_to_ray) - 1
sphere_to_ray = ray.origin - point(0,0,0)

The discriminant b² - 4ac tells you everything:

• < 0: ray misses the sphere (no real solutions)
• = 0: ray is tangent (one intersection)
• > 0: ray passes through (two intersections)

int intersectCount(double rox, double roy, double roz,
                   double rdx, double rdy, double rdz) {
    double a = rdx*rdx + rdy*rdy + rdz*rdz;
    double b = 2*(rdx*rox + rdy*roy + rdz*roz);
    double c = rox*rox + roy*roy + roz*roz - 1;
    double disc = b*b - 4*a*c;
    if (disc < 0) return 0;
    return 2;
}

The Intersection Points

If disc >= 0:

t1 = (-b - sqrt(disc)) / (2*a)
t2 = (-b + sqrt(disc)) / (2*a)

The hit is the smallest non-negative t — the closest visible intersection.

Your Task

Implement intersectCount that returns the number of intersections (0 or 2).

Expected output:

2
0