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The ray is parameterized as X = P + t * D where P is a point on the ray, t is a real value and t >= 0, and D is the ray direction which isn't necessarily unit-length. Given a sphere |X - C|^2 = r^2 we want to find the minimum distance of the sphere to the ray X. If the ray does not intersect the sphere, the closest point on the ray to the sphere must be the projection of C onto the ray or the ray starting point P, depending on the position of the ray to the sphere. The sphere center projected onto the ray is Cproj = P + u * D where u = (D * (C - P)) / (D * D) Let the point R be the closest point on the ray to the sphere. R is defined as R = Cproj if u > 0 R = P if u <= 0 Let E = (R - C) / |R - C|, the normalized distance vector from C to R, then Q = C + radius * E is the closest point on the sphere to the ray, and the minimum distance is therefore d = |R - Q| = |R - C| - r The ray intersects the sphere if |R - C| <= r. The points of intersection can be determined as follows. Let s = sqrt(r^2 - |Cproj - C|^2) and D' the normalized direction vector of the ray, then the two potential points of intersection are P1 = Cproj + s * D' P2 = Cproj - s * D' P1 will always be an intersection point. To determine if P2 is actually an intersection point with the sphere, it must be tested if P2 lies on the ray.
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Copyright © by Martin Ecker |