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The line is parameterized as X = P + t * D where P is a point on the line, t is a real value and D is the line 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 line X. If the line does not intersect the sphere, the closest point on the line to the sphere must be the projection of C onto the line. Cproj = P + u * D where u = (D * (C - P)) / (D * D) Let E = (Cproj - C) / |Cproj - C|, the normalized distance vector from C to Cproj, then Q = C + radius * E is the closest point on the sphere to the line, and the minimum distance is therefore d = |Cproj - Q| = |Cproj - C| - r The line intersects the sphere if |Cproj - 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 line, then the two points of intersection are: P1 = Cproj + s * D' P2 = Cproj - s * D'
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Copyright © by Martin Ecker |