Line–sphere intersection

In analytic geometry, a line and a sphere can intersect in three ways: no intersection at all, at exactly one point, or in two points. Methods for distinguishing these cases, and determining equations for the points in the latter cases, are useful in a number of circumstances. For example, this is a common calculation to perform during ray tracing (Eberly 2006:698).

Calculation using vectors in 3D

In vector notation, the equations are as follows:

Equation for a sphere

  • - center point
  • - radius
  • - points on the sphere

Equation for a line starting at

  • - distance along line from starting point
  • - direction of line (a unit vector)
  • - origin of the line
  • - points on the line

Searching for points that are on the line and on the sphere means combining the equations and solving for , involving the dot product of vectors:

Equations combined
The form of a quadratic formula is now observable. (This quadratic equation is an example of Joachimsthal's Equation .)
Note that is a unit vector, and thus . Thus, we can simplify this further to
  • If the value under the square-root ( ) is less than zero, then it is clear that no solutions exist, i.e. the line does not intersect the sphere (case 1).
  • If it is zero, then exactly one solution exists, i.e. the line just touches the sphere in one point (case 2).
  • If it is greater than zero, two solutions exist, and thus the line touches the sphere in two points (case 3).

See also


  • David H. Eberly (2006), 3D game engine design: a practical approach to real-time computer graphics, 2nd edition, Morgan Kaufmann. ISBN 0-12-229063-1
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