Line–sphere intersection

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thumb|350px|right|The three possible line-sphere intersections:">
1. No intersection.
2. Point intersection.
3. Two point 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.

In vector notation, the equations are as follows:

Equation for a sphere:$leftVert\; mathbf\{x\}\; -\; mathbf\{c\}\; ightVert^2=r^2$:*$mathbf\{c\}$ - center point:*$r$ - radius

Equation for a line starting at (0,0,0):$mathbf\{x\}=dmathbf\{l\}$:*$d$ - distance along line from starting point:*$mathbf\{l\}$ - direction of line (a unit vector)

Solving for $d$:

:Equations combined::$leftVert\; dmathbf\{l\}-mathbf\{c\}\; ightVert^2=r^2$:Expanded::$d^2mathbf\{l\}^2-2d(mathbf\{l\}cdotmathbf\{c\})+mathbf\{c\}^2=r^2$:Rearranged::$d^2mathbf\{l\}^2-2d(mathbf\{l\}cdotmathbf\{c\})+mathbf\{c\}^2-r^2=0$:Quadratic formula and simplified. (This quadratic equation is an example of Joachimsthal's Equation).::$d=frac\{(mathbf\{l\}cdotmathbf\{c\})\; pm\; sqrt\{(mathbf\{l\}cdotmathbf\{c\})^2-mathbf\{l\}^2(mathbf\{c\}^2-r^2)\{mathbf\{l\}^2\}$:Note that $mathbf\{l\}$ is a unit vector, and thus $mathbf\{l\}^2=1$. Thus, we can simplify this further to::$d=(mathbf\{l\}cdotmathbf\{c\})\; pm\; sqrt\{(mathbf\{l\}cdotmathbf\{c\})^2-mathbf\{c\}^2+r^2\}$

*If the value under the square-root ($(mathbf\{l\}cdotmathbf\{c\})^2-mathbf\{c\}^2+r^2$) 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).

ee also

*Analytic geometry
*Line-plane intersection
*Line of intersection between two planes