Gauss's principle of least constraint

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 (Eberly 2006:698).

Calculation using vectors in 3D

In vector notation, the equations are as follows:

Equation for a sphere

${\Vert 𝐱-𝐜\Vert }^2=r^2$

• $𝐜$ - center point
• $r$ - radius
• $𝐱$ - points on the sphere

Equation for a line starting at $𝐨$

$𝐱=𝐨+d𝐥$

• $d$ - distance along line from starting point
• $𝐥$ - direction of line (a unit vector)
• $𝐨$ - origin of the line
• $𝐱$ - points on the line

Searching for points that are on the line and on the sphere means combining the equations and solving for $d$:

Equations combined

${\Vert 𝐨+d𝐥-𝐜\Vert }^2=r^2\iff (𝐨+d𝐥-𝐜)\cdot (𝐨+d𝐥-𝐜)=r^2$

Expanded

$d^2(𝐥\cdot 𝐥)+2d(𝐥\cdot (𝐨-𝐜))+(𝐨-𝐜)\cdot (𝐨-𝐜)=r^2$

Rearranged

$d^2(𝐥\cdot 𝐥)+2d(𝐥\cdot (𝐨-𝐜))+(𝐨-𝐜)\cdot (𝐨-𝐜)-r^2=0$

The form of a Quadratic formula is now observable. (This quadratic equation is an example of Joachimsthal's Equation [1].)

$a{d}^2+bd+c=0$

where

• $a=𝐥\cdot 𝐥$
• $b=2(𝐥\cdot (𝐨-𝐜))$
• $c=(𝐨-𝐜)\cdot (𝐨-𝐜)-r^2$

Simplified

$d=\frac{-(𝐥\cdot (𝐨-𝐜))\pm \sqrt{(𝐥\cdot (𝐨-𝐜){)}^2-{𝐥}^2((𝐨-𝐜{)}^2-r^2)}}{{𝐥}^2}$

Note that $𝐥$ is a unit vector, and thus ${𝐥}^2=1$. Thus, we can simplify this further to

$d=-(𝐥\cdot (𝐨-𝐜))\pm \sqrt{(𝐥\cdot (𝐨-𝐜){)}^2-(𝐨-𝐜{)}^2+r^2}$

• If the value under the square-root ($(𝐥\cdot (𝐨-𝐜){)}^2-(𝐨-𝐜{)}^2+r^2$) 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.

See also

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

References

• 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