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

$\left\Vert \mathbf {x} -\mathbf {c} \right\Vert ^{2}=r^{2}$

• $\mathbf {c}$ - center point
• $r$ - radius
• $\mathbf {x}$ - points on the sphere

Equation for a line starting at $\mathbf {o}$

$\mathbf {x} =\mathbf {o} +d\mathbf {l}$

• $d$ - distance along line from starting point
• $\mathbf {l}$ - direction of line (a unit vector)
• $\mathbf {o}$ - origin of the line
• $\mathbf {x}$ - 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

$\left\Vert \mathbf {o} +d\mathbf {l} -\mathbf {c} \right\Vert ^{2}=r^{2}\Leftrightarrow (\mathbf {o} +d\mathbf {l} -\mathbf {c} )\cdot (\mathbf {o} +d\mathbf {l} -\mathbf {c} )=r^{2}$

Expanded

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

Rearranged

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

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

$ad^{2}+bd+c=0$

where

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

Simplified

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

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

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

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

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