Line–sphere intersection

In vector notation, the equations are as follows:

Equation for a sphere

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

:*$\backslash mathbf\{x\}$ : points on the sphere

:*$\backslash mathbf\{c\}$ : center point

:*$r$ : radius of the sphere

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

:$\backslash mathbf\{x\}=\backslash mathbf\{o\}\; +\; d\backslash mathbf\{u\}$

:*$\backslash mathbf\{x\}$ : points on the line

:*$\backslash mathbf\{o\}$ : origin of the line

:*$d$ : distance from the origin of the line

:*$\backslash mathbf\{u\}$ : direction of line (a non-zero vector)

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

:Equations combined

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

:Expanded and rearranged:

::$d^2(\backslash mathbf\{u\}\backslash cdot\backslash mathbf\{u\})+2d[\backslash mathbf\{u\}\backslash cdot(\backslash mathbf\{o\}-\backslash mathbf\{c\})]+(\backslash mathbf\{o\}-\backslash mathbf\{c\})\backslash cdot(\backslash mathbf\{o\}-\backslash mathbf\{c\})-r^2=0$

:The form of a quadratic formula is now observable. (This quadratic equation is an instance of Joachimsthal's equation.{{cite web | url=http://mathworld.wolfram.com/JoachimsthalsEquation.html | title=Joachimsthal's Equation }})

::$a\; d^2\; +\; b\; d\; +\; c\; =\; 0$

:where

:*$a=\backslash mathbf\{u\}\backslash cdot\backslash mathbf\{u\}=\backslash left\backslash Vert\backslash mathbf\{u\}\backslash right\backslash Vert^2$

:*$b=2[\backslash mathbf\{u\}\backslash cdot(\backslash mathbf\{o\}-\backslash mathbf\{c\})]$

:*$c=(\backslash mathbf\{o\}-\backslash mathbf\{c\})\backslash cdot(\backslash mathbf\{o\}-\backslash mathbf\{c\})-r^2=\backslash left\backslash Vert\backslash mathbf\{o\}-\backslash mathbf\{c\}\backslash right\backslash Vert^2-r^2$

:Simplified

::$$

d = \frac{-2[\mathbf{u}\cdot(\mathbf{o}-\mathbf{c})] \pm \sqrt{(2[\mathbf{u}\cdot(\mathbf{o}-\mathbf{c})])^2-4\left\Vert\mathbf{u}\right\Vert^2(\left\Vert\mathbf{o}-\mathbf{c}\right\Vert^2-r^2)}}{2 \left\Vert\mathbf{u}\right\Vert^2 }

:Note that in the specific case where $\backslash mathbf\{u\}$ is a unit vector, and thus $\backslash left\backslash Vert\backslash mathbf\{u\}\backslash right\backslash Vert^2=1$, we can simplify this further to (writing $\backslash hat\{\backslash mathbf\{u\}\}$ instead of $\backslash mathbf\{u\}$ to indicate a unit vector):

::$\backslash nabla=[\backslash hat\{\backslash mathbf\{u\}\}\backslash cdot(\backslash mathbf\{o\}-\backslash mathbf\{c\})]^2-(\backslash left\backslash Vert\backslash mathbf\{o\}-\backslash mathbf\{c\}\backslash right\backslash Vert^2-r^2)$

::$d=-[\backslash hat\{\backslash mathbf\{u\}\}\backslash cdot(\backslash mathbf\{o\}-\backslash mathbf\{c\})]\; \backslash pm\; \backslash sqrt\{\backslash nabla\}$

:*If , then it is clear that no solutions exist, i.e. the line does not intersect the sphere (case 1).

:*If $\backslash nabla\; =\; 0$, then exactly one solution exists, i.e. the line just touches the sphere in one point (case 2).

:*If $\backslash nabla\; >\; 0$, two solutions exist, and thus the line touches the sphere in two points (case 3).

• {{Section link|Intersection_(geometry)#A_line_and_a_circle}}
• Analytic geometry
• Line–plane intersection
• Plane–plane intersection
• Plane–sphere intersection

{{Reflist}}

{{DEFAULTSORT:Line-sphere intersection}}

Category:Analytic geometry

Category:Geometric algorithms

Category:Geometric intersection

Category:Spherical geometry