Plane–sphere intersection

Calculating a plane–sphere intersection using algebra. One application of this is the calculation of ray intersections in ray tracing.

Equation for a plane

$xp_{a}+yp_{b}+zp_{c}+p_{d}=0$

• $p_{?}$ - plane

Equation for a line

$x=s_{x}+dv_{x}$

$y=s_{y}+dv_{y}$

$z=s_{z}+dv_{z}$

• $d$ - distance along line from starting point
• $s_{?}$ - starting point of line
• $v_{?}$ - direction of line

Solving for $d$

Equations combined

$(s_{x}+dv_{x})p_{a}+(s_{y}+dv_{y})p_{b}+(s_{z}+dv_{z})p_{c}+p_{d}=0$

Expanded

$s_{x}p_{a}+dv_{x}p_{a}+s_{y}p_{b}+dv_{y}p_{b}+s_{z}p_{c}+dv_{z}p_{c}+p_{d}=0$

Factored

$d(v_{x}p_{a}+v_{y}p_{b}+v_{z}p_{c})+(s_{x}p_{a}+s_{y}p_{b}+s_{z}p_{c}+p_{d})=0$

Manipulated

$d(v_{x}p_{a}+v_{y}p_{b}+v_{z}p_{c})=-s_{x}p_{a}-s_{y}p_{b}-s_{z}p_{c}-p_{d}$

$d={\frac {-s_{x}p_{a}-s_{y}p_{b}-s_{z}p_{c}-p_{d}}{v_{x}p_{a}+v_{y}p_{b}+v_{z}p_{c}}}$