Calculations for a ray–sphere intersection using algebra and the quadratic equation. One application of this is the calculation of ray intersections in ray tracing.

Equation for a sphere

$(x-s\_x)^2+(y-s\_y)^2+(z-s\_z)^2=s\_r^2$

• $s\_?$ - center point and radius

Equation for a line starting at (0,0,0)

$x=d*l\_x$

$y=d*l\_y$

$z=d*l\_z$

• $d$ - distance along line from starting point
• $l\_?$ - direction of line

Solving for $d$:

1. Equations combined: $(d*l\_x-s\_x)^2+(d*l\_y-s\_y)^2+(d*l\_z-s\_z)^2=s\_r^2$
2. Expanded: $d^2l\_x^2-2dl\_xs\_x+s\_x^2+d^2l\_y^2-2dl\_ys\_y+s\_y^2+d^2l\_z^2-2dl\_zs\_z+s\_z^2=s\_r^2$
3. Factored: $d^2(l\_x^2+l\_y^2+l\_z^2)+d(-2l\_xs\_x-2l\_ys\_y-2l\_zs\_z)+s\_x^2+s\_y^2+s\_z^2-s\_r^2=0$
4. Put into quadratic formula and simplified: $d=\backslash frac\{l\_xs\_x+l\_ys\_y+l\_zs\_z\; \backslash pm\; \backslash sqrt\; \{(l\_xs\_x+l\_ys\_y+l\_zs\_z)^2-(l\_x^2+l\_y^2+l\_z^2)(s\_x^2+s\_y^2+s\_z^2-s\_r^2)\}\}\{l\_x^2+l\_y^2+l\_z^2\}$

If only the fact of if the ray and sphere intersect is needed:

$(l\_xs\_x+l\_ys\_y+l\_zs\_z)^2\; \backslash ge\; (l\_x^2+l\_y^2+l\_z^2)(s\_x^2+s\_y^2+s\_z^2-s\_r^2)$

The above is true if the line and sphere intersect.