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Sphere intersection

We wish to find out when a ray hits a sphere. Our spheres will be defined by the implicit equation of a sphere $$(\mathbf{p - c}) \cdot (\mathbf{p - c}) - r^2 = 0$$

where $\mathbf{p})$ is the a possible position on the sphere, $\mathbf{c})$ is the center of the sphere, and $r$ is the radius.

Given the parametric equation of a ray $$\mathbf{r} = \mathbf{e} + t\mathbf{d}\ $$ we can solve for the intersection by setting the equations equal to one another. Put the ray equation in place of the $\mathbf{p}$ in the sphere equation results in:

$$(\mathbf{e} + t\mathbf{d} - \mathbf{c}) \cdot (\mathbf{e} + t\mathbf{d} - \mathbf{c}) - r^2 = 0;$$

This expands to:

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

This result is solvable using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

where

$$a = \mathbf{d} \cdot \mathbf{d})$$

$$b = 2 \mathbf{d} \cdot (\mathbf{e} - \mathbf{c})$$

$$c = (\mathbf{e} - \mathbf{c}) \cdot (\mathbf{e} - \mathbf{c}) - r^2 = 0$$

We can then solve for t: $$t = \frac{ - \mathbf{d} \cdot (\mathbf{e} - \mathbf{c}) \pm \sqrt{ ( \mathbf{d} \cdot (\mathbf{e} - \mathbf{c}))^2 - (\mathbf{d} \cdot \mathbf{d}) ( (\mathbf{e} - \mathbf{c}) \cdot (\mathbf{e} - \mathbf{c}) - r^2) }}{ (\mathbf{d} \cdot \mathbf{d}) }$$