More than just hit detection, we’ve moved into the realm of reflection, lighting and shading. This is mainly about projecting vectors on other vectors, using dot products \(\overrightarrow{v1} \cdot \overrightarrow{v2}\) to measure the length of the projection. For that we need to find normal vectors of surfaces, and a light source.

The point light source is simple enough, just a position and an intensity.

\[light = (\dot{p}, i_{rgb})\]

The normal at a point of a unit sphere is also simple, it’s the same as the vector from the center to the point on the sphere.

\[\overrightarrow{v}_{normal} = \dot{p}_{sphere} - \dot{p}_{origin}\]

That’s in object space. The actual calculation in world space is a bit more convoluted, because we have transforms attached to the unit spheres, so a few inverse transforms are needed to counter the deformation.

def normal_at(world_point)
  object_point = @inverse * world_point
  object_normal = object_point - Point::ZERO
  world_normal = @inverse_transpose * object_normal

With that in place we can calculate the shade where the ray hits a sphere, by using the Phong reflection model.