Day 5, Assume nothing

Today I Learned that I shouldn’t have assumed the sphere would ever need to change from being a unit sphere, so intersecting gets even simpler:

\[\|\dot{O} + t\overrightarrow{D}\|^2 - 1 = 0\]

def intersect(ray)
  l = ray.origin - Point::ZERO
  d = ray.direction
  a = d.dot(d)
  b = 2 * d.dot(l)
  c = l.dot(l) - 1

  intersections = quadratic(a, b, c).map { |t| Intersection.new(t, self) }
  Intersections.new(*intersections)
end

You might wonder why subtract the zero point from the ray origin. \(x - 0 = x\), right? Wrong! \(\dot{p_1} - \dot{p_0} = \overrightarrow{v_1}\), subtracting points yields a vector, due to that neat trick I mentioned on the 0:th day, and \(v_1 \cdot v_2\) is only valid for vectors, not points.

See, a point has w = 1, and a vector has w = 0

\[\begin{bmatrix} x1 \\ y1 \\ z1 \\ 1 \end{bmatrix} - \begin{bmatrix} x0 \\ y0 \\ z0 \\ 1 \end{bmatrix} = \begin{bmatrix} x1 \\ y1 \\ z1 \\ 0 \end{bmatrix}\]

Like magicth.