The content shown in this window is the results of computing whether the selected point on the board is in the shadow of the selected object. As noted from the discussion page, we use the parameteric equations of the ray (line) drawn from the selected point and in the direction of the light. In the case of the sphere, we place the parametric equations into the equation of the sphere and test to see if the resulting quadratic equation in the parameter t has a solution. We make our decision based on the discriminant obtained through quadratic formula.
The methods for testing in the case of the other two geometries is covered in the discussion page.