Recall that we use the convention that \(\vec{\ell}\) points towards the infinitely distant directional light source, while a point light source is specified by the position of the source \(\vec{s}\).
Given a ray (\(\mathbf{p} + t \mathbf{d}\)) and the t at which it intersects a surface, calculate a unit vector giving the direction from the surface towards:
point light source at position \(\vec{s}\)
a directional light source with direction \(\vec{\ell}\)
Suppose, in 2D, a unit-length surface has its normal pointing directly towards a directional light source (i.e., \(\vec{n} = \vec{\ell}\)). In this configuration, suppose the “brightness” of the surface is 1 - that is, 1 unit of light was delivered to 1 unit of surface. Now suppose the normal \(\vec{n}\) makes an angle \(\theta\) with the light direction \(\vec{\ell}\) (assume \(-90 \le \theta < 90\)) . In terms of \(\theta\), how much brightness is delivered to the unit-length surface now?
A scene contains a unit sphere centered at the origin. A directional light with direction \(\vec{\ell}\) shines on the scene.
Give the coordinates of the brightest point on the sphere.
Describe (implicitly, i.e., by specifying a constraint satisfied by) the set of points on the sphere where the amount of light hitting the sphere goes from nonzero to zero.