Apparent contour

Let ${\displaystyle \mathbf {c} }$ be a point light source, ${\displaystyle S}$ a surface with unit normals ${\displaystyle \mathbf {n} }$, and ${\displaystyle P}$ a unit sphere centred on ${\displaystyle \mathbf {c} }$. Define two sets:

$${\displaystyle {\begin{aligned}\Gamma &=\{\mathbf {r} \in S:(\mathbf {r} -\mathbf {c} )\cdot \mathbf {n} =0\}\\\gamma &=\{\mathbf {p} :p=(\mathbf {r} -\mathbf {c} )/\|\mathbf {r} -\mathbf {c} \|,\mathbf {r} \in \Gamma \}\end{aligned}}}$$

${\displaystyle \Gamma }$ is called the contour generator, and ${\displaystyle \gamma }$ the apparent contour. The definition can be generalised to project onto a plane, or have parallel light rays.

The apparent contour can be considered as the envelope of projections of a family of curves on the surface.

The apparent contour can have cusps when the ray has higher order contact with the surface. This occurs when the ray is in asymptotic direction in a hyperbolic region of the surface. Higher singularities occur when the ray is tangent to a parabolic line or flat umbilic on the surface. Only a limited ^[4]