2. Variations

• If there are several objects, it is necessary to see where the ray hits them all, mathematically, and then take $Q$ to be the first hit (lowest $t$). If the surface at $Q$ is diffusely reflective, you have to trace the ray from $Q$ in the direction of the light source to see if hits another object (so $Q$ would be in a shadow).

• If one or more objects are not convex, then it is possible that n$\cdot$   s could be positive at $Q$ and yet the ray from $Q$ to the light source could hit another part of the same object, so you need to check for that too.

• If the surface of the object is specular-reflective (mirror-like) at $Q$, then it is necessary to bounce the ray off it by using the law of reflection (which involves the ray direction and n). Then follow that ray for possible reflections off other objects, and so on, until you get to a diffusely reflective surface.

• A transparent medium such as glass partly reflects and partly refracts. The proportions of each depend on the angle of incidence. It is necessary to trace both the reflected ray by the method just described and the refracted ray by using Snell's Law, which involves the the ray direction, optical density of the medium, and n. Then keep going with each of the two rays. The refracted ray inside the medium might hit the surface again at an angle that produces total internal reflection, or it might be partly reflected and partly refracted again and then go on to hit other objects, and so on--all this to find the brightness of one pixel!

• If the surfaces are described parametrically instead of being easy objects like spheres defined by relational equations $F(x,y,z)=c$, then finding where the ray hits the surface can be much harder.

• If the lighting source is at a finite point, the method is the same except that s will depend on $Q$. You may wish also to include a factor for the fact that brightness of a light source is inversely proportional to the square of the distance it is away.

• The methods described so far produce strong shadows. Often an ambient light source is added as well, meaning a light source that illuminates all objects uniformly, with no particular direction, the way light bouncing off white walls might. (``Ambient'' means ``surrounding''. In calculus you may have seen Newton's law of cooling, which can be described as telling how the temperature of a hot object approaches the ``ambient temperature'', meaning the air temperature of the surrounding room.)

• If the light source is in color, then you can consider its red, green, and blue components separately and treat each as if they were monochrome.

• The Phong formula can also include any additional constant factors you want, say representing the fact that a diffusely reflecting surface may not be fully reflective. (Some surfaces could be darker than others.)

• If an object has color, in the Phong formula you can introduce an extra factor that depends on color (red, green, or blue) and expresses the property that some colors are reflected more than others.

• Phong's original formula allows a power of n$\cdot$   s, say $($n$\cdot$   s$) ^ r$, where $r$ is a constant $\geq 1$. Because n$\cdot$   s is between 0 and 1, a higher power has the effect of a sharper falloff of brightness for normal directions not pointing near the light source.

• Some books use a setup with the $z$ coordinate axis running the other way, so that the viewpoint is on the negative $z$ axis. Such a coordinate system is left-handed, but the math is the same idea as described in §. Another possibility is to put the viewpoint at the origin and the viewplane at $z = H$; in this case the ray from the viewpoint to $P$ has the easy equation x$= {\frac{\displaystyle t}{\displaystyle P}}$.