Specular reflection is when the reflection is stronger in one viewing direction, i.e., there is a bright spot, called a specular highlight. This is readily apparent on shiny surfaces. For an ideal reflector, such as a mirror, the angle of incidence equals the angle of specular reflection, as shown below.

So if ** R** is the direction of specular
reflection and ** V** is the direction of the viewer (located at the
View Reference Point or **VRP**), then for an ideal reflector the
specular reflection is visible only when ** V** and ** R** coincide. For
real objects (not perfect reflectors) the specular reflectance
can be seen even if ** V** and ** R** don't coincide, i.e., it is visible
over a range of a values (or a cone of values). The shinier the
surface, the smaller the f range for specular visibility. So a
specular reflectance model must have maximum intensity at **R**, with
an intensity which decreases as f(a).

This is an empirical model, which is not
based on physics, but physical observation. Phong observed that
for very shiny surfaces the specular highlight was small and the
intensity fell off rapidly, while for duller surfaces it was
larger and fell off more slowly. He decided to let the reflected
intensity be a function of (cos a)^{n} with n >= 200 for a shiny surface
and n small for a dull surface. For a perfect reflector n equals
infinity, and for a piece of cardboard n equals 0 or 1. In the
diagram below we can see how the function (cos a)^{n} behaves
for different values of n.

Specular reflection is also a function of the light incidence angle q. An example is glass which has almost no specular reflectance for q = 0 degrees but a very high specular reflectance for q > 80 degrees. Some substances, such as copper, actually change color with change in the incidence angle, as shown in the following plot of the reflectance curve as a function of the incident angle for copper. .

A full specular reflectance function is the
Bi-directional Reflectance Distribution Function (BRDF). For
glass the BRDF at 0 degrees incidence equals 0 and for light
incident at 90 degrees, it equals 1. Since for many materials the
BRDF is approximately constant, Phong called this term the
specular coefficient (ks) and assumed it was constant. Then,
since cos a = **V·R**, a complete illumination
intensity model for reflection including diffuse reflection from
ambient light and a point light source, and the Phong model for
specular reflection is:

**I = ka * Ia + (Ip /(d)) [kd * (N·L)
+ ks * (V·R) ^{n}]**

For color there will be versions of the
above equation for Red, Green, and Blue components. The
coefficient of specular reflection **ks** is usually not the same as
the coefficient of diffuse reflection **kd** or the ambient reflection **ka**. The assumption is often
made that the specular highlights are determined by the color of
the light source, not the material, e.g., ** ks ^{R}** =

If the point light source is far from the
surface then **N·L **is constant across a planar
surface, e.g., across one planar polygon. Similarly if the VRP is
far from the surface then **V·R** is constant
across the surface.