Phong Shading Example

One variant of Phong's lighting model is:

                     m                                 n
        I = Ia*Ka + SUM (Ii)*((Kd * N.Li) + Ks * (R.Li) )
                    i=1


	I    = final intensity (the color we want to calculate) : (RGB triple)
	Ia   = ambient light in the scene : (RGB triple)
	Ka   = ambient color of the object : (RGB triple)
	m    = number of lights in the scene : (integer)
	Ii   = intensity of light i : (RGB triple)
	Kd   = diffuse color of the object : (RGB triple)
	Ks   = specular color of the object : (RGB triple)
	N    = surface normal : (Normalized Vector)
	Li   = direction to light i : (Normalized Vector)
	Ks   = specular color of the object : (RGB triple)
	V    = view direction : (Normalized Vector, from the viewer to the object)
	R    = the reflection vector of V about N. R = (-2*N.V)*N + V : (Normalized Vector)
	n    = phong constant of the object : (integer)

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Example:

	MATERIAL PROPERTIES:
	  Ka   = [0.1 0.1 0.1]
	  Kd   = [0.9 0.0 0.0]
	  Ks   = [0.5 0.5 0.4]
	  n    = 10

	SURFACE NORMAL:
	  N    = [0 0 1]  

	VIEW DIRECTION:
	  V    = [0.23570 0.23570 -0.94281]

	AMBIENT LIGHT:
	  Ia   = [1 1 1]

	SCENE LIGHTS: (m=2)
	  LIGHT1
	  L1   = [0.57735 0.57735 0.57735]
	  I1   = [0.5 0.5 0.5]

	  LIGHT2
	  L2   = [0 0 -1]
	  I2   = [0.5 0.5 0.5] 
 
	---------------------------------------------------------------

	Step 1: Add ambient term
	
	  I = Ia * Ka = [1 1 1]*[0.1 0.1 0.1] = [0.1 0.1 0.1]

	---------------------------------------------------------------

	Step 2: Calculate R

	  R = (-2*N.V)*N + V  
	    = (-2 * [0 0 1].[0.23570 0.23570 -0.94281]) * [0 0 1] + [0.23570 0.23570 -0.94281]
	    = (-2 * -0.94281) * [0 0 1] + [0.23570 0.23570 -0.94281]
	    = [0 0 1.88562] + [0.23570 0.23570 -0.94281]
	    = [0.23570 0.23570 0.94281]

	---------------------------------------------------------------

	Step 3: Add contribution from lights

	  LIGHT1:

	  I += (I1)*((Kd * N.L1) + Ks * (R.L1)^n)

	       N.L1      = [0 0 1].[0.57735 0.57735 0.57735] = 0.57735
	       Kd * N.L1 = [0.9 0.0 0.0] * 0.57735 = [0.51961 0.0 0.0]
	       R.L1      = [0.23570 0.23570 0.94281].[0.57735 0.57735 0.57735] = 0.81649
	       (R.L1)^n  = 0.81649 ^ 10 = 0.13167
	       Ks * (R.L1)^n = [0.5 0.5 0.4] * 0.13167 = [0.06583 0.06583 0.05267]

	    += [0.5 0.5 0.5] * ([0.51961 0.0 0.0] + [0.06583 0.06583 0.05267])
	    += [0.5 0.5 0.5] * [0.58544 0.06583 0.05267]
	    += [0.29272 0.03292 0.02634]

	  I  = [0.39272 0.13292 0.12634]


	  LIGHT2:

	  I += (I1)*((Kd * N.L1) + Ks * (R.L1)^n)

	       N.L2 = [0 0 1].[0 0 -1] = -1

	       Since this dot product is negative, this light does not fall on the
	       surface, and thus does not contribute to the final intensity.  Note also
	       that if R.L1 < 0, it should be set to 0.

	---------------------------------------------------------------

	So the final intensity in this example is:

	  I = [0.39272 0.13292 0.12634]