• To include a term depending on the distance to the light source, if we denote by
• d = d (light source, Q)
• I = k_aI_a + factor (k_dI_d + k_sI_s)
• where factor = 1/(a+bd + cd²)
• The factor is know as the quadric attenuation term (see chapter 6 in the textbook).

• For 2 vectors A=(x_a, y_a, z_a), B =(x_b, y_b, z_b), by definition, their scalar product is:
• AB = x_ax_b+ y_ay_b+ z_az_b =|A| |B| cos (A, B)
• If |A|=1, |B|=1, we can compute the cosine of the angle between the 2 vectors as:
• cos (A, B) = AB = x_ax_b+ y_ay_b+ z_az_b.

Normals for a Triangle Strip

• The orientation of N depends on the order in which we multiply the vectors.
• For a triangle strip, we must insure that the normals are consistently oriented.
• If P₁, P₂, P₃, and P₄ are 4 points of the triangle strip in sequence, then we compute
• N₁ = P₁P₂ x P₁P₃
• N₂ = P₂P₄ x P₂P₃ = P₁₊₁P₃₊₁ x P₁₊₁P₂₊₁

• For a sphere centered in O, the normal to the surface in the point Q is OQ.
• For a cylinder around the z axis, the normal in the point Q=(x, y, z) is N=(x, y, 0).
• For any parameteric curve C(x(s), y(s), z(s)),
• T = (x'(s), y'(s), z'(s)) is tangent to the curve, and
• N = (x"(s), y"(s), z"(s)) is perpendicular to the curve.

Normal for a Parametric Surface

• For a parameteric surface S(x(s, t), y(s, t), z(s, t)),
• T_s = (¶_s x(s, t), ¶_s y(s, t), ¶_s z(s, t)) is tangent to the surface,
• T_t = (¶_t x(s, t), ¶_t y(s, t), ¶_t z(s, t)) is another tangent to the surface,

Normal for an Implicit Surface

For any surface defined implicitly by F(x, y, z) = c (constant), we can compute the normal to this surface using the gradient of the function F:

For a sphere: F(x, y, z) = x² + y² + z² = 1
The gradient of the function is (2x, 2y, 2z). If we normalize it, we obtain (x, y, z).