C481 Computing the Normals

The Angle Between 2 Vectors

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.

Computing the Normal Vector
For a triangle defined by P₁(x₁, y₁, z₁), P₂(x₂, y₂, z₂), and P₃(x₃, y₃, z₃), and i=(1,0,0), j=(0,1,0), k=(0,0,1):

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

Examples of the Normal Vector

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 = (delta _s x(s, t), delta _s y(s, t), delta_s z(s, t)) is tangent to the surface,
T_t = (delta_t x(s, t), delta_t y(s, t), delta_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).