B.1.1 Spherical Coordinates

Given a point specified in Cartesian coordinates (x, y, z)^T, the spherical coordinates (ρ, θ, φ) consisting of a length and two angles are

Given length ρ, azimuth angle φ, and elevation angle θ (assuming y is up) the Cartesian coordinates of the corresponding unit vector are

For unit vectors, ρ is equal to 1.

B.1.2 Linear Interpolation of 3D Vectors

Each vector component is interpolated separately, and then the resulting interpolated vector is renormalized. Renormalization is required if the interpolated vectors need to have unit length. Linearly interpolating a vector V using a single parameter α along a line with vectors A and B at the endpoints:

Linearly interpolating a vector V using two barycentric parameters α and β over a triangle with vectors A, B, C at the vertices:

Note: When using linear vector interpolation care must be taken to handle or report the case where one or more vector components interpolate to zero or a number so small that an excessive amount of accuracy is lost in the representation. In cases where linear interpolation is inadequate, spherical interpolation (SLERP) using quaternions may be a preferable method.

B.1.3 Barycentric Coordinates

Given a triangle with vertices A, B, C, the barycentric coordinates of the point P inside triangle ABC are α, β, γ, with P = αA + βB + γC and α + β + γ = 1. Let area (XYZ) be the area of the triangle with vertices X, Y, Z. Then

and the barycentric coordinates are the ratios of the areas of the subtriangles formed by the interior point P to the entire triangle ABC:

B.2.1 Orthographic Projection

The call glOrtho(l, r, b, t, u, f) generates R, where

R is defined as long as l ≠ r, t ≠ b, and n ≠ f.

B.2.2 Perspective Projection

The call glFrustum(l, r, b, t, n, f) generates R, where

R is defined as long as l ≠ r, t ≠ b, and n ≠ f.

B.2.3 Perspective z-Coordinate Transformations

The z value in eye coordinates, z_eye, can be computed from the window coordinate z value, z_window, using the near and far plane values, near and far, from the glFrustum command and the viewport near and far values, far_vp and near_vp, from the glDepthRange command using the equation

The z-window coordinate is computed from the eye coordinate z using the equation

B.2.4 Alternative Perspective Projection

The call gluPerspective(fov_y, ar, n, f) generates R, where

where . R is defined as long as ar ≠ 0, sin(fov_y) ≠ 0, and n ≠ f.

B.4.1 Scaling

The call glScalef(s_x, s_y, s_z) generates S, where

B.4.2 Translation

The call glTranslatef(t_x, t_y, t_z) generates T, where

B.4.3 Rotation

The call glRotatef(θ, V_x, V_y, V_z) generates R, where

and

e_m, a_m, d_m, and s_m are the material emissive, ambient, diffuse, and specular reflectances

a_sc is the scene ambient intensity

, and are the ambient, diffuse, and specular intensities for light source i

sh_i is the specular exponent for light source i

att_i is the distance attenuation for light source i

d_i is the distance between the surface point and light source i

is the constant attenuation for light source i

is the linear attenuation for light source i

is the quadratic attenuation for light source i

sd_i is the spotlight direction unit vector for light source i

se_i is the spotlight exponent for light source i

co_i is the spotlight cutoff angle for light source i

B.8.1 Taylor Series Expansion

The Taylor series expansion of a function f(x) about the point a is

where f^(j) denotes the jth derivative of f(x). The Maclaurin series expansion is the special case of the Taylor series expansion about the point a = 0.

B.8.2 Newton-Raphson Method

The Newton-Raphson method for obtaining a root of the function f(x) uses an initial estimate x₀ and the recurrence

(B.1)

The approximation for the reciprocal of a number a is the root of the equation f(x) = 1/x−a. Substituting f(x) in Equation B.1 gives

and the reciprocal square root of a number a is the root of the equation f(x) = 1/x² − a. Substituting f(x) in Equation B.1 gives

B.8.3 Hypotenuse

A rough approximation of the 2D hypotenuse or length function suitable for level-of-detail computations (Wu, 1988), is