Computing Quaternion Conjugate and Norm in GLSL

This article explains how to represent quaternions in OpenGL Shading Language (GLSL) and provides the mathematical formulas and code implementations for computing their norm and conjugate, specifically focusing on unit quaternions commonly used for 3D rotations.

Quaternion Representation in GLSL

In GLSL, a quaternion \(q = w + xi + yj + zk\) is typically stored using a four-component floating-point vector (vec4). The standard convention assigns the imaginary vector components \((x, y, z)\) to the .xyz swizzle and the real scalar component \(w\) to the .w component:

// q = vec4(x, y, z, w) where xyz is the vector part and w is the scalar part
vec4 q;

Computing the Quaternion Norm

The Euclidean norm (or magnitude) of a quaternion is the square root of the sum of the squares of its components:

\[\|q\| = \sqrt{x^2 + y^2 + z^2 + w^2}\]

In GLSL, this can be calculated directly using the built-in length() function:

float quatNorm(vec4 q) {
    return length(q);
}

Alternatively, the squared norm can be computed efficiently using the dot product:

float quatNormSq(vec4 q) {
    return dot(q, q);
}

For a unit quaternion (representing a valid 3D orientation), the norm is mathematically equal to \(1.0\):

\[\|q\| = 1.0\]

If numerical drift occurs over time, a quaternion can be restored to unit length using the GLSL normalize() function:

vec4 unit_q = normalize(q);

Computing the Quaternion Conjugate

The conjugate of a quaternion \(q = (x, y, z, w)\), denoted as \(q^*\), negates the imaginary vector components while keeping the real scalar component unchanged:

\[q^* = (-x, -y, -z, w)\]

In GLSL, the conjugate is formulated as:

vec4 quatConjugate(vec4 q) {
    return vec4(-q.xyz, q.w);
}

This operation can also be implemented using component-wise multiplication:

vec4 quatConjugate(vec4 q) {
    return q * vec4(-1.0, -1.0, -1.0, 1.0);
}

Unit Quaternion Inverse Relationship

For any general quaternion, the multiplicative inverse is defined as:

\[q^{-1} = \frac{q^*}{\|q\|^2}\]

Because a unit quaternion satisfies \(\|q\| = 1.0\) (and therefore \(\|q\|^2 = 1.0\)), the inverse of a unit quaternion is identical to its conjugate:

\[q^{-1} = q^*\]

In GLSL shader pipelines, reversing a rotation represented by a unit quaternion requires only the conjugate operation vec4(-q.xyz, q.w), avoiding computationally expensive division or square root operations.