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Quaternion to Rotation Matrix

Related Topics: Quaternion, OpenGL Rotation About Arbitrary Axis
Download: quaternion.zip

Multiplying Quaternions implies a rotation of a vector in 3D space and it is commonly used in 3D computer graphics algorithms because it is simpler and cheaper than the matrix multiplication. This page is focused on how to convert a Quaternion rotation to a 4x4 rotation matrix form using Quaternion algebra. Note that OpenGL accepts only 4x4 matrix to transform vertices.
4x4 rotation matrix from Quaternion

Overview

Diagram of quaternion rotation
qpq* means rotating P to P' along R axis by θ

Suppose there is a 3D vector, a vector to rotate and rotate it along an arbitrary axis (unit vector) rotating axis by angle rotation angle.

The vector vector p can be converted as Quaternion form;
(We only use x, y and z components for vector p.) quaternion p

And, the Quaternion representation for the rotation axis vector vector r and the half rotation angle rotation angle can be written as;
quaternion q
(Note that we only uses half angle to define the rotation quaternion q because we are going to multiply q twice.)

To rotate the quaternion p, we simply multiply the quaternion q. However, the length of p is changed during multiplication, so we multiply q* (conjugate of q) again at the back of p in order to cancel out the length changes. Since we multiply q twice, front and back, we only need a half rotation angle for q. Therefore, the final equation for the rotation in Quaternion becomes rotation in Quaternion.

This special double multiplication is called "conjugation by q". Here explains why this double multiplication is required and the proof of the length conservation.

Quaternion to Matrix

In this section, we calculate rotation in Quaternion from arbitrary Quaternions, p and q. Let's say p and q are defined as below, and q is normalized (unit quaternion);

quaternion p

quaternion p

For a typical usage, rotating a 3D vector, we can set the scalar component to 0 for p. It makes the multiplication simpler, but here we use a generic 4-component quaternion for p.

First we multiply p by q (applying q on p);
quaternion qp

The above equation is simplified by Quaternion quadrantal (perpendicular) vectors property.
quaternion quadrantal vectors

Let's set each component of quaternion qp to quaternion s, quaternion x, quaternion y and quaternion z respectively.
quaternion components

Then, multiply the conjugate of q at the back of the previous multiplication.
qpq*

Since the equation becomes too long, we compute each term and component independently and combine them later. This page provides the complete computation of qpq*. You may skip to the final result of qpq* below.

quaternion terms

quaternion terms

quaternion terms

quaternion terms

The scalar component of qpq*;
quaternion s

The x component of qpq*;
quaternion x

The y component of qpq*;
quaternion y

The z component of qpq*;
quaternion z

Finally, combine s, x, y and z components into the rotation quaternion, qpq*;
quaternion qpq*

Since q is a unit quaternion, substitute the followings to the above equation to simplify it a little further.
unit quaternion unit quaternion

The final simplified rotation quaternion qpq* becomes;
unit quaternion

Now, we only take the x, y and z compoments (without i, j and k), and convert it to a matrix form. It becomes multiplying a 3x3 matrix to a 3D vector vector p to transform.
matrix form

The 3x3 matrix itself is the rotation matrix equivalent to the quaternion rotation;
3x3 rotation matrix

Or, as 4x4 matrix;
4x4 rotation matrix

Example: Rotation with Quaternion

example example

Download: quaternion.zip

This example provides C++ Quaternion class and tests rotating a vertex, (1, 2, 3) along a rotation axis, (0.57735, 0.57735, 0.57735) by 45 degree. We need to convert the vertex and rotation axis to quaternion forms first, then rotate it with quaternion multiplication, rotation in Quaternion. The Quaternion class provides getMatrix() to convert the rotation quaternion to 4x4 matrix. It verifies the rotation result by comparing with rotation matrix.


Vector3 v(1, 2, 3);                             // 3D vertex to rotate
Vector3 r(0.57735f, 0.57735f, 0.57735f);        // rotation axis (unit vector)
float a = 45.0f;                                // rotation angle in degree

// convert to quaternions
Quaternion p = Quaternion(0, v.x, v.y, v.z);    // quaternion form of v
Quaternion q = Quaternion(r, a * 0.5f * D2R);   // rotation quaternion with half-angle
Quaternion c = q;                               // copy of q
c.conjugate();                                  // q* (conjugate of q)

// rotate p by multiplying qpq*
Quaternion p2 = q * p * c;

// vector part of p2 contains the rotated 3D vertex
Vector3 v2(p2.x, p2.y, p2.z);                   // quaternion to vector
std::cout << "v2: " << v2 << std::endl;         // print the result

// OR, convert quaternion to 4x4 roatation matrix
Matrix4 m = q.getMatrix();
v2 = m * v;                                     // rotation using matrix instead

// OR, use matrix rotation directly
Matrix4 m
m.rotate(a, r);                                 // rotate A degree along R axis
v2 = m * v;                                     // rotation using matrix instead

If there is a sequence of multiple rotations, we can simply multiply the quaternions one after another, similar to matrix transformations. Multiplication of 2 quaternions requires 16 products and 12 additions. On the contrary, multiplying two 4x4 matrices needs 64 products and 48 additions.


// 3 rotations in quaternion form
Quaternion qx = Quaternion(Vector3(1, 0, 0), 22.5f * D2R); // 45 degree about x-axis
Quaternion qy = Quaternion(Vector3(0, 1, 0), 22.5f * D2R); // 45 degree about y-axis
Quaternion qz = Quaternion(Vector3(0, 0, 1), 22.5f * D2R); // 45 degree about z-axis

// compose multiple rotations, order is qz -> qy -> qx
Quaternion q = qx * qy * qz;
std::cout << "q: " << q << std::endl;         // print the result

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