Quaternion

A rotation represented as a four component vector modeled after the XNA Game Studio 4 (Microsoft.Xna.Framework.Quaternion) math library.

A quaternion is a very efficient and compact method for working with 3D rotation. A quaternion is a 4-dimensional value and only has physical meaning when it's normalized. In computer graphics, they are used to represent 3D rotations as a 4-vector (i.e. 4 float values) instead of requiring a 3x3 matrix (i.e. 9 float values). They are extremely useful implementing cameras and animation where a quaternion can smoothly interpolate between 3D rotations while avoiding the gimbal lock problem common to Euler angles. They are also more compact the the 9 elements required for a 3D rotation matrix.

Header

#include <SimpleMath.h>

Initialization

using namespace DirectX::SimpleMath;

Quaternion q;                     // Creates the identity quaternion [0, 0, 0, 1]
Quaternion q(0, 0, 0, 1);         // Creates a quaternion [0, 0, 0, 1]
Quaternion q( Vector3(0,0,0), 1); // Creates a quaternion [0, 0, 0, 1]
Quaternion q( Vector4(0,0,0,1) ); // Creates a quaternion [0, 0, 0, 1]

float arr[4] = { 0, 0, 0, 1 };
Quaternion q(arr);                // Creates a quaternion [0, 0, 0, 1]

Fields

x vector component of the quaternion
y vector component of the quaternion
z vector component of the quaternion
w scalar component of the quaternion

Methods

Comparison operators: == and !=
Assignment operators: =, +=, -=, *=, /=
Unary operators: +, -
Binary operators: +, -, *, /
Length
LengthSquared
Normalize: Normalizes the quaternion. Note that only normalized quaternions correspond to 3D rotations.
Conjugate: Computes the conjugate of a quaternion. This result is Quaternion(-x, -y, -z, w). Note for a normalized quaternion, this is the inverse.
Inverse: Computes the inverse of a quaternion including normalization.
Dot
RotateTowards
ToEuler: Computes rotation about y-axis (y), then x-axis (x), then z-axis (z). The return value is compatible with one of the overloads of CreateFromYawPitchRoll.

Statics

CreateFromAxisAngle: Creates a quaternion representing a rotation of a given angle (in radians) around an arbitrary axis vector.
CreateFromYawPitchRoll: Creates a quaternion representation a rotation about y-axis (yaw), then x-axis (pitch), then z-axis (roll) given in radians.

The original D3DXmath library took the rotations in the the Yaw, Pitch, Roll order and that order was replicated in XNA Game Studio. In DirectXMath, the order was normalized to Roll (X), Pitch (Y), Yaw (Z) for the parameters, but the application of the rotations is in the same order.

CreateFromRotationMatrix
Concatenate: Concatenates two quaternion rotations. Note: Concatenate(q1,q2) is equivalent to q2*q1.
Lerp: Linear interpolation
Slerp: Spherical linear interpolation

For interpolating between arbitrary 3D rotations, the slerp is the gold-standard. For small differences, however, lerp is much faster and almost identical.