Quaternions¶
A quaternion represents a rotation around some axis and have two benefits over rotation matrices.
First, a quaternion requires only 4 numbers and thus can be stored in a vector (ℝ4), while a rotation matrix requires 16 numbers (ℝ4×4). A quaternion thus needs much less space to be stored.
Second, quaternions are not affected by singularities (a degree of freedom is lost when 'gimbal lock' occurs) like Euler angles are, which are needed to construct a rotation matrix.
The function Identity returns a quaternion that does not perform
any rotation.
Chaining rotations¶
Multiple rotations can be chained with the * operator:
Rotation_A : constant Quaternion := R ((1.0, 0.0, 0.0, 0.0), To_Radians (30.0));
Rotation_B : constant Quaternion := R ((1.0, 0.0, 0.0, 0.0), To_Radians (60.0));
Rotation_C : constant Quaternion := Rotation_B * Rotation_A;
The difference between two rotations is another rotation and can be
obtained with the function Difference:
Difference (Rotation_A, Rotation_C) is equivalent to Rotation_B.
The functions Conjugate and Inverse return the conjugate and inverse
of a quaternion.
Length and normalization¶
The function Norm returns the magnitude of the quaternion.
A unit quaternion has a magnitude equal to 1.
The function Normalize will normalize a quaternion and return
a unit quaternion. The function Normalized will return True
or False depending on whether the quaternion is normalized or not.
Axis and angle¶
The function To_Axis_Angle converts a quaternion to a Axis_Angle,
while From_Axis_Angle converts the Axis_Angle back to a quaternion.
An Axis_Angle consists of an Axis, a normalized Direction vector,
and an Angle in radians.
Alternatively, the function R can return a quaternion given a
separate Axis and Angle.
Rotations of vectors¶
The function R can return a quaternion describing the rotation
from one vector to another.
The function Rotate rotates and returns a vector according to a given rotation.
Interpolation¶
Function Slerp returns the interpolated unit quaternion on the shortest
arc between two normalized quaternions, while Lerp returns the
interpolation on the chord between two normalized quaternions.