How do you find the rotation of a quaternion?
Any quaternion can be written as q=cos(ϕ/2)+sin(ϕ/2)k, where k is a unit vector representing the axis of rotation and ϕ is the angle in radians. Following the logic, the quaternion q2=cos(π/4)+sin(π/4)k represents rotation about the z-axis and not y-axis.
How do you calculate quaternion?
To use quaternions for rotation, you need to:
- Identify the vector defining the axis of rotation.
- If needed, find its unit equivalent.
- The quaternion of rotation is q = cos(θ/2) + (xₐ*i + yₐ*j + zₐ*k) * sin(θ/2) , where:
- If needed, rotate v using the formula qᵥ’ = q * qᵥ * q⁻¹ , where:
Why do quaternions represent rotation?
The representation of a rotation as a quaternion (4 numbers) is more compact than the representation as an orthogonal matrix (9 numbers). Furthermore, for a given axis and angle, one can easily construct the corresponding quaternion, and conversely, for a given quaternion one can easily read off the axis and the angle.
How do you rotate a vector by quaternion?
you can solve for the rotation angle using the axis-angle form of quaternions: θ = 2 cos − 1 ( a ) . q rv = θ sin ( θ 2 ) [ b , c , d ] .
How do i find the rotation between 2 vectors?
First step, you want to find the angle between the two vectors using the dot product. Next, to find the axis of rotation, use the cross product. Knowing that the cross product will yield a vector perpendicular to both u and v , crossing them in either order will give an appropriate axis.
Why do quaternions have 4 components?
Four values make up a quaternion, namely x, y, z and w. Three of the values are used to represent the axis in vector format, and the forth value would be the angle of rotation around the axis.” So you could think of it as the rotation of the rotation, in simple terms!
Are quaternions faster than rotation matrices?
For quaternions versus a 3×3 rotation matrix, the quaternion has the advantage in size (4 scalars vs. 9) and speed (quaternion multiplication is much faster than 3×3 matrix multiplication).
What is quaternion XYZW?
A quaternion is a set of 4 numbers, [x y z w], which represents rotations the following way: // RotationAngle is in radians x = RotationAxis. x * sin(RotationAngle / 2) y = RotationAxis. y * sin(RotationAngle / 2) z = RotationAxis. z * sin(RotationAngle / 2) w = cos(RotationAngle / 2)
How to calculate the rotation of a quaternion in G3?
Think about quaternions as planes. In G3 a quaternion is a bivector. The general rotation formula is V′ = exp(bϕ / 2)Vexp( − bϕ / 2) where b is a unit bivector s.d. b2 = − 1. Then we have a generalization of the de Moivre formula exp( ± bϕ / 2) = cos(ϕ / 2) ± bsin(ϕ / 2)
How do you represent a quaternion in math?
Therefore, the easiest way to represent a quaternion is to imagine the rotation of a given angle around a given vector. The following figure illustrates the rotation of angle θ around vector V → defined by 3 scalars ( V x, V y and V z) : The quaternion associated to this transformation is given by:
How do you express a quaternion from Euler angles?
It can be expressed from Euler angles as on this online visualization . Therefore, the easiest way to represent a quaternion is to imagine the rotation of a given angle around a given vector. The following figure illustrates the rotation of angle θ around vector V → defined by 3 scalars ( V x, V y and V z) :
What does the quaternion B Q a represent?
The quaternion B Q A represents the transformation from frame A and frame B. This simple example shows the rotation of an ellipse defined in the plane { X → , Y → }. The green and red points are respectively the initial and final position of the ellipses. The black dash-dotted line is the rotation axis.