In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space.

Étienne Brûlé

The quaternions almost form a field. They have the basic operations of addition and multiplication, and these operations satisfy the associative laws, (p + q) + r = p + (q + r), (pq)r = p(qr).

Quaternions only require 4 numbers (3 if they are normalized. The Real part can be computed at run-time) to represent a rotation where a matrix requires at least 9 values.

Complex algebra is 2D and what is known as quaternion algebra jumps to 4D. Using 1,i,j, and k as the base (where complex uses 1 and i (or j if you are an EE)) which results in a 4-axis space.

noun. a group or set of four persons or things. Bookbinding. four gathered sheets folded in two for binding together.

A 4-D being would be a god to us. It would see everything in our world. It could even look inside your stomach and remove your breakfast without cutting through your skin, just like you could remove a dot inside a circle by moving it up into the third dimension, perpendicular to the circle, without breaking the circle.

eul = quat2eul( quat , sequence ) converts a quaternion into Euler angles. The Euler angles are specified in the axis rotation sequence, sequence . The default order for Euler angle rotations is “ZYX” .

Unit quaternions, known as versors, provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation about an arbitrary axis.

Radians are only important when using with Sin and Cos functions as their input is in radians instead of degrees. Finally, quaternions are similar to Euler angles in that they express 3d angles.

a quaternion is a complex number with w as the real part and x, y, z as imaginary parts. If a quaternion represents a rotation then w = cos(theta / 2), where theta is the rotation angle around the axis of the quaternion. If w is -1 the quaternion defines +/-2pi rotation angle around an undefined axis v = (0,0,0).

A Euler Angle is an angle that you would think of as between 0 degrees and 360 degrees. transform. rotation is NOT a Euler Angle, it is a Quaternion. 3D Euler angles are represented by a Vector3 in Unity. // using Quaternion.

Having given a Quaternion q, you can calculate roll, pitch and yaw like this: var yaw = atan2(2.0*(q.y*q.z + q.w*q.x), q.w*q.w – q.x*q.x – q.y*q.y + q.z*q.z); var pitch = asin(-2.0*(q.x*q.z – q.w*q.y)); var roll = atan2(2.0*(q.x*q.y + q.w*q.z), q.w*q.w + q.x*q.x – q.y*q.y – q.z*q.z);

Adding a fourth rotational axis can solve the problem of gimbal lock, but it requires the outermost ring to be actively driven so that it stays 90 degrees out of alignment with the innermost axis (the flywheel shaft).

One way to do it, which is pretty easy to visualize, is to apply the rotation specified by your quaternion to the basis vectors (1,0,0), (0,1,0), and (0,0,1). The rotated values give the basis vectors in the rotated system relative to the original system. Use these vectors to form the rows of the rotation matrix.

The Quaternion structure is used to efficiently rotate an object about the (x,y,z) vector by the angle theta, where: w = cos(theta/2)

You can use Quaternion. Slerp() to limit the rotation of an object, for instance: transform. localRotation = Quaternion.

With quaternions, it’s as simple as multiplication. Typically you will take the orientation you have (as a quaternion) and just multiply by the rotation (another quaternion) you want to apply.

Representing rotations using quaternions Negating q results in a negative rotation around the negative of the axis of rotation, which is the same rotation represented by q (Eq.

or if you want to use Quaternions use the “*” operator to multiply them together:

1. var rotationAngle : float = 90;
2. var quatA = Quaternion. AngleAxis(rotationAngle , Vector3.
3. var quatB = Quaternion.
4. var quatC = quatA * quatB;
5. //this will be rotate 90 degrees around the z axis and 90 degrees around the y axis.
6. transform.

The inverse of a quaternion is equivalent to its conjugate, which means that all the vector elements (the last three elements in the vector) are negated. The rotation also uses quaternion multiplication, which has its own definition.

A pure quaternion is defined as a quaternion with a zero for the scalar value (q0=0). A standard 3D vector can be readily stored in a pure quaternion. Pure quaternions can then be used to rotate vectors or transform the vector coordinates between different rotated reference frames.

An identity quaternion is a quaternion that doesn’t change any quaternion it is multiplied with, thus 1 + 0i + 0j + 0k or 1 . An identity quaternion is thus a rotation of nothing.

The algebra of quaternions is the unique associative non-commutative finite-dimensional normed algebra over the field of real numbers with an identity. The algebra of quaternions is a skew-field, that is, division is defined in it, and the quaternion inverse to a quaternion X is ¯X/N(X).