Rotation Sphere

R3 – Rotation Definition and Quaternions

What is Rotation in R3?

The R3 rotation operation is a fundamental aspect of the Wave Number system. The facts expressed in its axioms support this. A rotation moves a point, represented by Opposite Values, to a new location, also represented by Opposite Values, through circular movement around an axis. This axis is represented by a pair of Opposite Values. It remains fixed during the rotation.

Typically, one of the pair of Opposite Values that defines the axis is the origin (0, 0, 0), but any pair of Opposite Values can define an axis.

Syntax

Rotation in R3 requires an amount of rotation in degrees or radians, an axis, and a starting point. It is expressed as follows, where ^? represents the axis of rotation. The default axis is the z-axis, described by the pair of Opposite Values (0, 0, 0) and (0, 0, j^{^}) :

(Amount of rotation) ↺^? (Opposite Value).

The sample calculations, the formula and the calculator assume that the axis is formed by the origin and a given Opposite Value. If an axis does not pass through the origin, it needs to be translated to the origin, the rotation performed and the result translated back by the original translation in reverse.

Examples of R3 Rotation

Consider the expression $a$ ↺ $b$ . Here, $b$ is the operand, $a$ is the operator, and ↺ denotes the rotation operation.

For π^{^}/2 ↺ 1^{^} = i^{^}:

This means taking a point located at 1^{^} on the x-axis and rotating it clockwise by 90^{^o} around the z-axis defined by (0, 0, 0), (0, 0, j^{^}).
The point moves from (1^{^}, 0, 0) to (0, i^{^}, 0), placing it at i^{^} on the y-axis.

For π^v/2 ↺ 1^{^} = i^v:

This means taking a point located at 1^{^} on the x-axis and rotating it clockwise by 90^vo around the z-axis defined by (0, 0, 0), (0, 0, j^{^}).
The point moves from (1^{^}, 0, 0) to (0, i^v, 0), placing it at i^v on the y-axis.

Try these rotations using our Wave Number Rotation Ball.

Pythagoras Theorem

To calculate the distance from the origin, or the radius, of any point (x, y, z), use Pythagoras’ theorem:

Distance $= \sqrt{|x^2| + |y^2| + |z^2|}$ .

Quaternions

The quaternion number system facilitates the performance of rotations in three dimensions. Quaternions are used in this section to verify the accurate calculation of R3 rotations within the Wave Number system.

The coordinates of a point in quaternion notation are expressed as

$(a + bi + cj + dk)$

where $a$ is a scalar, and the terms $b$ i, $c$ j and $d$ k represent rotations around the x, y and z-axes, respectively. Quaternions extend complex numbers and are based on a four-dimensional sphere used to represent three-dimensional rotations. Despite their complexity and challenging visualization, quaternions effectively support three-dimensional multiplication and rotation. They are very hard to visualise and understand, but the following excellent video from 3Blue1Brown may help – Visualising Quaternions.

To convert a 3D point into a quaternion, assign it a scalar of 0 and use its Cartesian coordinates as the coefficients for i, j and k.

Quaternion multiplications can be performed either singly or in a chain. However, the outcome is a quaternion representing four dimensions, rather than a set of three-dimensional Cartesian coordinates. To convert this four-dimensional result back into three-dimensional coordinates, a method known as the quaternion sandwich is employed. This involves performing the original quaternion multiplications in reverse order using the conjugates of the quaternions.

The quaternion sandwich process, for a chain of two rotations, starts by multiplying a point $P$ first by quaternion $q1$ and then by quaternion $q2$ . Subsequently, the product is multiplied by the conjugate of $q$ 1, ( $q$ 1′). This process is represented by the formula: