R3 – Quaternion Comparison
Comparison of Wave Number and Quaternion Multiplication Tables: An earlier post makes a detailed comparison of the Wave Number Multiplication table and the Quaternion Multiplication table. The only difference occurs when Quaternions are multiplied together, as in i*i, = -1, but in R3, 1^*1^ = 1^. In R3, multiplying 1^ and 1^ = 1^ and multiplying 1^ and 1v = 1v, leaving the operand […]
R3 – Complex Rotations – Comparing Wave Numbers and Quaternion Efficiency
Introduction This comparison of Wave Numbers and Quaternions evaluates their efficiency based on the rotation operation due to its computational complexity. Computer games and other high quality graphics frequently use rotation operations to implement the physics of movement, making this an important consideration for performance. Unit of Rotation The magnitude and unit rotation of a […]
R3 – Complex Rotations – Example 2
This post gives a second example of a complex R3 rotation. It uses the Wave Number Rotation formula adapted from Rodrigues’ formula and shows the Quaternion equivalent. Wave Number Rotation Formula Example 2 Take for example the rotation of 60^o ↺(4v+ 5iv + 6j^) (1v + 2i^ + 3jv). Here θ = 60^o and v = (1v + 2i^ + 3jv). ur is […]
R3 – Complex Rotations – Example 3
This post provides an example of a rotation chain involving three consecutive rotations. It uses the Wave Number Rotation formula adapted from Rodrigues’ formula and shows the Quaternion equivalent. Wave Number Rotation Formula Example 3 Rotation 1 In this example, 3 rotations take place one after the other: Calculate the chain of rotations using the […]
R3 – Complex Rotations – Example 1
Up to this point, we’ve focused on unitary rotations around a single axis on the unit sphere. Now, let’s delve into more complex rotations in R3, where the operand can be any combination of Opposite Values with any degree of rotation. This post introduces the Wave Number Rotation formula, based on Rodrigues’ formula for calculating […]
R3 – Rotation – Multiplication Table
Orthogonal Rule The layout of the R3 axes is defined as part of the Wave Number axioms. The Axioms of Rotation provide the Orthogonal rule which states that multiplication by a unitary is the equivalent of rotation around the unitary’s axis by 90o. This allows for the derivation of the R3 unitary multiplication table, shown below, from rotations […]
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 […]