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 – Rotication
This post defines the new concept of the R3 rotication operation. It also provides examples. What is Rotication in R3? The rotication operation is grounded in the principles outlined in its axioms. Rotication consists of two steps. The first step is identical to rotation, resulting in a circular movement of a point. The second step takes […]
R3 – Multiplication – Dot Product
Dot product multiplication in R3 uses the same approach as in classical mathematics. The dot product of the points and is written as and is given by the formula where a flip () precedes the Counters when the Opposite Value is or : For example: Euclidean Geometry Wikipedia describes dot product as follows: […]
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 […]
R2 – Multiplication – Dot Product
Dot product multiplication in R2 uses the same approach as in classical maths by using the formula: Dot product of a.b = (ax*bx) + (ay*by) where ax, bx, ay and by are the Counters of the Opposite Values a and b. For example: The dot product is based on the Counters of the Opposite Values […]