R2 – Rotication

This post defines the new concept of the R2 rotication operation. It also provides examples.

What is Rotication in R2?

The R2 rotication operation is supported by the facts expressed in its axiom definition of operations. Rotication is a two-step operation. The first part of the operation is the same as rotation and results in a circular movement of a point. In the second part, the result of the rotation is multiplied by the magnitude of single point axis of rotation. This results in a linear movement of a point. Consequently, the new point is on the same line from the origin as that of a pure rotation.  

Typically the single point axis is the origin (0, 0), but any Opposite Value can define a single point axis.

Rotication requires an amount of rotation in degrees or radians, a single point axis and a starting point. Rotication is expressed as follows where ^? represents the single point axis around which the rotication takes place. The default is (0, 0) :

(Amount of rotication) R↺^? (Opposite Value).

The calculation of R2 rotication uses the magnitude of the single point axis at (xa, ya), √(|xa|² + |ya|²) and the Wave Number rotation formula described earlier:

• x’ = √(|xa|² + |ya|²) * ((|x| + ^–|xa|)cosθ +
  • (^–|y| + |ya|)sinθ + |xa|)*^{^}
• y’ = √(|xa|² + |ya|²) * ((|y|+ ^–|ya|)cosθ +
  • (|x| + ^–|xa|)sinθ + |xa|)*i^{^}

Here x and y are the counters of the point to be rotated. The coordinates xa and ya are the counters of the single point axis around which the rotation takes place. Finally, θ is the angle of rotation.

Examples

The result of a R2 rotication is 0 when it is around the origin as the magnitude of the origin is 0. For example:

• π^{^}/2 ↺ (3^{^} + 4i^v) – around the origin
  • => x’ = √(0² + 0²) * ((3 +^–0)cos(π^{^}/2) + (^––4 +0)sinπ^{^}/2 + 0)*^{^}
    • = 0
  • => y’ = √(0² + 0²) * ((^–4 +^–0)cos(π^{^}/2)^{^} + (3 +^–0)sinπ^{^}/2 + 0)i^{^}
    • = 0

• π^{^}/2 ↺^{(7v, 8iv)} (3^{^}+ 4i^v) – around point (7^v, 8i^v)
  • =>x’ = √(7² + 8²) * ((3+^{– –}7)cos(π^{^}/2) + (^{– –}4 +^–8)sinπ^{^}/2 + ^–7)*^{^}
    • = 10.63(10*0 + ^–4*1 + ^–7)*^{^} 
    • = 10.63*11^v = 116.93^v
  • =>y’= √(7² + 8²) * ((^–4 +^{– –}8)cos(π^{^}/2)+(3+^{– –}7)sinπ^{^}/2 + ^–8)*i^{^}
    • = 10.63(4*0 + 10*1 +  ^–8)*i^{^}
    • = 10.63*2i^{^} = 21.26i^{^}
  • =   116.93^v + 21.26i^{^}

Next: Roticvision

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