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 the result of the rotation and multiplies it by the magnitude of the axis of rotation, leading to a linear movement of the point. Thus, the new point remains on the same line from the origin as that of a pure rotation.

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

Syntax

Rotication requires an amount of rotation, an axis, and a starting point. Rotication is expressed as follows, where ^? represents the axis around which the rotication takes place. The default axis is formed by (0, 0, 0) and (0, 0, j^{^}) :

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

In sample calculations, formulae, and the calculator, the axis is assumed to be formed by the origin and the given Opposite Value. If an axis does not pass through the origin, it needs to be translated to the origin, the rotation performed, and then the result translated back by reversing the original translation.

R3 rotication is calculated using the magnitude of the axis, √(x²_a + y²_a + z²_a) and Rodrigues’ formula as follows:

Formula

R3 rotication is calculated using the magnitude of the axis, $\sqrt{(|x_a|^2 + |y_a|^2 + |z_a|^2)}$ and the Wave Number formula as follows:

$v_{Rotication} = \sqrt{(|x_a|^2 + |y_a|^2 + |z_a|^2)} * ((u_r \cdot v)u_r + sin\theta(u_r \times v) + cos\theta((u_r \times v)\times u_r))$

where $x_a, y_a$ and $z_a$ are the coordinates of the point that forms the axis with the origin.

Examples

Rotication is the same as Rotation when the magnitude of the axis is 1. Example 1 shows the dot product and cross product calculations.

Rotication where Magnitude of Axis is 1

π^{^}/2 R↺ (1^{^}, 0, 0) = √(1² + 0² + 0²) ((ur.v)ur) + (sinπ^{^}/2(ur x v)) + (cosπ^{^}/2)(ur x v) x ur)
- = 1*(((0, 0, j^{^}) . (1^{^}, 0, 0))*(0, 0, j^{^}) + 1* ((0, 0, j^{^}) x (1^{^}, 0, 0)) + 0*((0, 0, j^{^}) x (1^{^}, 0, 0)) x (0, 0, j^{^}))
- = 1*(0*(0, 0, j^{^})) + 1* (i^{^}) + 0
- = i^
- Note that ur = (0, 0, j^{^})
π^{^}/2 R↺^y (1^{^}, 0, 0) = (0, 0, j^v)
π^{^}/2 R↺^x (1^{^}, 0, 0) = (1^{^}, 0, 0)
π^v/2 R↺^z (1^{^}, 0, 0) = (0, i^v, 0)
π^v/2 R↺^y (1^{^}, 0, 0) = (0, 0, j^{^})
π^v/2 R↺^x (1^{^}, 0, 0) = (1^{^}, 0, 0)
π^{^}/2 R↺^x (0, i^{^}, 0) = (0, 0, j^{^})
π^{^}/2 R↺^y (0, 0, j^{^}) = (1^{^}, 0, 0)
π^{^}/2 R↺^z (0, i^{^}, 0) = (1^{^v}, 0, 0)

Rotication where Magnitude of Axis > 1

The following rotications show the effect of the magnitude of the axis when it is greater than 1 on the result. Example 1 shows the dot product and cross product calculations.

2π^{^}/3 R↺^{(^ + 2i^ + j^)} (1^v + 1i^v + j^{^}) = (2.6^{^} + 2.96i^v + 1.58j^v)
- = √(1² + 2² + 1²) ((ur.v)*ur) + (sin2/3π^{^}(ur x v)) + (cos2/3π^{^})*(ur x v) x ur)
- = 2.449*(((0.408^{^}, 0.816i^{^}, 0.408j^{^}).(1^v, i^v, j^{^}) )*(0.408^{^}, 0.816i^{^}, 0.408j^{^})
  - + 0.866* ((0.408^{^}, 0.816i^{^}, 0.408j^{^}) x (1^v, i^v, j^{^}))
  - + ^–.5*((0.408^{^}, 0.816i^{^}, 0.408j^{^}) x (1^v, 1i^v, j^{^}) ) x (0.408^{^}, 0.816i^{^}, 0.408j^{^}) )
- = 2.449*(^–0.816*(0.408^{^}, 0.816i^{^}, 0.408j^{^})
  - + 0.866* (1.225^{^}, 0.816i^v, 0.408j^{^})
  - + ^–.5*((1.225^{^}, 0.816i^v, 0.408j^{^}) x (0.408^{^}, 0.816i^{^}, 0.408j^{^}) )
- = 2.449*((0.333^v, 0.667i^v, 0.333j^v) + (1.06^{^}, 0.707i^v, 0.354j^{^}) + ^–.5*(0.667^v, 0.333i^v, 1.333j^{^})
- = 2.449*((0.333^v, 0.667i^v, 0.333j^v) + (1.06^{^}, 0.707i^v, 0.354j^{^}) + (0.333^v, 0.167i^{^}, 0.667j^v)
- = 2.449*(1.061^{^}, 1.207i^v, .646j^v)
- = (2.6^{^} + 2.96i^v + 1.58j^v)
- Note that ur = (0.408^{^}, 0.816i^{^}, 0.408j^{^})

More Examples

π^{^}/2 R↺^4v (0, 0, 2j^v) = (0, 8i^v, 0)
60^o R↺^{(4^{^} +5i^{^} + 6j^{^})} (1^{^} + 2i^{^} + 3j^{^})
- = (14.28^{^} + 12.7i^{^} +26.7j^{^})
60^o R↺^{(4^v+ 5i^v + 6j^{^})} (1^v + 2i^{^} + 3j^v)
- = (3.69^{^} + 0.024i^{^} + 32.64j^v)
π^{^}/4 R↺^{(^{^} + 2i^{^} + 3j^{^})}(1.061^{^}+1.207i^v+ 0.646j^v)
- = (4.2^{^} + i^v +4.83j^v)
π^{^}/3 R↺^{(4^ + 5i^ +6j^)}(1.122^{^} + 0.268i^v + 1.292j^v)
- = (0.32^v + 7.82i^{^} + 13.03j^v)

Next: Roticvision

Previous: Rotvision

Share to: