Euler Rotation

Definitions

Euler - Pronounced Oiler; discovered that as 1 is continuously compounded, it reaches a number, e. However, Euler rotation is taking a 3-dimensional axes set and rotating them about the z-axis followed by a forward or backward rotation to create a new axes set. For objects that inherit the property 'eulerrotation', the format is supposedly "Δα Δβ Δγ". Read further for a better explanation.

θ: greek letter theta
φ: greek letter phi
theta and phi are commonly used in trigonometry for angles (but phi is most commonly used as a constant: 1.618...).
α: 1^st greek letter alpha. For our purposes, it will represent The angle between the original x axis and the normal vector⁽¹⁾.
β: 2^nd greek letter beta. For our purposes, it will represent the forward or backward "tilt".
γ: 3^rd greek letter gamma. For our purposes, it will represent the angle between the normal vector⁽¹⁾ and the new x axis.

Therefore, images and text could be rotated, something that could not be done before, so that said image or text would have a three-dimensional effect. If all said is true, eulerrotation = "0 -pi/3 0"; for text would project it in a Star Wars-like fashion.

sin(θ) = y / r (Δy of a point from the center on a circle with radius r)
cos(θ) = x / r (Δx of a point from the center on a circle with radius r)
sin(θ ± φ) = sin(θ) cos(φ) ± cos(θ) sin(φ)
-cos(θ ± φ) = -cos(θ) cos(φ) ± sin(θ) sin(φ)

In addition (not really useful):

tan(θ) = sin(θ) / cos(θ)
cot(θ) = cos(θ) / sin(θ)
sec(θ) = 1 / cos(θ)
csc(θ) = 1 / sin(θ)

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⁽¹⁾ The normal vector is the vector perpendicular to the original z and the new z axes. This can be calculated through cross product.

Rotation Matrices

Note: The following example uses MATLAB notation for the matrix transformation.

[cos(γ) sin(γ) 0; -sin(γ) cos(γ) 0; 0 0 1] * ...
[1 0 0; 0 cos(α) sin(α); 0 -sin(α) cos(α)] * ...
[cos(β) sin(β) 0; -sin(β) cos(β) 0; 0 0 1] * ...
([x0 y0 z0]') = [x y z]'