Here's a challenging query for you. I'm currently utilizing three.js and facing the need to compute a specific vector.
Given one vector referred to as C and a rotation denoted as (r) in the form of an euler, I am now seeking to derive a second vector indicated as (X). This second vector should, when subtracted by itself with the applied euler rotation, result in the original vector C.
In essence, this can be represented as:
X - X(r) = C
To transform the Euler angles, convert them into a 3x3 matrix using three.js. It's important to note that three.js utilizes Tait-Bryan angles instead of Proper Euler angles. Construct the necessary matrix R by combining these axis rotation matrices (refer to this link) in the order of Rz * Ry * Rx.
Remember that X = I * X where I stands for the 3x3 identity matrix.
The equation can be expressed as (I - R) * X = C, and is easily inverted as X = inverse(I - R) * C (Matrix3.getInverse).
UPDATE: steps to calculate R:
Utilize standard rotation matrices:
For the Euler angles euler = (x, y, z), the calculation becomes R = Rz(z) * Ry(y) * Rx(x):
var c_x = Math.Cos(euler.x), s_x = Math.Sin(euler.x);
var Rx = (new Matrix3()).set(
1, 0, 0,
0, c_x, -s_x,
0, s_x, c_x
);
// proceed likewise with Ry and Rz
Ultimately:
var R = Rz.multiply(Ry.multiply(Rx));
var ImR = (new Matrix3()).set(
1.0 - R.elements[0], -R.elements[1], -R.elements[2],
-R.elements[3], 1.0 - R.elements[4], -R.elements[5],
-R.elements[6], -R.elements[7], 1.0 - R.elements[8]
);
// perform multiplication of ImR.getInverse() with C to derive X
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