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I have 4 vertices in 3d space that form a rectangle. My goal is to rotate these vertices around the normal of the rectangle.

So what I need is:

1. How to calculate the normal vector of the 3d rectangle (origin in the center of the rect)


2. How can I rotate the vertices around this normal vector in angles (e.g. 5 degrees)

Thx for your help.

Let's say your four vertices are
$$bbox{ vec{v}_1 = (x_1, y_1, z_1) } , quad
bbox{ vec{v}_2 = (x_2, y_2, z_2) } , quad
bbox{ vec{v}_3 = (x_3, y_3, z_3) } , quad
bbox{ vec{v}_4 = (x_4, y_4, z_4) } $$
with $vec{v}_1$ and $vec{v}_3$ diagonally opposite each other in the rectangle.

The center of the rectangle is at $vec{c}$,
$$bbox{ vec{c} = frac{vec{v}_1 + vec{v}_2 + vec{v}_3 + vec{v}_4}{4}
= left ( frac{x_1 + x_2 + x_3 + x_4}{4}, frac{ y_1+y_2+y_3+y_4 }{4}, frac{z_1+z_2+z_3+z_4}{4} right ) }$$
and the normal of the plane that passes through vertices $vec{v}_1$, $vec{v}_2$, and $vec{v}_3$ is
$$bbox{ vec{p} = (vec{v}_3 - vec{v}_2) times (vec{v}_1 - vec{v}_2) }$$
that is,
$$bbox{ begin{aligned}
vec{p} &= ( x_p ,, y_p ,, z_p ) \
x_p &= (y_3 - y_2)(z_1 - z_2) - (z_3 - z_2)(y_1 - y_2) \
y_p &= (z_3 - z_2)(x_1 - x_2) - (x_3 - x_2)(z_1 - z_2) \
z_p &= (x_3 - x_2)(y_1 - y_2) - (y_3 - y_2)(x_1 - x_2) \
end{aligned} }$$

In practice, you'll want to use the unit normal, the above scaled to length 1,
$$bbox{ hat{n} = ( x_n ,, y_n ,, z_n ) = left (
frac{x_p}{sqrt{x_p^2 + y_p^2 + z_p^2}} ,;
frac{y_p}{sqrt{x_p^2 + y_p^2 + z_p^2}} ,;
frac{z_p}{sqrt{x_p^2 + y_p^2 + z_p^2}} right ) }$$

For the rotation around that unit normal, you can use Rodrigues' rotation formula. To rotate $vec{v}$ by $theta$ around axis $hat{n}$, to $vec{v}^prime$,
$$bbox{ vec{v}^prime = vec{v} costheta + (hat{n} times vec{v}) sintheta + hat{n} ( hat{n} cdot vec{v} )(1 - costheta) }$$
where $cdot$ denotes dot product, and $times$ vector cross product.

In OP's case, the rotation is around $vec{c}$:
$$bbox{ vec{v}^prime = vec{c} + (vec{v} - vec{c}) costheta + bigr ( hat{n} times (vec{v} - vec{c}) bigr ) sintheta + hat{n} bigr ( hat{n} cdot ( vec{v} - vec{c} ) bigr ) (1 - costheta) }$$

or in notation perhaps more familiar to programmers,

x_new = x_c + cos(theta)*(x_v - x_c) + sin(theta)*( y_c*z_n - y_n*z_c + y_n*z_v - y_v*z_n) + (cos(theta) - 1)*x_n*( x_c*x_n - x_n*x_v + y_c*y_n - y_n*y_v + z_c*z_n - z_n*z_v )

y_new = y_c + cos(theta)*(y_v - y_c) + sin(theta)*(-x_c*z_n + x_n*z_c - x_n*z_v + x_v*z_n) + (cos(theta) - 1)*y_n*( x_c*x_n - x_n*x_v + y_c*y_n - y_n*y_v + z_c*z_n - z_n*z_v )

z_new = z_c + cos(theta)*(z_v - z_c) + sin(theta)*( x_c*y_n - x_n*y_c + x_n*y_v - x_v*y_n) + (cos(theta) - 1)*z_n*( x_c*x_n - x_n*x_v + y_c*y_n - y_n*y_v + z_c*z_n - z_n*z_v )

where $vec{v}$ = (x_v, y_v, z_v) is the point to rotate, $hat{n}$ = (x_n, y_n, z_n) is the unit vector representing the rotation axis, $vec{c}$ = (x_c, y_c, z_c) is the center of rotation, and theta is the angle rotated.

You can optimize the expression quite a bit by calculating many of the terms into temporary variables, so you don't do the same operations again and again needlessly.

If we have $vec{a} = ( a_x , a_y , a_z )$ and $vec{b} = ( b_x , b_y , b_z )$, then dot product is
$$bbox{ vec{a} cdot vec{b} = a_x b_x + a_y b_y + a_z b_z }$$
and yields a scalar; and cross product is
$$bbox{ vec{a} times vec{b} = ( a_y b_z - a_z b_y ,, a_z b_x - a_x b_z ,, a_x b_y - a_y b_x ) }$$
and yields a vector.