在 3d 中使用倾斜坐标系旋转一个点

Question

I try to convert an given point in the normal coordinate system, to an tilted coordinate system.

What i have given: The point (2;0;0) in the "normal coordinate system"

With the old coordinate system:

X

X3;Y3;Z3 the vectors for the local z axis you get from the vector product of x vector (1) and y vector (2)

The tilted coordinate system is at 45 degree tilted coord sys and X3: 0 Y3: 1 Z3: 0 The coord sys is tilted 45°, but i don´t have that info. a picture about the tilted coord sys

I want to know where the point (2;0;0) getting moved to.

There are many sources about 2D tilting, but not about 3D. I am trying to calculate it "by hand" but also try to write an c# programm about it. So if you just can answer me with an c# function you can do it.

I tried to work with euler angles and looked at quaternions, but i couldn´t get the right calculations with the calculator. And the fact that i don´t have the angle, just the vectors of the new coordinate system is hard for me to imagine how to deal with it.

I think this formula can be useful for me, but i dont have the rotation angle to fill. https://en.wikipedia.org/wiki/Rotation_matrix#Rotation_matrix_from_axis_and_angle

I hope you understood my question, otherwise feel free to ask and i try to explain it more/better.

Answer 1

As Thomas stated out, i had to use the change of basis matrix. The gauss elimination algo helped me there.