what is the parametric form of rotating a curve around the $x$,$y$,$z$ axis?

Question

If I have a curve $C(u)$ defined in the xy plane by $C(u)=(X(u),Y(u),0)$. What will be the parametric form of rotating the curve around the z-axis ?

I have looked around and saw that for the rotation in the x-axis the formula would be :

$(X(u),Y(u)sin(v),Z(u)cos(v))$

And rotation around the y-axis will be of the form:

$(X(u)cos(v),Y(u),Z(u)sin(v))$

Are these formulas correct?

Answer 1

You just multiply the vector with the 3D rotational matrices https://en.m.wikipedia.org/wiki/Rotation_matrix

For example a rotation $\theta_x$ around $x$ would give:

$$ \begin{bmatrix} X(u)\\ Y(u)\cos(\theta_x) - Z(u) \sin(\theta_x)\\ Y(u) \sin(\theta_x) + Z(u) \cos(\theta_x) \end{bmatrix} $$