Circle Parametric equation in $3D$ space?

Question

What are the parametric equations of a circle in $x z$ plane with a rotation a round $z$-axis ? so if

$x = r * \cos(\theta)$

$z = r * \sin(\theta)$

what should $y =$ ??

Answer 1

The circle

$$ p = (r\cos(t),0,r\sin(t)) $$

rotated around the $z$ axis is built with the rotation matrix

$$ R(\theta) = \left( \begin{array}{ccc} \cos (\theta ) & -\sin (\theta ) & 0 \\ \sin (\theta ) & \cos (\theta ) & 0 \\ 0 & 0 & 1 \\ \end{array} \right) $$

In new coordinates reads

$$ p\cdot R(\theta) = (r \cos (\theta ) \cos (t),-r \sin (\theta ) \cos (t),r \sin (t)) $$

Attached a rotated circle (red) by $\frac{\pi}{3}$

Answer 2

$HINT:$

You can see geometrically that :

any point $P(x,0,z)$ in the $XY-plane$,

if rotated about the $Z-axis$ by an angle $α$ with the $XZ- plane$ changes to :

$(x-x.cosα, x.sinα, z)$.

so if your original coordinates are:

$(r.cosθ,0,r.sinθ)$,(in the $XZ$-plane)

they will change to:

$x=r.cosθ-r.cosθ.cosα$

$y=r.cosθ.sinα$

$z=r.sinθ$