# Circle Parametric equation in $3D$ space?

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 $y =$ ??

• If I'm understanding your description properly, it sounds like $y=0$, so your parametric representation is: $(r\cos\theta,0,r\sin\theta)$ – G Tony Jacobs May 13 '18 at 16:02
• yes, when y = 0 the circle is in xz plane, now I want to rotate it around z-axis ? – Mostafa Said May 13 '18 at 16:13
• I see... it sounds as if you'll need one parameter, $\theta$, to trace out your circle, and another parameter, $t$, to make it rotate? – G Tony Jacobs May 13 '18 at 16:29
• You should start from the parametrization of a sphere in $\mathbb{R}^3$, and then fix the polar angle to the desired value – user438666 May 13 '18 at 16:30
• G Tony, --> thats exactly what i am trying to do, so $theta$ for the circle and t for the tilt – Mostafa Said May 13 '18 at 16:41

The circle

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

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

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

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

Attached a rotated circle (red) by $\frac{\pi}{3}$ • great visuals, thanks a lot :) – Mostafa Said May 13 '18 at 19:14

$HINT:$

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:

$x=r.cosθ-r.cosθ.cosα$

$y=r.cosθ.sinα$

$z=r.sinθ$

• thanks a lot, works :) could you tell where is the source of such equations ? – Mostafa Said May 13 '18 at 17:13
• you can take the coordinates of a point (x,0,z). Rotate it about z-axis in a rough diagram by an angle α, and use trignometric ratios to find the final directed distances from x,y,z axis. Hence finding the final coordinates.(sorry i cannot attach a photo yet) – kadoodle May 14 '18 at 14:36
• Thanks a lot, thats really helpful – Mostafa Said May 15 '18 at 14:44