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 $y =$ ??

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

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

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 $\frac{\pi}{3}$

enter image description here

  • $\begingroup$ great visuals, thanks a lot :) $\endgroup$ – Mostafa Said May 13 '18 at 19:14


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:




  • $\begingroup$ thanks a lot, works :) could you tell where is the source of such equations ? $\endgroup$ – Mostafa Said May 13 '18 at 17:13
  • $\begingroup$ 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) $\endgroup$ – kadoodle May 14 '18 at 14:36
  • $\begingroup$ Thanks a lot, thats really helpful $\endgroup$ – Mostafa Said May 15 '18 at 14:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.