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 $y =$ ??
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$\begingroup$ 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 JacobsCommented May 13, 2018 at 16:02
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$\begingroup$ yes, when y = 0 the circle is in xz plane, now I want to rotate it around z-axis ? $\endgroup$– MostafaBakrCommented May 13, 2018 at 16:13
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$\begingroup$ 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 JacobsCommented May 13, 2018 at 16:29
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$\begingroup$ You should start from the parametrization of a sphere in $\mathbb{R}^3$, and then fix the polar angle to the desired value $\endgroup$– user438666Commented May 13, 2018 at 16:30
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$\begingroup$ G Tony, --> thats exactly what i am trying to do, so $theta$ for the circle and t for the tilt $\endgroup$– MostafaBakrCommented May 13, 2018 at 16:41
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2 Answers
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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 $\frac{\pi}{3}$
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$\begingroup$ great visuals, thanks a lot :) $\endgroup$ Commented May 13, 2018 at 19:14
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$HINT:$
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:
$x=r.cosθ-r.cosθ.cosα$
$y=r.cosθ.sinα$
$z=r.sinθ$
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$\begingroup$ thanks a lot, works :) could you tell where is the source of such equations ? $\endgroup$ Commented May 13, 2018 at 17:13
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$\begingroup$ 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) $\endgroup$– kadoodleCommented May 14, 2018 at 14:36
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$\begingroup$ Thanks a lot, thats really helpful $\endgroup$ Commented May 15, 2018 at 14:44