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 $y =$ ??
$\begingroup$
$\endgroup$
5
-
$\begingroup$ 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 JacobsMay 13, 2018 at 16:02
-
$\begingroup$ yes, when y = 0 the circle is in xz plane, now I want to rotate it around z-axis ? $\endgroup$– MostafaBakrMay 13, 2018 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 JacobsMay 13, 2018 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 $\endgroup$– user438666May 13, 2018 at 16:30
-
$\begingroup$ G Tony, --> thats exactly what i am trying to do, so $theta$ for the circle and t for the tilt $\endgroup$– MostafaBakrMay 13, 2018 at 16:41
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
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 $\frac{\pi}{3}$
$\begingroup$
$\endgroup$
3
$HINT:$
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:
$x=r.cosθ-r.cosθ.cosα$
$y=r.cosθ.sinα$
$z=r.sinθ$
-
$\begingroup$ thanks a lot, works :) could you tell where is the source of such equations ? $\endgroup$ May 13, 2018 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) $\endgroup$– kadoodleMay 14, 2018 at 14:36
-