Wikipedia:Reference desk/Archives/Mathematics/2012 August 29

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August 29

Interesting curves where θ = f(r) ?

I'm testing a program I've written which (among other things) graphs polar equations where the angle is a function of the radius. So far all I've tested is θ = r, which is just a basic spiral. Anyone have anything more interesting for me to try ? StuRat (talk) 12:41, 29 August 2012 (UTC)

You can try $r^{2},\ {\sqrt {r}},\ \exp(r)$. If you had $r=f(\theta )$ you would have some nicer options. -- Meni Rosenfeld (talk) 14:21, 29 August 2012 (UTC)

I have both. For r = f(θ) I found some nice roses and heart plots, the last of which looks like something right off a valentine. StuRat (talk) 14:34, 29 August 2012 (UTC)

theta = 2 pi r/(r-1) Count Iblis (talk) 16:00, 29 August 2012 (UTC)

theta = 2 pi sin(r) Count Iblis (talk) 16:10, 29 August 2012 (UTC)

I like that last one, since it's the only one which isn't some type of spiral. StuRat (talk) 03:29, 30 August 2012 (UTC)

$\theta =\pm \arccos {\frac {r^{2}+d^{2}-m^{2}}{2rd}}$ for $0<m<d$ is a circle with radius m and the center d apart from the coordinate system's pole. --CiaPan (talk) 06:14, 30 August 2012 (UTC)

How can the center be a single variable "d" ? Isn't it always 2 coords in 2D, either (x,y) or (r,θ) ? StuRat (talk) 10:13, 30 August 2012 (UTC)

Sorry for my poor English, I meant the distance from the pole to the circle's center equals d. CiaPan (talk) 11:12, 30 August 2012 (UTC)

Thanks for the clarification. StuRat (talk) 18:58, 30 August 2012 (UTC)

Rose (mathematics) expands on the roses mentioned above.--Salix (talk): 10:33, 30 August 2012 (UTC)

Conic sections have general equation in polar coordinates
    $r={\frac {l}{1+e\cos \theta }}$
(with the pole being the section's focus), see Conic section#Polar coordinates. Solving for θ results in
    $\theta =\pm \arccos {\frac {l-r}{er}}$
(except the circle, which has e=0). --CiaPan (talk) 11:48, 30 August 2012 (UTC)

Isn't that a division by zero error ? StuRat (talk) 18:58, 30 August 2012 (UTC)

Congratulations! You've discovered the asymptote through numerical experimentation! Nimur (talk) 20:48, 30 August 2012 (UTC)

Asymptotic to a circle ? StuRat (talk) 06:42, 31 August 2012 (UTC)