# Cardioid-looking curve, does it have a name?

The curve, given in polar coordinates as $$r(\theta)=\sin(\theta)/\theta$$ is plotted below.

This is similar to the classical cardioid, but it is not the same curve (the curve above is not even algebraic, I believe). Does this curve have a name? Does it show up somewhere? This curve has the property that it solves $$\mathrm{Im}(1/z+\log(z))=0$$, if this perhaps rings a bell.

This particular curve arises in some research I am working on at the moment, and it would be great if it perhaps connects to some classical area.

Edit: Thanks for the great references! As a reward, here is a more artistic rendering of the shape using a type of complex dynamical systems.

• Check mathcurve.com. At least 100 named curves. Sep 15, 2020 at 20:17
• That was a great resource! Sep 15, 2020 at 20:30
• For people too lazy to type, and because I like the enthusiasm of the name, a clickable version of @skbmoore's link: Encyclopédie des formes mathématiques remarquables. Sep 15, 2020 at 20:38
• It seems to be a limacon.
– user165496
Sep 17, 2020 at 10:59
• @user165496 --- the limaçon has a different equation, in particular it lacks the infinite series of windings that the cochleoid has: en.wikipedia.org/wiki/Limaçon Sep 17, 2020 at 11:55

Not an answer - just want to note that the curve has more hidden branches. They can be seen looking at the parametric equation $$x=\frac{\sin(\varphi)}\varphi,\quad y=\frac{1-\cos(\varphi)}\varphi$$ where $$\varphi=2\theta$$
• This is indeed true: say,$$x=t+\cos(t),y=-\sin(t),z=1+t$$is a helix, and projecting it from the point $(1,0,1)$ to the plane $z=0$ gives$$(\frac{1-\cos(t)}t,\frac{\sin(t)}t,0).$$Explicitly, to project you must send $(x,y,z)$ to $((x-z)/(1-z),y/(1-z),0)$; then just substitute the above parametric expressions for $t$. Sep 18, 2020 at 8:59