# What is the equation of the following polar curve?

I am trying to plot the following curve. It has 3 leaves, each leaf is identical and 120 degrees apart. It is traced as shown in the attached numbers. My attempt is $r(\theta)=1-0.6\sin(3\theta)$ but I have no idea how to adjust it to resemble the curve above.

\documentclass[pstricks]{standalone}
\usepackage{pst-plot}
\begin{document}
\begin{pspicture}[showgrid](-3,-3)(3,3)
\psplot[algebraic,polarplot,linecolor=red,plotpoints=100]{0}{Pi 2 mul}{1-.6*sin(3*x)}
\end{pspicture}
\end{document} # Question

What is the equation of the polar curve or parametric curve (or any kind of curve) given above?

• Why do you think it is a polar curve? That is, why do you think it has a polar equation form? It does not appear to have one, to my eye. Would a parametric form be just as useful for you? – Matthew Conroy Dec 5 '17 at 4:55
• It looks like a Hypotrochoid to me but I can't get the parameters right to get three leaves. – Ross Millikan Dec 5 '17 at 6:04
• A close, but not perfect, fit is (a projection of) the trefoil knot. It can be parametrized simply. – mephistolotl Dec 5 '17 at 6:26
• $(\cos t+\frac34\cos(2t), \sin t-\frac34\sin(2t))$. Further reading: Hypotrochoid – Rahul Dec 5 '17 at 7:18
• @Rahul: Your answer deserves 25 points. – Money Oriented Programmer Dec 5 '17 at 8:16

## 2 Answers

Probably not what you are looking for, but, with $$f(t) = \frac{1}{3}t+\frac{3}{2}\cos\left(\frac{1}{2}t\right)-\frac{1}{5}\sin t,$$ the curve $$x(t)=\int_{0}^t \cos(f(u)) \, du, \,\, y(t) = \int_{0}^t \sin(f(u)) \, du$$ looks like this: which looks like your curve if you squint.

UPDATE: With $$f(t) = \frac{1}{3}t+\cos\left(\frac{1}{2}t\right)-\frac{1}{5}\sin t,$$ the curve is this: • @ArtificialStupidity I improved it! – Matthew Conroy Dec 5 '17 at 6:58

The Adobe's logo like curve

$$(\cos t+\frac34\cos(2t), \sin t-\frac34\sin(2t))$$

mentioned in Rahul's comment is the most similar to my requirement.

\documentclass[pstricks]{standalone}
\usepackage{pst-plot}
\begin{document}
\multido{\i=0+1}{51}{%
\begin{pspicture}[showgrid=false](-2,-2)(2,2)
\rput{36}(0,0){\psparametricplot[algebraic,polarplot,plotpoints=100,linecolor=red]
{0}{Pi 2 mul 50 div \i\space mul}{cos(t)+3*cos(2*t)/4|sin(t)-3*sin(2*t)/4}}
\end{pspicture}}
\end{document} 