Curves in polar coordinates

Question

Why there are different units used for rectangular and polar coordinates?

\begin{tikzpicture}[scale=2]
\draw[->] (-1,0) -- (1,0);
\draw[->] (0,-1) -- (0,1);
\draw node [red] at (-1,.25) {\scriptsize{Kardioida $r=5-5\sin \theta$}};
\draw[color=red,domain=0:6.28,samples=200,smooth] plot (canvas polar cs:angle=\x r,radius=      {5-5*sin(\x r)});    %r = angle en radian
\end{tikzpicture} 

Answer 1

When you use canvas polar cs, the radius is taken to be in TeX points and 5pt is not very big. To get the plot in the "natural" coordinates, use the xy polar syntax or use the implicit form, (<angle>:<radius>). What is perhaps a bit confusing about the implicit form is that it is a shorthand for both canvas polar cs and for xy polar cs. The rule is that if you supply an explicit length, it is canvas, but if you don't then it is xy.

Thus the lower two examples produce something that is on the right scale (I shrank the size to get it to fit on the page):

\documentclass{article}
%\url{http://tex.stackexchange.com/q/65446/86}

\usepackage{tikz}

\begin{document}
\begin{tikzpicture}[scale=2]
\draw[->] (-1,0) -- (1,0);
\draw[->] (0,-1) -- (0,1);
\draw node [red] at (-1,.25) {\scriptsize{Kardioida $r=5-5\sin
\theta$}};
\draw[color=red,domain=0:6.28,samples=200,smooth] plot (canvas polar
cs:angle=\x r,radius=      {5-5*sin(\x r)});    %r = angle en radian
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\draw[->] (-1,0) -- (1,0);
\draw[->] (0,-1) -- (0,1);
\draw node [red] at (-1,.25) {\scriptsize{Kardioida $r=5-5\sin
\theta$}};
\draw[color=red,domain=0:6.28,samples=200,smooth] plot (canvas polar
cs:angle=\x r,radius=      {.5cm-.5cm*sin(\x r)});    %r = angle en
radian
\end{tikzpicture}

\begin{tikzpicture}[scale=2]
\draw[->] (-1,0) -- (1,0);
\draw[->] (0,-1) -- (0,1);
\draw node [red] at (-1,.25) {\scriptsize{Kardioida $r=5-5\sin
\theta$}};
\draw[color=red,domain=0:6.28,samples=200,smooth] plot (xy polar
cs:angle=\x r,radius=      {.5-.5*sin(\x r)});    %r = angle en radian
\end{tikzpicture} 
\end{document}

Answer 2

In PSTricks, we can use \psset{runit=1.2cm,unit=\psrunit} to change the radial unit and rectangular unit. In this example, I made all equal.

\documentclass[border=12pt,pstricks]{standalone}

\usepackage{pst-plot}

\psset{runit=1.2cm,unit=\psrunit}

\begin{document}
\begin{pspicture}(-3.5,-3.5)(3.5,3.5)
\psaxes[axesstyle=polar,subticklinestyle=dashed,subticks=2,labelFontSize=\scriptstyle](3,3)
\psplot[polarplot,algebraic=true,linecolor=red,linewidth=2pt,plotpoints=2000]{0}{TwoPi}{1.5-1.5*sin(3*x)}
\end{pspicture}
\end{document}

Edit:

An animation might help us to choose a good constant for the function.

\documentclass[border=12pt,pstricks]{standalone}

\usepackage{pstricks-add}

\psset{runit=1cm,unit=\psrunit}

\begin{document}
\multido{\i=1+1}{10}{%
\begin{pspicture}(-3.5,-3.5)(3.5,3.5)
\psaxes[axesstyle=polar,subticklinestyle=dashed,subticks=2,labelFontSize=\scriptstyle](3,3)
\psplot[polarplot,algebraic,linecolor=red,linewidth=2pt,plotpoints=2000]{0}{TwoPi}{1.5-1.5*sin(\i*x)}
\rput(0,-3.5){\scriptsize Kardioida \ifnum\i=1\relax$r=1.5-1.5\sin \theta$\else$r=1.5-1.5\sin(\i \theta)$\fi}
\end{pspicture}}
\end{document}