polar coordinate system

The polar coordinate system specifies a point in a plane using two coordinates:

the point's distance from the pole, and
the point's direction relative to the polar axis.

Formally,
a point is specified as $(r, θ)$ , where

$r$ is the radius, i.e. the length of the segment to pole $O$ ,
$θ$ is the azimuth, i.e. the angle between radius $r$ and polar axis $L$ .

Visually,

\documentclass[tikz,margin={2cm 0cm}]{standalone}
%
\usepackage{tkz-euclide}
%
\definecolor{radiuscolor}{HTML}{8A5CF5}
\definecolor{anglecolor}{HTML}{E9973F}
\definecolor{raycolor}{HTML}{E0DE71}
%
\begin{document}
\begin{tikzpicture}
    \tkzDefPoint(0,0){O}
    \tkzDefPoint(40:2){A}
    %
    \tkzCalcLength(O,A)\tkzGetLength{radius}
    \tkzDefPoints{\radius/0/I}
    \tkzDefLine[perpendicular=through A](I,O) \tkzGetPoint{X_t}
    \tkzInterLL(O,I)(X_t,A) \tkzGetPoint{X}
	%
    \tkzPicAngle["$\theta$",anglecolor,draw,thin,angle radius=1cm](X,O,A)
	%
	\tkzDrawLine[raycolor,thick,-Latex,add=0 and 0.125](O,I)
		\tkzLabelLine[raycolor,pos=1.25](O,I){$L$}
	%
     \tkzDrawSegment[radiuscolor, thick](O,A)
        \tkzLabelSegment[radiuscolor,above left](O,A){$r$}
    %
    \tkzDrawPoints(O,A)
    \tkzLabelPoint[below](O){$O$}
    \tkzLabelPoint[above right](A){$(r,\theta)$}
\end{tikzpicture}
\end{document}

#WIP
shifted circle: $r = 2 a \cos θ = 2 a \sin θ$
cardioid: $r = 2 a (1 - \cos θ) = 2 a (1 - \sin θ)$
limacon: $r = b + a \cos θ = b + a \sin θ$
rose: $r = a \cos (k θ) = a \sin (k θ)$
spiral: $r = a θ$