Cartesian coordinate system

#LaTeX

The Cartesian coordinate system specifies each point in a plane uniquely using two coordinates, each corresponding to signed distances from the origin along its fixed axes.

Formally,
a point is specified as $(x, y)$ , where

$x$ is the distance from the origin along the $x$ -axis, called the abscissa,
$y$ is the distance from the origin along the $y$ -axis, called the ordinate,

Visually,

\documentclass[tikz,margin={2cm 0cm}]{standalone}
\usepackage{tkz-euclide}
%
\definecolor{xcolor}{HTML}{45CF6E}
\definecolor{xpcolor}{HTML}{53DFDD}
\definecolor{ycolor}{HTML}{FB464C}
\definecolor{ypcolor}{HTML}{FB99CD}
%
\begin{document}
\begin{tikzpicture}
    \tkzDefPoint(0,0){O}
    \tkzDefPoint(1,0){X}
    \tkzDefPoint(0,1){Y}
    \tkzDefPoint(1,1){P}
    %
    \tkzMarkRightAngle(X,O,Y)
	\tkzDrawLine[xcolor,thick,Latex-Latex,add=1 and 1](O,X)
	\tkzDrawLine[ycolor,thick,Latex-Latex,add=1 and 1](O,Y)
    %
    \tkzDrawSegment[ypcolor,dashed](X,P)
	    \tkzLabelLine[ypcolor,pos=0.5,right](X,P){$y$}
    \tkzDrawSegment[xpcolor,dashed](Y,P)
	    \tkzLabelLine[xpcolor,pos=0.5,above](Y,P){$x$}
    %
    \tkzDrawPoints(O,P)
    \tkzLabelPoint[below left](O){$O$}
    \tkzLabelPoint[above right](P){$(x,y)$}
\end{tikzpicture}
\end{document}

The Cartesian coordinate system generalizes to an arbitrary number of dimensions via the conventions of a vector space.

In 3D, the $z$ -component is called the applicate.

The two axes divide the plane into four regions, called quadrants:

\documentclass[tikz,margin={2cm 0cm}]{standalone}
\usepackage{tkz-euclide}
%
\definecolor{xcolor}{HTML}{45CF6E}
\definecolor{ycolor}{HTML}{FB464C}
%
\begin{document}
\begin{tikzpicture}
    \tkzDefPoint(0,0){O}
    \tkzDefPoint(1,0){X}
    \tkzDefPoint(0,1){Y}
    %
    %\tkzMarkRightAngle(X,O,Y)
	\tkzDrawLine[xcolor,thick,Latex-Latex,add=1.5 and 0.5](O,X)
	\tkzDrawLine[ycolor,thick,Latex-Latex,add=1.5 and 0.5](O,Y)
    %
    \tkzDrawPoints(O)
    %
    \tkzDefPoint(1,1){A}
	    \tkzLabelPoint[below left](A){$I$}
	\tkzDefPoint(-1,1){B}
		\tkzLabelPoint[below right](B){$II$}
	\tkzDefPoint(-1,-1){C}
		\tkzLabelPoint[above right](C){$III$}
	\tkzDefPoint(1,-1){D}
		\tkzLabelPoint[above left](D){$IV$}
\end{tikzpicture}
\end{document}

Cartesian coordinates are named for their inventor Rene Descartes.

Historically, they were the first connection of geometric shapes to algebraic equations.
For example, we might express a circle as the set of points

{(x, y) \in X \times Y : x^{2} + y^{2} = r^{2}}

Visually,

\documentclass[tikz,margin={4cm 0cm}]{standalone}
\usepackage{tkz-euclide}
%
\definecolor{xcolor}{HTML}{45CF6E}
\definecolor{xpcolor}{HTML}{53DFDD}
\definecolor{ycolor}{HTML}{FB464C}
\definecolor{ypcolor}{HTML}{FB99CD}
%
\begin{document}
\begin{tikzpicture}[scale=1.4]
    \tkzDefPoint(0,0){O}
    \tkzDefPoint(1,0){X}
    \tkzDefPoint(0,1){Y}
    \tkzDefPoint(1,1){R}
	%
	\tkzMarkRightAngle(R,X,O)
	\tkzDrawLine[xcolor,thick,Latex-Latex,add=2 and 1](O,X)
	\tkzDrawLine[ycolor,thick,Latex-Latex,add=2 and 1](O,Y)
	%
	\tkzDrawSegment[ypcolor,dashed](X,R)
	    \tkzLabelLine[ypcolor,pos=0.5,right](X,R){$y$}
    \tkzDrawSegment[xpcolor,dashed](Y,R)
	    \tkzLabelLine[xpcolor,pos=0.5,above](Y,R){$x$}
	%
    \tkzDrawCircle(O,R)
    \tkzDrawSegment[dashed,gray,thick](O,R)
    \tkzDrawCircle[black,thick](O,R)
    \tkzDrawPoint(O)
    %
    \tkzLabelSegment(O,R){$r$}
\end{tikzpicture}
\end{document}