Plane

Euclidean plane

A Euclidean plane is a Euclidean space of dimension two, denoted ${\mathbb{E}}^2$. It is a geometric space in which two real numbers are required to determine the position of each point.

A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane.

Cartesian coordinate system

This is a coordinate system which can specify each point on a plane by a pair/tuple of real numbers called coordinates. These coordinates are the signed distance relative to the origin of the axes, which is the point at which two fixed perpendicular lines intersect.

\begin{document}
\begin{tikzpicture}[scale=1.5]
    % Draw the grid
    \draw[black!80] (-3.5,-3.5) grid (3.5,3.5);
    
    % Draw axes
    \draw[black!40,thick,->] (-3.5,0) -- (3.5,0) node[right] {$x$};
    \draw[black!40,thick,->] (0,-3.5) -- (0,3.5) node[above] {$y$};
    
    % Add axis labels
    \foreach \x in {-3,-2,-1,1,2,3}
        \draw (\x,0.1) -- (\x,-0.1) node[below] {\x};
    \foreach \y in {-3,-2,-1,1,2,3}
        \draw (0.1,\y) -- (-0.1,\y) node[left] {\y};
        
    % Plot points and their coordinates
    % Origin point (0,0) in purple
    \fill[purple] (0,0) circle (2pt);
    \node[purple] at (0.5,0.3) {$(0,0)$};
    
    % Point (-3,1) in red with dashed lines
    \fill[red] (-3,1) circle (2pt);
    \draw[red,dashed] (-3,1) -- (-3,0);
    \draw[red,dashed] (-3,1) -- (0,1);
    \node[red] at (-3,1.3) {$(-3,1)$};
    
    % Point (2,3) in green with dashed lines
    \fill[green!70!black] (2,3) circle (2pt);
    \draw[green!70!black,dashed] (2,3) -- (2,0);
    \draw[green!70!black,dashed] (2,3) -- (0,3);
    \node[green!70!black] at (2,3.3) {$(2,3)$};
    
    % Point (-1.5,-2.5) in blue with dashed lines
    \fill[blue] (-1.5,-2.5) circle (2pt);
    \draw[blue,dashed] (-1.5,-2.5) -- (-1.5,0);
    \draw[blue,dashed] (-1.5,-2.5) -- (0,-2.5);
    \node[blue] at (-1.5,-2.8) {$(-1.5,-2.5)$};
\end{tikzpicture}
\end{document}

Geometry of the plane

The plane is the natural home of two-dimensional geometry: circles, triangles, polygons, graphs of functions, and regions with Area. Once axes are chosen, the displacement from $p=(x_1,y_1)$ to $q=(x_2,y_2)$ is the vector

$$q-p=(x_2-x_1,y_2-y_1),$$

and the Euclidean distance is

$$d(p,q)=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}.$$

The dot product

$$\mathbf{u}\cdot \mathbf{v}=u_x{v}_x+u_y{v}_y$$

measures alignment and gives angles through

$$\mathbf{u}\cdot \mathbf{v}=\Vert \mathbf{u}\Vert \Vert \mathbf{v}\Vert \cos \theta .$$

This links the plane to Trigonometry and to projections.

A plane is flat: triangle angles sum to $180^{\circ }$, parallel lines stay parallel, and intrinsic Curvature is zero. Curved surfaces can still look planar when zoomed in, which is the key local idea behind Manifolds. In physics, planar coordinates are often used as approximations for motion, fields, images, detector screens, or local patches of larger curved systems.