Point

Def

A point is the abstract specification of an exact position in a space. It can be seen as a zero dimensional object - so a pure geometric line segment has zero width and zero length.

Axiom

There is exactly one straight Line that passes through two distinct points.

Diagram showing a bunch of points on a cartesian plane:

\begin{document}
\begin{tikzpicture}[scale=2]
    % Draw the axes
    \draw[thick] (-2,0) -- (2,0);
    \draw[thick] (0,-2) -- (0,2);
    
    % Add minimal labels
    \foreach \x in {-2,-1,1,2}
        \draw (\x,0.1) -- (\x,0) node[below, black!40] {\x};
    \foreach \x in {-2,-1,1,2}
        \draw (0.1,\x) -- (0,\x) node[left, black!40] {\x};
    
    % Random points (coordinates approximated from the image)
    \foreach \point in {
        (-1.8,0.8), (-1.6,-1.2), (-1.4,0.4), (-1.2,-0.8),
        (-1,1.2), (-0.8,-1.4), (-0.6,0.6), (-0.4,-0.4),
        (-0.2,1.6), (0,0), (0.2,-1.6), (0.4,1.4),
        (0.6,-0.6), (0.8,1.8), (1,-1.2), (1.2,0.8),
        (1.4,-0.8), (1.6,1.2), (1.8,-1.4), (-1.7,-0.7),
        (-1.3,1.5), (-0.9,-1.1), (-0.5,1.7), (-0.1,-1.3),
        (0.3,1.3), (0.7,-0.9), (1.1,1.1), (1.5,-0.5),
        (1.9,0.9), (-1.9,-1.5), (-1.5,1.9), (-1.1,-1.7),
        (-0.7,1.1), (-0.3,-1.9), (0.1,0.7), (0.5,-1.5),
        (0.9,1.5), (1.3,-1.1), (1.7,0.5)
    } {
        \fill[red] \point circle (0.5pt);
    }
\end{tikzpicture}
\end{document}

Coordinates and role in geometry

A point becomes calculable once a coordinate system is chosen. On a Line, one number is enough; in a Plane, two numbers such as $(x,y)$ are needed; in Space (3D), three numbers such as $(x,y,z)$ are needed. The point itself is not the coordinate tuple: coordinates are labels that depend on the chosen axes and origin.

Points are the atoms of elementary geometry. A curve can be thought of as a continuous set of points, a surface as a two-dimensional collection of points, and a solid region as a three-dimensional collection of points. Distances between points require a metric, such as the Euclidean distance in Euclidean spaces.

In physics, a particle’s position at an instant is represented by a point or position vector. Motion then becomes a path through points, and Velocity describes how that position changes with time.