Definition:Coordinate System/3-Space

Definition

In physics and applied mathematics, it is usual for the coordinate system under discussion to be considered as superimposed on ordinary space of $3$ dimensions.

Cartesian

The special case of Cartesian $3$-space is usually considered separately.

Every point in ordinary $3$-space can be identified uniquely by means of an ordered triple of real coordinates $\tuple {x, y, z}$, as follows:

Identify one distinct point in space as the origin $O$.

Let $3$ distinct planes be constructed through $O$ such that all are perpendicular.

Each pair of these $3$ planes intersect in a straight line that passes through $O$.

Let $X$, $Y$ and $Z$ be points, other than $O$, one on each of these $3$ lines of intersection.

Then the lines $OX$, $OY$ and $OZ$ are named the $x$-axis, $y$-axis and $z$-axis respectively.

Select a point $P$ on the $x$-axis different from $O$.

Let $P$ be identified with the coordinate pair $\tuple {1, 0}$ in the $x$-$y$ plane.

Identify the point $P'$ on the $y$-axis such that $OP' = OP$.

Identify the point $P$ on the $z$-axis such that $OP = OP$.

Curvilinear

Recall that we can define Cartesian $3$-space by means of perpendicular planes.

Let us superimpose onto this coordinate system $3$ other one-parameter families of surfaces.

In each of these families, the surfaces need not be parallel and they need not be planes.

These families are such that every point in (ordinary) $3$-dimensional space can then be uniquely identified as the intersection of $3$ such surfaces: one surface from each family.

Hence every point is identified as the ordered triple $\tuple {q_1, q_2, q_3}$, where $q_1$, $q_2$ and $q_3$ are the parameters of each of the families.

Hence we can describe a curvilinear coordinate system.