Definition:Cartesian 3-Space/Orientation/Left-Handed

Definition


Consider a Cartesian $3$-Space.

Let the $x$-axis, $y$-axis and $z$-axis be defined.

Let a point $P$ be identified on the $x$-axis, different from $O$, with the coordinate pair $\tuple {1, 0}$ in the $x$-$y$ plane.

Let the point $P'$ be identified on the $y$-axis such that $OP' = OP$.

It remains to identify the point $P$ on the $z$-axis such that $OP = OP$.

The orientation of the $z$-axis is determined by the position of $P''$ relative to $O$.

The Cartesian $3$-Space is defined as left-handed when $P''$ is located as follows.

Let the coordinate axes be oriented as follows:

Imagine being positioned, standing on the $x$-$y$ plane at $O$, and facing along the $x$-axis towards $P$, with $P'$ on the left.

Then $P''$ is then one unit below the $x$-$y$ plane.

Hence, let the coordinate axes be oriented as follows:


 * Let the $x$-axis increase from West to East.
 * Let the $y$-axis increase from South to North.

Then the $z$-axis increases from above to below.

Simiarly, let the $x$-$y$ plane be identified with the plane of the page or screen such aligned perpendicular to the line of sight such that:


 * the $x$-axis increases from left to right.
 * the $y$-axis increases from bottom to top.

Then the $z$-axis increases from in front to behind (that is, from closer in to further away).