Editing Cartesian coordinate system (section)

==Orientation and handedness==
{{Main|Orientability}}
{{See also|Right-hand rule|Axes conventions}}

===In two dimensions===
[[File:Rechte-hand-regel.jpg|left|thumb|The [[right-hand rule]] ]]

Fixing or choosing the ''x''-axis determines the ''y''-axis up to direction. Namely, the ''y''-axis is necessarily the [[perpendicular]] to the ''x''-axis through the point marked 0 on the ''x''-axis. But there is a choice of which of the two half lines on the perpendicular to designate as positive and which as negative. Each of these two choices determines a different orientation (also called ''handedness'') of the Cartesian plane.

The usual way of orienting the plane, with the positive ''x''-axis pointing right and the positive ''y''-axis pointing up (and the ''x''-axis being the "first" and the ''y''-axis the "second" axis), is considered the ''positive'' or ''standard'' orientation, also called the ''right-handed'' orientation.

A commonly used mnemonic for defining the positive orientation is the ''[[right-hand rule]]''. Placing a somewhat closed right hand on the plane with the thumb pointing up, the fingers point from the ''x''-axis to the ''y''-axis, in a positively oriented coordinate system.

The other way of orienting the plane is following the ''left-hand rule'', placing the left hand on the plane with the thumb pointing up.

When pointing the thumb away from the origin along an axis towards positive, the curvature of the fingers indicates a positive rotation along that axis.

Regardless of the rule used to orient the plane, rotating the coordinate system will preserve the orientation. Switching any one axis will reverse the orientation, but switching both will leave the orientation unchanged.

===In three dimensions===
[[File:Cartesian coordinate system handedness.svg|left|thumb|Fig. 7 – The left-handed orientation is shown on the left, and the right-handed on the right.]]
[[File:Right hand cartesian.svg|thumb|Fig. 8 – The right-handed Cartesian coordinate system indicating the coordinate planes]]

Once the ''x''- and ''y''-axes are specified, they determine the [[line (geometry)|line]] along which the ''z''-axis should lie, but there are two possible orientations for this line. The two possible coordinate systems, which result are called 'right-handed' and 'left-handed'.<ref>{{harvnb|Anton|Bivens|Davis|2021|p=[https://books.google.com/books?id=001EEAAAQBAJ&pg=PA657 657]}}</ref> The standard orientation, where the ''xy''-plane is horizontal and the ''z''-axis points up (and the ''x''- and the ''y''-axis form a positively oriented two-dimensional coordinate system in the ''xy''-plane if observed from ''above'' the ''xy''-plane) is called '''right-handed''' or '''positive'''.

[[File:3D Cartesian Coodinate Handedness.jpg|thumb|3D Cartesian coordinate handedness]]

The name derives from the [[right-hand rule]]. If the [[index finger]] of the right hand is pointed forward, the [[middle finger]] bent inward at a right angle to it, and the [[thumb]] placed at a right angle to both, the three fingers indicate the relative orientation of the ''x''-, ''y''-, and ''z''-axes in a ''right-handed'' system. The thumb indicates the ''x''-axis, the index finger the ''y''-axis and the middle finger the ''z''-axis. Conversely, if the same is done with the left hand, a left-handed system results.

Figure 7 depicts a left and a right-handed coordinate system. Because a three-dimensional object is represented on the two-dimensional screen, distortion and ambiguity result. The axis pointing downward (and to the right) is also meant to point ''towards'' the observer, whereas the "middle"-axis is meant to point ''away'' from the observer. The red circle is ''parallel'' to the horizontal ''xy''-plane and indicates rotation from the ''x''-axis to the ''y''-axis (in both cases). Hence the red arrow passes ''in front of'' the ''z''-axis.

Figure 8 is another attempt at depicting a right-handed coordinate system. Again, there is an ambiguity caused by projecting the three-dimensional coordinate system into the plane. Many observers see Figure 8 as "flipping in and out" between a [[wikt:convex|convex]] cube and a [[wikt:concave|concave]] "corner". This corresponds to the two possible orientations of the space. Seeing the figure as convex gives a left-handed coordinate system. Thus the "correct" way to view Figure 8 is to imagine the ''x''-axis as pointing ''towards'' the observer and thus seeing a concave corner.
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