Cartesian coordinate system

A glide reflection is the composition of a reflection across a line followed by a translation in the direction of that line. It can be seen that the order of these operations does not matter (the translation can come first, followed by the reflection).

With this trick, the composition of affine transformations is obtained by multiplying the augmented matrices.

Effect of applying various 2D affine transformation matrices on a unit square (reflections are special cases of scaling)

The augmented matrix that represents the composition of two affine transformations is obtained by multiplying their augmented matrices.

Some affine transformations that are not Euclidean transformations have received specific names.

A shearing transformation will push the top of a square sideways to form a parallelogram. Horizontal shearing is defined by:

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.

Fig. 7 – The left-handed orientation is shown on the left, and the right-handed on the right.
Fig. 8 – The right-handed Cartesian coordinate system indicating the coordinate planes.

Cartesian coordinates are an abstraction that have a multitude of possible applications in the real world. However, three constructive steps are involved in superimposing coordinates on a problem application.

In engineering projects, agreement on the definition of coordinates is a crucial foundation. One cannot assume that coordinates come predefined for a novel application, so knowledge of how to erect a coordinate system where there previously was no such coordinate system is essential to applying René Descartes' thinking.

While spatial applications employ identical units along all axes, in business and scientific applications, each axis may have different units of measurement associated with it (such as kilograms, seconds, pounds, etc.). Although four- and higher-dimensional spaces are difficult to visualize, the algebra of Cartesian coordinates can be extended relatively easily to four or more variables, so that certain calculations involving many variables can be done. (This sort of algebraic extension is what is used to define the geometry of higher-dimensional spaces.) Conversely, it is often helpful to use the geometry of Cartesian coordinates in two or three dimensions to visualize algebraic relationships between two or three of many non-spatial variables.