English: In PGA, all elements are transformations, and all can be helpfully considered by visualizing planes that could be used to construct them.

Individual planes are the basic elements. Lines can be constructed from two intersecting planes, and points from three intersecting planes.

Adapted with permission from a picture made by Steven De Keninck.

Planes also perform planar reflections, and all transformations in PGA can be created by composing these planar reflections. Translations can be made from composing two reflections in parallel planes; rotations from two planes at an angle; rotoreflections and transflections with three; and screw motions with four planar reflections.

A line is actually a special case of a rotation - it is one where the two planes are orthogonal. A line is thus able to perform a "line reflection", eg a reflection of everything through itself, which has the same effect as a 180 degree rotation around itself. The same applies to lines. Thus any line can be visualized as two orthogonal planes, and any point as three orthogonal planes.