What is the relationship between map projections and planar (i.e. state plane and UTM) coordinates?

Question

What is the relationship between map projections and planar (i.e. state plane and UTM) coordinates?

Answer 1

Map projections are sets of mathematical models which transform spherical coordinates (such as latitude and longitude) to planar coordinates (x and y). In the process, data which actually lie on a sphere are projected onto a flat plane or a surface. That surface can be converted to a planar section without stretching.

Planar coordinates are added to projections to give the user a sense of distance and direction.

The state plane and UTM projections encompass both the map projection and planar coordinates.

For example, the Universal Transverse Mercator (UTM) system is based on a series of 60 transverse Mercator projections, in which different areas of the earth fall into different 6-degree zones. Within each zone, a local coordinate system is defined, in which the X-origin is located 500,000 m west of the central meridian, and the Y-origin is the south pole or the equator, depending on the hemisphere. The State Plane system also defines both projection and coordinate system.