More Geospatial Terminology Confusion

I was recently asked to clarify the difference between a coordinate system and a map projection.  This can be confusing, especially because some geospatial software (e.g., ENVI) handles unprojected coordinate systems exactly the same way that it handles map projections, without distinguishing between the two.

The geographic coordinate system.  From R. Knippers.

The following are some definitions that I hope will help to clarify this situation:

A coordinate system is any fixed reference framework superimposed onto the surface of an area and used to designate the location of features within it.  There can be three-dimensional geographic coordinates systems and two-dimensional projected coordinate systems.

A geographic coordinate system is a three dimensional reference framework with which objects on the earth’s surface are located.  A geographic coordinate system includes a datum, spheroid, units of measure and a prime meridian.  Geographic coordinate systems typically use degrees of latitude and longitude as the units of measure.

A spheroid or ellipsoid is a (technically oblate-spheroidal) model of the earth’s shape.

A datum is a practical application of a spheroid, linking the spheroid to a particular portion of the earth’s surface.  It is necessary, when defining a geographic or projected coordinate system, to specify exactly where that system considers the earth’s surface to be.  Note that the latitude and longitude of a location on the earth will change depending on which datum is used for the measurement.

A prime meridian is the line of longitude at which we consider the longitude to be zero degrees.  The prime meridian is arbitrary.  An international conference in 1884 decided that the modern Prime Meridian passes through the Royal Observatory, Greenwich in southeast London, as well as the north pole and the south pole.

A map projection, or projected coordinate system is a two-dimensional reference framework onto which the three-dimensional earth’s surface is projected.  This is necessary if we wish to show the earth’s surface in a two-dimensional representation, such as a map sheet or a computer screen.  Representing the earth's surface in two dimensions using a map projection causes distortion in the shape, area, distance, or direction of the data.  It may help to understand this if you imagine taking the peel from an orange, and spreading it onto a flat paper.  There will be gaps where the peel has to split to become flat. A projected coordinate system is always based on a geographic coordinate system, which is in turn based on a spheroid.  In a projected coordinate system, locations are identified by x,y coordinates on a grid. Each position has two values that reference it to the origin of the grid. One specifies its horizontal position and the other its vertical position. The two values are often called the easting (x coordinate) and northing (y coordinate).  There are lots of different types of map projections.

An excerpt from "What your favorite map projection says about you" by Randall Munroe.

What other explanations have you found helpful in understanding the relationships between types of coordinate systems that can be used to specify locations on the earth?