A map projection constructed in the usual way people are taught map projections are constructed. This writeup is a general discussion of the topic; for a list of projections see map projection.

When people first started realizing that the Earth was round, they also realized the difficulties this would create for creating maps.  There being no algebra or calculus at the time, the ancients had to make do with geometry.   The idea was to project the Earth's surface onto a shape that could be "developed" (i.e. cut up an rolled out without any distortion).

There are really only three such shapes:

However, geometry can only go so far.   Although it's possible to choose a point of perspective to directly project the Earth's surface onto one of these three shapes and thus create some map projections,  many others cannot.  However, the idea of projection onto a surface is a good way to learn.

A map projection created by projecting the Earth's surface onto a plane has a peculiar property:  There is a "central" point where all angles using that point (aka azimuths from that point) as the vertex are correct!   For this reason, such a projection is called an azimuthal projection.

A projection created by projecting onto a cone is a conic projection; a projection created by projecting onto a cylinder is a cylindrical projection.

If you project the Earth's surface onto a plane, cone, or cylinder, most of the projected map is going to be distorted.  However, a few points will not.

If you place a cone on top of a sphere, it will touch the sphere at a circle.  Since this circle is identical on both the sphere and the cone, the corresponding circular arc on the map will be accurate.   When applied to a map, such an arc is called a "standard line" or a "standard parallel".

However, the cone (plane, cylinder) can be embedded in the cylinder.  This means that the cone and the sphere intersect at two circles (one circle for a plane).   This gives us two standard lines to work with; not only that, we can adjust the standard lines on a given map so that the areas of less distortion cover as much of the map as possible.

A projection with one standard line (standard point for azimuthal projections) is called a tangent projection because the surface is tangent to the sphere.

A projection with two standard lines (standard circle for azimuthal projections) is called a secant projection because cross sections of the cone-sphere combination show a circle with a secant line going through it.

The formulae for each of these projections has two steps:

The most complicated part of projection takes a point's latitude and converts it into the polar-coordinate radius.  The polar angle is a linear transformation of longitude.   When you project the entire surface of the Earth, all or part of a disk or ring appears. The proportion of a disk that appears is known as the projection's "cone constant" (which is also the coefficient in the linear transformation of longitude).
• An azimuthal projection has a cone constant of 1 and shows a whole disk.
• A conic projection has a cone constant between 0 and 1.
• A cylindrical projection has a cone constant of 0.
All right, I've misled you a bit here: A cone constant of 0 makes using the formulae for conic projections impossible.  Special formulae exist to directly calculate rectangular coordinates for cylindrical projections.  However, the plane and the cylinder can be considered limiting forms of the cone (as its cone constant changes), and the formulae for a given cylindrical projection are limiting forms of the more general formulas for the corresponding conic projection, given a cone constant of 0.

A projection's cone constant is dependent upon the basic property the map projection is designed for, and the one or two "standard lines" that cartographer has selected.

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