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 conesphere 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 polarcoordinate 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.