For almost 500 years, it has been conclusively established that the Earth is essentially a sphere, although there were a number of intellectuals nearly 2,000 years earlier who were convinced of this. Even to the scholars who considered the Earth flat, the skies appeared hemispherical, however. It was established at an early date that attempts to prepare a flat map of a surface curving in all directions leads to distortion of one form or another.

A map projection is a device for reproducing all or part of a round body on a flat sheet. Since this cannot be done without distortion, the cartographer must choose the characteristic that is to be shown accurately at the expense of others, or a compromise of several characteristics. There is literally an infinite number of ways in which this can be done, and several hundred projections have been published, most of which are rarely used novelties. Most projections may be infinitely varied by choosing different points on the Earth as the center or as a starting point. Manifold, for example, allows the setting of the projection’s center point for most projections.

It cannot be said that there is one "best" projection for mapping. It is even risky to claim that one has found the "best" projection for a given application, unless the parameters chosen are artificially constricting. Even a carefully constructed globe is not the best map for most applications because its scale is by necessity too small, a straightedge cannot be satisfactorily used on it for measurement of distance, and it is awkward to use in general.

The characteristics normally considered in choosing a map projection are as follows:

**Area** - Many map projections
are designed to be **equal-area**,
so that a coin, for example, on one part of the map covers exactly the
same area of the actual Earth as the same coin on any other part of the
map. Shapes, angles, and scale must be distorted on most parts of such
a map, but there are usually some parts of an equal-area map which are
designed to retain these characteristics correctly, or very nearly so.
Less common terms used for equal-area projections are **equivalent**,
**homolographic**, **authalic**,
and **equiareal**.

**Shape** - Many of the most common
and most important projections are **conformal**
or **orthomorphic**, in that normally
the shape of every small feature of the map is shown correctly. On a conformal
map of the entire Earth there are usually one or more "singular"
points at which shape is still distorted. A large landmass must still
be shown distorted in shape, even though its small features are shaped
correctly. An important result of conformality is that relative angles
at each point are correct, and the local scale in every direction around
any one point is constant. Consequently, meridians intersect parallels
at right (90 degree) angles on a conformal projection, just as they do
on the Earth. Areas are generally enlarged or reduced throughout the map,
but they are relatively correct along certain lines, depending on the
projection. Nearly all large-scale maps of the Geological Survey and other
mapping agencies throughout the world are now prepared on a conformal
projection.

**Scale** - No map projection shows
scale correctly throughout the map, but there are usually one or more
lines on the map along which the scale remains true. By choosing the locations
of these lines properly, the scale errors elsewhere may be minimized,
although some errors may still be large, depending on the size of the
area being mapped and the projection. Some projections show true scale
between one or two points and every other point on the map, or along every
meridian. They are called **equidistant**
projections.

**Direction** - While conformal
maps give the relative local directions correctly at any given point,
there is one frequently used group of map projections, called **azimuthal**
or **zenithal**, on which the directions
or azimuths of all points on the map are shown correctly with respect
to the center. One of these projections is also equal-area, another is
conformal, and another is equidistant. There are also projections on which
directions from two points are correct, or on which directions from all
points to one or two selected points are correct, but these are rarely
used.

**Special Characteristics** - Several
map projections provide special characteristics that no other projection
provides. On the Mercator projection, all rhumb lines, or lines of constant
direction, are shown as straight lines. On the Gnomonic projection, all
great circle paths - the shortest routes between points on a sphere -
are shown as straight lines. On the Stereographic, all small circles,
as well as great circles, are shown as circles on the map. Some newer
projections are specially designed for satellite mapping. Less useful
but mathematically intriguing projections have been designed to fit the
sphere conformally into a square, an ellipse, a triangle, or some other
geometric figure.

**Method of Construction** - In
the days before ready access to computers, ease of construction was of
greater importance. Some projections have become popular simply because
they are easy to compute. With the advent of computers, very complicated
formulas can be handled as routinely as simple projections in the past.

While the above features should ordinarily be considered in choosing a map projection, they are not so obvious in recognizing a projection. In fact, if the region shown on a map is no much larger than the United States, for example, even a trained eye cannot often distinguish whether the map is equal-area or conformal. It is necessary to make measurements to detect small differences in spacing or location of meridians and parallels, or to make other tests. The type of construction of the map projection is more easily recognized with experience, if the projection falls into one of the common categories.

A **developable** surface is one
that can be transformed to a plane without distortion. There are three
types of developable surfaces onto which most of the map projections used
by USGS and other agencies are at least partially geometrically projected.
They are the **cylinder**, the **cone**, and the **plane**.
Actually all three are variations of the cone. A cylinder is a limiting
form of a cone with an increasingly sharp point or apex (i.e., drawn out
to infinity). As the cone becomes flatter, its limit is a plane.

**Regular Cylindrical Projection**
- If a cylinder is wrapped around the globe representing the Earth, so
that its surface touches the Equator throughout its circumference, the
meridians of longitude may be projected onto the cylinder as equidistant
straight lines perpendicular to the Equator, and the parallels of latitude
marked as lines parallels to the Equator, around the circumference of
the cylinder and mathematically spaced for certain characteristics. When
the cylinder is cut along some meridian and unrolled, a cylindrical projection
with straight meridians and straight parallels results. The Mercator projection
is the best-known example.

**Regular Conic Projection** - If
a cone is placed over the globe, with its peak or apex along the polar
axis of the Earth and with the surface of the cone touching the globe
along some particular parallel of latitude, a conic (or conical) projection
can be produced. This time the meridians are projected onto the cone as
equidistant straight lines radiating from the apex, and the parallels
are marked as lines around the circumference of the cone in planes perpendicular
to the Earth’s axis, spaced for the desired characteristics.

When the cone is cut along a meridian, unrolled, and laid flat, the meridians remain straight radiating lines, but the parallels are now circular arcs centered on the apex. The angles between meridians are shown smaller than the true angles.

**Polar Azimuthal Projection** -
A plane tangent to one of the Earth’s poles is the basis for polar azimuthal
projections. In this case, the group of projections is named for the function,
not the plane, since all common tangent-plane projections of the sphere
are azimuthal. The meridians are projected as straight lines radiating
from a point, but they are spaced at their true angles instead of the
smaller angles of the conic projections. The parallels of latitude are
complete circles, centered on the pole.

On some important azimuthal projections, such as the Stereographic (for the sphere) the parallels are geometrically projected from a common point of perspective; on others, such as the Azimuthal Equidistant, they are non-perspective.

The concepts outlined above may be modified in two ways, which still provide cylindrical, conic, or azimuthal projections (although the azimuthals retain this property precisely only for the sphere, not for ellipsoidal Earth models):

- The cylinder or cone may be secant to or cut the globe at two parallels
instead of being tangent to just one. This conceptually provides
**two standard parallels**(as can be specified in some Manifold projections); but for most conic projections this construction is not geometrically correct. The plane may likewise cut through the globe at any parallel instead of touching a pole. Those Manifold projections which allow secant projection surfaces will allow the setting of additional**standard parallels**beyond what is required for the simple tangent form of the projection. - The axis of the cylinder or cone can have a direction different from that of the Earth’s axis, while the plane may be tangent to a point other than a pole. This type of modification leads to important oblique, transverse and Equatorial projections, in which most meridians and parallels are no longer straight lines or arcs of circles. What were standard parallels in the normal orientation now become standard lines not following parallels of latitude.

Transverse Cylindrical Projection

Oblique Cylindrical Projection

Oblique Azimuthal

Some other projections in common use resemble one or another of these
categories only in some respects. The **Sinusoidal**
projection is called **pseudocylindrical**
because its latitude lines are parallel and straight, but its meridians
are curved. The **Polyconic** projection
is projected onto cones tangent to each parallel of latitude, so the meridians
are curved, not straight. Still others are more remotely related to cylindrical,
conic, or azimuthal projections, if at all.

Manifold includes a vast array of different projections that are named
mostly in accordance as described in US Geological Survey bulletins, which
tend to follow international cartographic practise for the names of the
most common projections. In addition, **EPSG**
codes are used to name projections.

Many of the "standard" projections allow the use of various projection parameters as described above. Some countries have standardized on the use of a particular projection for mapping their countries that has acquired a local name when used with the locally-preferred set of parameters. Manifold includes these alternate names as choices for many of the more well known national projections.

Datum names used within Manifold originate primarily either with **EPSG** or with the U.S. National Geospatial
Administration (formerly the National Imagery and Mapping Agency) the
official keeper of such data for the U.S. government. Datum names
local to other countries are usually taken from the analogous government
organizations in those countries.