Definitions

# Transverse Mercator projection

The transverse Mercator projection is an adaptation of the Mercator projection. Both projections are cylindrical and conformal. However, in the transverse Mercator, the cylinder is rotated 90° (transverse) relative to the equator so that the projected surface is aligned to a central meridian rather than to the equator, as is the case with the equatorial Mercator projection.

The projection introduces little distortion in the narrow region close to the tangent or secant points on the model globe. Scale 5° away from the central meridian is less than 0.4% greater than scale at the central meridian, and is about 1.54% at an angular distance of 10°. This low level of distortion, combined with the conformal property which it inherits from the Mercator projection, make the transverse Mercator projection ideal for mapping regions with a narrow longitudinal extent, such as Chile.

## Forms of the transverse Mercator projection

In constructing a map on any projection, a sphere is normally chosen to model the earth when the extent of the mapped region exceeds a few hundred kilometers in length in both dimensions. For maps of smaller regions, an ellipsoidal model is chosen if greater accuracy is required. The transverse Mercator projection comes in both forms. Work has also progressed on forms for irregular celestial bodies.

Regardless of the developing model, the transverse Mercator projection is characterized by two conditions: the map is conformal, and distances along the central meridian are proportionally correct. That is, the scale is constant along the central meridian. The projected surface can be tangent to the model of the Earth, which produces a map that is true to scale along this line. Or, the scale factor can be reduced in order to balance out the distortion over the mapped region. In this secant case, two paths of true scale flank the central meridian. These paths are straight lines that run parallel to the central meridian in the spherical model. They are curves in the ellipsoidal model, approximately straight and parallel when sufficiently close to the central meridian.

### Spherical

The spherical form of the transverse Mercator projection was presented by Johann Heinrich Lambert in 1772. Distortion of scale increases entirely as a function of distance from the central meridian, approaching infinite as the extent of the map approaches the entire sphere. Thus the entire sphere cannot be shown. The spherical version sees some use, but its utility is often exceeded by other considerations in small-scale mapping, such as preserving area.

### Ellipsoidal

The ellipsoidal transverse Mercator, developed from an ellipsoidal model of the Earth, was presented by mathematician Carl Friedrich Gauss (Gauß) in 1822 and further analyzed by L. Krüger in the early 20th century. In Europe, the ellipsoidal form is sometimes referred to as the Gauss-Krüger or Gauss conformal projection. Its distortion is a function of latitude, longitude, and eccentricity of the ellipsoid, rather than just distance away from the central meridian. While the complete spherical form is infinite in extent, the ellipsoidal form is finite.

The ellipsoidal form with a reduced scale factor has been the most widely used projection in geodetic mapping since the mid twentieth century. It is employed by most national mapping systems. The UTM, for example, uses the secant case, applying a scale factor of 0.9996 along the central meridian.

The ellipsoidal form is computationally difficult to work with: the condition of constant scale implies that the vertical (or northing) coordinate on the central meridian is proportional to rectifying latitude, whereas the conformal condition means that it is convenient to start with conformal latitude. Since these notions of latitude do not coincide, there is no simple way to compute the Cartesian coordinates of a point on the ellipsoidal transverse Mercator projection from its geodetic coordinates. Approximate formulas are generally found in form of convergent power series as a function of the Earth's eccentricity and distance from the central meridian. Exact formulas, involving reciprocal functions and complex incomplete elliptic integrals, could be given by extending to the complex domain the transformation of conformal latitude to rectifying latitude, but they are typically useful only for deriving the power series for numerical computations.