# Quartic authalic

## Description

The quartic authalic is a pseudocylindrical equal-area projection, created by modifying the Lambert azimuthal equal-area projection. The boundary meridians quite excessively bulge outward, producing considerable shape distortion near the map outline. The McBryde-Thomas flat-polar quartic projection is based on this projection.

The quartic authalic projection was independently presented by Karl Siemon in 1937 and Oscar S. Adams in 1945. The equations for an ellipsoid were developed at Esri. It is available in ArcGIS Pro 1.0 and later and in ArcGIS Desktop 8.0 and later.

## Projection properties

The subsections below describe the quartic authalic projection properties.

### Graticule

Quartic authalic is pseudocylindric projection. The equator and the central meridian are projected as straight lines, where the projected equator is about 2.22144 times as long as the central meridian. The other meridians are fourth-order algebraic curves, bulging away the central meridian and equally spaced. The parallels are unequally spaced like they are on the Lambert azimuthal equal-area projection in an equatorial aspect. All parallels are straight lines perpendicular to the central meridian. The poles are represented as points and they appear as protruding edges. The outline is similar to the sinusoidal projection. The graticule is symmetric across the equator and the central meridian.

### Distortion

Quartic authalic is an equal-area (equivalent) projection. Shapes, directions, angles, and distances are generally distorted. Scale is true along the equator and constant along any other given latitude. Distortions are moderate near the projection center. Bulging meridians produce considerable distortion toward the edge of the projection. Distortion values are symmetric across the equator and the central meridian.

## Usage

The quartic authalic projection is appropriate for thematic world maps although its use is not advisable.

## Variants

There are two variants available in ArcGIS:

• Quartic authalic uses the semimajor axis for the radius and equations for a sphere. It only supports spherical Earth models. It is available in ArcGIS Pro 1.0 and later and in ArcGIS Desktop 8.0 and later.
• Quartic authalic ellipsoidal supports spherical and ellipsoidal Earth models. It is available in ArcGIS Pro 1.2 and later and in ArcGIS Desktop 10.4 and later.

## Limitations

Only the quartic authalic ellipsoidal version supports projection for ellipsoids. The quartic authalic version does not maintain areas when an ellipsoid is used.

## Parameters

Quartic authalic parameters are as follows:

• False Easting
• False Northing
• Central Meridian

Quartic authalic ellipsoidal parameters are as follows:

• False Easting
• False Northing
• Central Meridian
• Latitude Of Origin

## Sources

Bugayevskiy, L. M. and Snyder, J. P. (1995). Map Projections: A Reference Manual. London: Taylor & Francis.

Snyder, J. P. (1993). Flattening the Earth. Two Thousand Years of Map Projections. Chicago and London: University of Chicago Press.

Snyder, J. P. and Voxland, P. M. (1989). An Album of Map Projections. U.S. Geological Survey Professional Paper 1453. Washington, DC: United States Government Printing Office.