# Polyconic projection

American polyconic projection of the world

Polyconic can refer either to a class of map projections or to a specific projection known less ambiguously as the American Polyconic. Polyconic as a class refers to those projections whose parallels are all non-concentric circular arcs, except for a straight equator, and the centers of these circles lie along a central axis. This description applies to projections in equatorial aspect.[1]

As a specific projection, the American Polyconic is conceptualized as "rolling" a cone tangent to the Earth at all parallels of latitude, instead of a single cone as in a normal conic projection. Each parallel is a circular arc of true scale. The scale is also true on the central meridian of the projection. The projection was in common use by many map-making agencies of the United States from the time of its proposal by Ferdinand Rudolph Hassler in 1825 until the middle of the 20th century.[2]

The projection is defined by:

$x = \cot(\varphi) \sin((\lambda - \lambda_0)\sin(\varphi))\,$
$y = \varphi-\varphi_0 + \cot(\varphi) (1 - \cos((\lambda - \lambda_0)\sin(\varphi)))\,$

where $\lambda$ is the longitude of the point to be projected; $\varphi$ is the latitude of the point to be projected; $\lambda_0$ is the longitude of the central meridian, and $\varphi_0$ is the latitude chosen to be the origin at $\lambda_0$. To avoid division by zero, the formulas above are extended so that if $\varphi = 0$ then $x = \lambda - \lambda_0$ and $y = 0$.