Central Conic

New in version 5.0.0.

This is central (centrographic) projection on cone tangent at :option:lat_1 latitude, identical with conic() projection from mapproj R package.

Classification

Conic

Available forms

Forward and inverse, spherical projection

Defined area

Global, but best used near the standard parallel

Alias

ccon

Domain

2D

Input type

Geodetic coordinates

Output type

Projected coordinates

Central Conic

proj-string: +proj=ccon +lat_1=52 +lon_0=19

Usage

This simple projection is rarely used, as it is not equidistant, equal-area, nor conformal.

An example of usage (and the main reason to implement this projection in proj4) is the ATPOL geobotanical grid of Poland, developed in Institute of Botany, Jagiellonian University, Krakow, Poland in 1970s [Zajac1978]. The grid was originally handwritten on paper maps and further copied by hand. The projection (together with strange Earth radius) was chosen by its creators as the compromise fit to existing maps during first software development in DOS era. Many years later it is still de facto standard grid in Polish geobotanical research.

The ATPOL coordinates can be achieved with with the following parameters:

+proj=ccon +lat_1=52 +lon_0=19 +axis=esu +a=6390000 +x_0=330000 +y_0=-350000

For more information see [Komsta2016] and [Verey2017].

Parameters

Required

+lat_1=<value>

Standard parallel of projection.

Optional

+lon_0=<value>

Longitude of projection center.

Defaults to 0.0.

Note

The default convention is to interpret this value as decimal degrees. To specify radians instead, follow the value with the “r” character.

Example: +lon_0=1.5708r

See Projection Units for more information.

+R=<value>

Radius of the sphere, given in meters. If used in conjunction with +ellps, +R takes precedence.

See Ellipsoid size parameters for more information.

+x_0=<value>

False easting.

Defaults to 0.0.

+y_0=<value>

False northing.

Defaults to 0.0.

Mathematical definition

Forward projection

\[r = \cot \phi_0 - \tan (\phi - \phi_0)\]
\[x = r \sin (\lambda\sin\phi_0)\]
\[y = \cot \phi_0 - r \cos (\lambda\sin\phi_0)\]

Inverse projection

\[y = \cot \phi_0 - y\]
\[\phi = \phi_0 - \tan^{-1} ( \sqrt{x^2+y^2} - \cot \phi_0 )\]
\[\lambda = \frac{\tan^{-1} \sqrt{x^2+y^2}}{\sin \phi_0}\]

Reference values

For ATPOL to WGS84 test, run the following script:

#!/bin/bash
cat << EOF | src/cs2cs -v -f "%E" +proj=ccon +lat_1=52 +lat_0=52 +lon_0=19 +axis=esu +a=6390000 +x_0=330000 +y_0=-350000 +to +proj=longlat
0 0
0 700000
700000 0
700000 700000
330000 350000
EOF

It should result with

1.384023E+01 5.503040E+01 0.000000E+00
1.451445E+01 4.877385E+01 0.000000E+00
2.478271E+01 5.500352E+01 0.000000E+00
2.402761E+01 4.875048E+01 0.000000E+00
1.900000E+01 5.200000E+01 0.000000E+00

Analogous script can be run for reverse test:

cat << EOF  | src/cs2cs -v -f "%E" +proj=longlat +to +proj=ccon +lat_1=52 +lat_0=52 +lon_0=19 +axis=esu +a=6390000 +x_0=330000 +y_0=-350000
24 55
15 49
24 49
19 52
EOF

and it should give the following results:

6.500315E+05 4.106162E+03 0.000000E+00
3.707419E+04 6.768262E+05 0.000000E+00
6.960534E+05 6.722946E+05 0.000000E+00
3.300000E+05 3.500000E+05 0.000000E+00