Peirce Quincuncial

The Peirce Quincuncial projection is a conformal map projection that transforms the circle of the northern hemisphere into a square, and the southern hemisphere split into four triangles arranged around the square to form a quincunx. The resulting projection is a regular diamond shape or can be rotated to form a square. The resulting tile can be infinitely tessellated. Though this implementation defaults to a central meridian of 0, it is more common to use a central meridian of around 25 to optimise the distortions. Peirce’s original published map from 1879 used a central meridian of approx -70. The diamond and square versions can be produced using the +shape=diamond and +shape=square options respectively. This implementation includes an alternative lateral projection which places hemispheres side-by-side (+shape=horizontal or +shape=vertical). Combined with a general oblique transformation, this can be used to produced a Grieger Triptychial projection (see example below).

Classification

Miscellaneous

Available forms

Forward spherical projection

Defined area

Global

Alias

peirce_q

Domain

2D

Input type

Geodetic coordinates

Output type

Projected coordinates

Peirce Quincuncial (Square)

proj-string: +proj=peirce_q +lon_0=25 +shape=square

Peirce Quincuncial (Diamond)

proj-string: +proj=peirce_q +lon_0=25 +shape=diamond

Peirce Quincuncial (Horizontal)

proj-string: +proj=peirce_q +lon_0=25 +shape=horizontal

Grieger Triptychial

proj-string: +proj=pipeline +step +proj=ob_tran +o_proj=peirce_q +o_lat_p=-45 +o_lon_p=45 +o_type=horizontal +o_scrollx=-0.25 +step +proj=affine +s11=-1 +s12=0 +s21=0 +s22=-1

Parameters

Note

All parameters are 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.

+shape=square/diamond/horizontal/vertical/nhemisphere/shemisphere

New in version 9.0.

Defaults to diamond.

Warning

This option was wrongly introduced introduced in 8.2.1 with the type name, which was inappropriate as it conflicted with the +type=crs general hint.

Indicates the shape of transformation applied to the southern hemisphere: square and diamond represent the traditional quincuncial form suggested by Peirce with the southern hemisphere divided into 4 triangles and reflected outward from the northern hemisphere. The square shape is rotated by 45 degrees to produce the conventional square presentation. The origin lies at the centre of the square or diamond.

By contrast, the horizontal and vertical forms reflect the southern hemisphere laterally across the x or y axis respectively to produce a rectangular form. The origin lies at the centre of the rectangle.

The other two types, nhemisphere and shemisphere, discard latitudes of less than 0 or more than 0, respectively, to allow single hemispheres to be selected. The origin lies at the centre of the square or diamond.

+scrollx=<value>

For horizontal shape allows a scalar circular scroll of resulting x coordinates to shift sections of the projection to the other horizontal side of the map.

Defaults to 0.0. Must be a scale between -1.0 and 1.0.

+scrolly=<value>

For vertical shape allows a scalar circular scroll of resulting y coordinates to shift sections of the projection to the other vertical side of the map.

Defaults to 0.0. Must be a scale between -1.0 and 1.0.

+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.