Hybrid Projections

Some map projections are simple (or not so simple) combinations of other projections. A projection due to Boggs, for example, is an average of the sinusoidal and Mollweide projections. Such hybrid projections can be created quite easily by reference to existing projections as shown below:

> mapcoords(Boggs,
input = [lambda,phi],
coords = [(`Maps/Sinusoidal`(lambda,phi,r) +
`Maps/Mollweide`(lambda,phi,r)) / 2],
params = [r],
view = [-180..180,-90..90,13,7,-180..180,-180..180]);

Warning: coordinates already exists, system redefined.

Warning: default information already exists, redefining.

> coordplot(Boggs);

Winkel's first projection is a combination of the sinusoidal and equirectangular projections:

> mapcoords(`Winkel I`,
input = [lambda,phi],
coords = [(`Maps/Sinusoidal`(lambda,phi,r)+`Maps/Equirectangular`(lambda,phi,r))/2],
params = [r],
view = [-180..180,-90..90,13,7,-180..180,-90.0001..90.0001]);

> changecoords(world[50],`Winkel I`, scaling=constrained);

Another hybrid projection developed by Goode and known as the Homolosine projection uses the sinusoidal projection up to a latitude of 40 degrees 44 minutes and the Mollweide projection north/south of that latitude (40/44 is the latitude at which the lengths of parallels is the same in both projections). The invocation of mapcoords needed to create this projection is shown below.

> mapcoords(Homolosine,
input = [lambda,phi],
coords = [`if`(abs(phi)<40+44/60+11.98/3600,`Maps/Sinusoidal`(lambda,phi,r),
`Maps/Mollweide`(lambda,phi,r))],
params = [r],
view = [-180..180,-90..90,13,7,-180..180,-180..180]);

Warning: coordinates already exists, system redefined.

Warning: default information already exists, redefining.

The coordinate system is

> coordplot(Homolosine,scaling=constrained);

This looks a little strange. However, it must be said that Goode himself used this projection in an interrupted form in which the shift in coordinate systems works to the advantage of the projection. The interrupted form is shown elsewhere.

A projection by G. Erdi-Krausz was used in Hungarian atlases. It is a combination of a Mollweide and flat-polar sinusoidal projections with their scales adjusted to match at the 60 degree parallel.

> mapcoords(`Erdi-Krausz I`,
input = [lambda,phi],
coords = [[sin(theta) = 0.8*sin(phi), theta = 0],
[0.96042*r*lambda*cos(theta),1.30152*r*theta]],
params = [r],
view = [-180..180,-90..90,13,7,-180..180,-90..90]):

> mapcoords(`Erdi-Krausz II`,
input = [lambda,phi],
coords = [[2*theta+sin(2*theta) = Pi*sin(phi), theta = 0],
[1.07023*r*lambda*cos(theta),r*(1.68111*sin(theta)-0.28549*'sign'(phi))]],
params = [r],
view = [-180..180,-90..90,13,7,-180..180,-90..90]):

> print(`Maps/Erdi-Krausz II`);

> mapcoords(`Erdi-Krausz`,
input = [lambda,phi],
coords = [`if`(abs(phi)<60,`Maps/Erdi-Krausz I`(lambda,phi,r),
`Maps/Erdi-Krausz II`(lambda,phi,r))],
params = [r],
view = [-180..180,-90..90,13,7,-180..180,-180..180]);

> changecoords(world[50],`Erdi-Krausz`);

>