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Mollweide and volcanoes

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Cartographic projections to image objects and phenomena on the whole Earth surface.



Elliptical projections with the centre on the Equator are commonly used for geological maps. But in this case the belt of volcanoes stretching along North American and East Eurasian coasts is interrupted. The outstanding specialist in geology V.V.Yarmolyuk has suggested use of nontraditional aspect of elliptical projection. It is caused by the fact that during recent years a large volume of data testifying to the prevalence of continental crust in the aquatoria of the Arctic Ocean was obtained. It has become a basis for joining North America and Eurasia within the framework of a common supercontinent with the centre in the Arctic Ocean. Normal aspect of elliptical projection does not give visual perception to the generality of these continents.

Mollweide projection (pseudocylindrical, equal-area, elliptical) in normal, transverse and oblique aspects.

Since the Earth surface for transverse and oblique aspects is interrupted not only by the meridian of the globe 180 degrees less (or more) than the central meridian but by the central meridian also it is necessary to prepare raster layers or line and area vector layers prior to transforming them to map projection. So we choose projection parameters using point layers, the layer of volcanoes for example. The layer was created in GIS GeoGraph 2.0 on the basis of data from the site http://vulcanism.ru/050507.html
While coordinate calculation we convert latitude and longitude to transverse or oblique coordinate system if necessary and use normal projection equations after.

Projection constants:
the Earth radius (6371116 m), longitude of the central meridian (east of Greenwich) in decimal degrees, latitude of the centre of projection in decimal degrees.
map centre (only if latitude of the centre of projection is equal to zero). Two variants of map centre are possible: “North Pole” and “South Pole”.

Input variables:
latitude and longitude (east of Greenwich) in decimal degrees.

Output variables:
- rectangular coordinate: distance to the right of the vertical line (Y axis)
- rectangular coordinate: distance above the horizontal line (X axis)
- are given in units of custom sphere radius.

Symbols in the following formulae:
R - the Earth radius
- - latitude of the centre of projection in radians
, if the North Pole is the centre of map, or , if the South Pole is the centre of map.
- inputted latitude converted to radians.
- inputted longitude converted to be east of the central meridian and then converted to radians.
- auxiliary coefficient for point location east or west of the central meridian .
- angular distance from the center of projection.
- azimuth as an angle measured clockwise from the north.
- latitude in the final coordinate system (normal, transverse or oblique aspects).
- longitude in the final coordinate system (normal, transverse or oblique aspects).
- approximately calculated function of latitude.
- small number used for definition of calculation interval.
- accuracy of calculation.
- small number used to rule out errors concerned with inaccuracies of calculation.

Normal aspect of projection , ().

and


Calculation of and to obtain transverse or oblique aspects.

If or , then , else

Calculation of and for

or and



If , then
If and or , then

Angular distance from the center of projection and azimuth are calculated from equations:





Since , and , we obtain:


atan2 ()

If or , then


Calculation of and for

or and



If , then
If and or , then

Angular distance from the center of projection and azimuth are calculated from equations:





Since , and , we obtain:


atan2 ()

If , then
If , then
If , Then


Calculation of rectangular coordinates in the final coordinate system





where is calculated approximately


If ,
If
else by applying the method of halving from equation: with the accuracy
If , then


References.
Соловьев М.Д. Математическая картография, М.: Недра, 1969– 288с.
Snyder John P., Philip M. Voxland. An Album of Map Projections U.S. Geological Survey professional paper 1453 (introduction by Joel L. Morrison), Washington, 1989 - 239p.

Program for calculation of rectangular coordinates

Mollweide projection
цилиндрическая проекция
longitude from to step
latitude from to step
radius
longitude of the central meridian
latitude of the centre of projection
map centre
show/hide:
volcanoes
continents
M.E. Fleis, M.M. Borisov, D.V. Bolodurin