MATHEMATICAL
CARTOGRAPHY
I.G.Plachinta
S.Ualihanov University.
dinarika@mail.ru
The given report is devoted to the solution of the
main question of cartography namely to creation of a cartographic projection
with the least distortions.
The purpose of work is development of a particular cartographical projection.
Problems:
· To consider features of available projections, to reveal defects of spatial modelling and of an arrangement of distortions.
· To develop figure for projecting which will be give easier for construction cartographic image with less distortions.
· To make calculations.
· Search of map projection possible versions.

Distortions of lengths on: 
Area
distortions 

parallels 
meridians 

Equidistant projection ¹1 
0,9956,48

12,6 
0,9981,65 
Equidistant projection ¹1 
0,953,32 
12,64 
0,951,14 
Equalarea projection ¹1 
0,9314,6 
0,352,6 
0,991,00 
Equalarea projection ¹1 
0,934,9 
0,722,91 
0,991,00 
Maps are object of research of
scientists as they store a significant volume of the information which quality
depends also on the chosen cartographic projection. However the selection of a
projection can be complicated. It is related to distortions of the basic
cartometric parameters. Therefore the given project puts
before itself the purpose to facilitate this selection through creation of more
perfect projection. The design concept is extremely simple and consists of
creating such projecting figure which gave in developing less deformed projection.
The projecting figure on a plan should be complex, that is consists of several
figures. It is a question about «cylindercone». The given figure adjoins to a sphere on three lines, accordingly gives
three parallels free of distortions.
Originality of a
projection consists in that «cylindercone» provides preservation of lengths along
meridians and parallels in different parts of a map. However, a problem of all
complex figures consists of gaps between the compound projecting figures,
arising while developing on a plane. But in this case, having applied some
mathematical proofs, these gaps can be eliminated. As a result, we receive
equidistant version ¹1 (on tangent «cylindercone»). Considering qualities of
the given projection, for reduction of distortions, it has been decided to
create equidistant version ¹2 (on secant «cylindercone»). Calculations have shown
that distortions of length and the areas have two maxima and two minima. Maxima
of distortions are on tropics (1,06 1,09) and polar lines (up to 14). Minima
where more often distortions are absent are on equator and 45 latitudes. During
the work equalarea projections have been created also. All the minimal and
maximal parameters of distortions are presented in the below table. During the
work the scientific monograph «Mathematical cartography» has been created and
provided with the copyright certificate given by Intellectual Property Committee
of the Ministry of Justice of the Republic Kazakhstan. It contains the detailed definition of the projection.
Scientific works of K.K.Grjunberg («Cartography with the Bases of Topography»), and K.A.Salishchev («Cartovedenie») have been used in the current research.
The represented project should stimulate development of a cartographic idea in Kazakhstan. Offered projections allow to look at designing of an ellipsoid in a new fashion, that is not through conventional figures (the cylinder, a cone), and through their complex analogues.