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Cartographical Projections of Triaxial Ellipsoid

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Cartographical Projections of Triaxial Ellipsoid



It is suggested the projections of triaxial ellipsoid derived without approximate calculations of series. Instead of this it is used program of calculation of the definite integral based on Gaussian quadrature rule. Today it is possible to calculate the equidistant along meridians cylindrical projections as well as the equidistant along meridians azimuthal projection.Cylindrical projection with right angle between meridian and parallel, Bugaevskiy projection and the conformal projection proposed by Carl Jacobi can also be calculated.
The Cylindrical projections are equidistant along the equator. The equidistant along meridians azimuthal projection has no distortion at the pole.

Projection Constants:
- semimajor equatorial axis (15000 meters for Eros)
- semiminor equatorial axis (7500 meters for Eros)
- polar semiaxis (7500 meters for Eros)
For Azimuthal projection the longitude of central meridian and center of a map (North Pole, South Pole, intersection of equator and prime meridian, intersection of equator and meridian 900) are inputted additionally.
For Cylindrical projection the cylinder tangent line (equator, prime meridian, meridian 900) are inputted additionally.
Horizontal or vertical offset is inputted in the same units as axes

Input variables
The latitude depends of the way of its definition and longitude from prime meridian.
Output variables
- projection’s coordinates (in the same units as axes). - horizontal to the right, - vertical up

Symbols are used in formulas
The variables used in software are given in brackets.
- quasi-geodesic latitude (the angle between the vertical to meridian ellipse in plane of meridian section and line of cross-section this plane and the equatorial plane) (y_i – in degrees, b_1 – in radians)
- geodesic latitude (the angle between the vertical to the triaxial ellipsoid and the equatorial plane) (y_i – in degrees, b_1 – in radians)
- planetocentric latitude (the angle between radii from the center of ellipsoid to a given point on the surface of ellipsoid and the equatorial plane) (y_i – in degrees, b_1 – in radians)
- longitude of prime meridian (x_i – in degrees, la – in radians)
- square of eccentricity of an arbitrary ellipse (e_el_2)
- square of eccentricity of the equatorial ellipse (e_1_2)
- square of eccentricity of the polar ellipse (section of the prime meridian) (e_2)
- major semi-axis of ellipse of the meridian section (d)
- square of eccentricity of ellipse of the meridian section (e_d_2)
xel , yel , zel - three-dimensional rectangular coordinates
E, G, F - Gauss coefficients of the first fundamental form, G0 - coefficient G at the equator
ω - the angle between meridian and parallel of planetocentric latitude on the surface of the triaxial ellipsoid
Kscale – scale factor along the direction perpendicular to meridian line for cylindrical projections and for azimuthal projections (but not at the pole) in the normal orientation; scale factor along the direction perpendicular to meridian of the transverse coordinate system for cylindrical projections and for azimuthal projections (but not at the center of projection) in the transverse orientation; ratio of similitude for Jacobi conformal projection.

Calculation of projection

Calculation of squared eccentricity of the equatorial ellipse


Calculation of squared eccentricity of the polar ellipse


With derived value of longitude






The cylindrical equidistant along meridians projection for quasi-geodesic latitude

, that are input parameters for a function of calculation of an integral: number of integrand «1» , , lower limit of integration is «0», upper limit of integration is «», the accuracy in radians is «»

, that are input parameters for a function of calculation of an integral: number of integrand «2», lower limit of integration is «0», upper limit of integration is «»,the accuracy in radians is «»

The cylindrical equidistant along meridians projection for geodesic latitude

, that are input parameters for a function of calculation of an integral: number of integrand «1», lower limit of integration is «0», upper limit of integration is «», the accuracy in radians is «»,
, that are input parameters for a function of calculation of an integral: number of integrand «3», lower limit of integration is «0», upper limit of integration is «», the accuracy in radians is «»


The cylindrical equidistant along meridians projection for planetocentric latitude

, that are input parameters for a function of calculation of an integral: number of integrand «1» , , lower limit of integration is «0», upper limit of integration is «», the accuracy in radians is «»
, that are input parameters for a function of calculation of an integral: number of integrand «1» , , lower limit of integration is «0», upper limit of integration is «», the accuracy in radians is «»

To control of correctness of calculation of , it is possible to define planetocentric latitude through quasi-geodesic latitude , with the use of number of integrand 2, with upper limit of integration . Where quasi-geodesic latitude is calculated from relation:


The azimuthal equidistant along meridians projection for planetocentric latitude

for the North pole and for the South pole, where
, that are input parameters for a function of calculation of an integral: number of integrand «1» , , lower limit of integration is «» for the North pole and «-» for the South pole, upper limit of integration is «», the accuracy in radians is «» и

Bugayevskiy projection for planetocentric latitude

, that are input parameters for a function of calculation of an integral: number of integrand «1» , , lower limit of integration is «0», upper limit of integration is «», the accuracy in radians is «»





, that are input parameters for a function of calculation of an integral: : number of integrand «4», lower limit of integration is «0», upper limit of integration is «», the accuracy in radians is «»


The Cylindrical projection with right angle between meridian and parallel for planetocentric latitude

, that are input parameters for a function of calculation of an integral: number of integrand «1» , , lower limit of integration is «0», upper limit of integration is«», the accuracy in radians is «»

At the prime meridian, ,

, that are input parameters for a function of calculation of an integral: number of integrand «4», lower limit of integration is «0», upper limit of integration is «», the accuracy in radians is «»

, that are input parameters for a function of calculation of an integral: number of integrand «5», lower limit of integration is «0», upper limit of integration is «», the accuracy in radians is «»

Jacobi conformal projection for planetocentric latitude.

, where u, v – the elliptic coordinates on the surface of the triaxial ellipsoid

References.
1. Бугаевский Л. М. Теория картографических проекций регулярных поверхностей. – М.: «Златоуст», 1999-144 с.
2. Калиткин Н. Н. Численные методы. – М.: Наука, 1978. – 512 с.
3. Никольский С. М. Квадратурные формулы. – М.: Наука, 1988.- 256 с.
4. Jacobi's Lectures on Dynamics: Second Edition (2009) Edited by: A. Clebsch Hindustan Book Agency
5. Каган В. Ф.. Основы теории поверхностей в тензорном изложении. Часть первая/При редакционном участии Г. Б. Гуревича. – М., ОГИЗ Гостехихдат, 1941. – 512 с.
6. Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. (1986) Integrals and Series, Vol. 1: Elementary Functions. New York: Gordon and Breach
7. Nyrtsov, M.V., Fleis, M.E., Borisov, M.M. Mapping asteroid 433 Eros with equidistant along meridians cylindrical and azimuthal projections of triaxial ellipsoid// Izvestia vuzov . Geodesy and aerophotography . – 2012. №1. P. 54-61

Program for calculation of rectangular coordinates

The cylindrical equidistant along meridians projection
quasi-geodesic latitude
geodesic latitude
planetocentric latitude
planetocentric through quasi-geodesic

The azimuthal equidistant along meridians
The cylindrical projection with right angle between meridian and parallel
Bugayevskiy projection
The azimuthal projection with right angle between meridian and parallel
Jacobi conformal projection
center of the map
tangent line
line of gap
longitude of the central meridian
parallel of circular points
longitude from to step
latitude from to step
semimajor equatorial axis
semiminor equatorial axis
polar semiaxis
x-shift
y-shift
accuracy
M.V.Nyrtsov, M.E. Fleis, M.M. Borisov