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Equal-Area Projections of The Triaxial Ellipsoid

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Equal-Area Projections of The Triaxial Ellipsoid (in work)



Equal-Area Projections of The Triaxial Ellipsoid are suggested. For derivation of the projections the program of calculation of the definite integral,based on Gaussian quadrature rule is used. Today it is possible to calculate the Equal-Area cylindrical projection as well as the Equal-Area azimuthal projection. The Cylindrical projection is equidistant along the cylinder tangent line. The azimuthal projection has no distortion at the pole. Ellipse of distortion (Tissot indicatrix) is showed in points with latitude 0, 30N, 60N, 30S, 60S and longitude 0, 30E, 60E, 90E, 120E.

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
Planetocentric latitude and longitude to the East 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.
- 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 from 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)
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.
Kmer – local linear scale along a meridian (along a meridian of the transverse coordinate system in the transverse orientation).
Kpar – local linear scale along a parallel (along a parallel of the transverse coordinate system in the transverse orientation.
ωproj – the angle between meridian and parallel on projection (between meridian and parallel of the transverse coordinate system in the transverse orientation).
θmax – the maximum angular deformation on the 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 Equal-Area 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 «10», lower limit of integration is «0», upper limit of integration is «», the accuracy in radians is «»


The azimuthal projection for planetocentric latitude




for the North pole and for the South pole, that are input parameters for a function of calculation of an integral: number of integrand «11» , , lower limit of integration is «» for the North pole and «-» for the South pole, upper limit of integration is «», the accuracy in radians is «» and


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. Каган В. Ф.. Основы теории поверхностей в тензорном изложении. Часть первая/При редакционном участии Г. Б. Гуревича. – М., ОГИЗ Гостехихдат, 1941. – 512 с.
5. Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. (1986) Integrals and Series, Vol. 1: Elementary Functions. New York: Gordon and Breach
6. 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 Equal-Area projection
The Azimuthal Equal-Area projection
center of the map
tangent line
линия разрыва
longitude of the central meridian
параллель круговых точек
longitude from to step
latitude from to step
semimajor equatorial axis
semiminor equatorial axis
polar semiaxis
x-shift
y-shift
accuracy
М.В.Нырцов, М.Э.Флейс, М.М. Борисов