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Cartographic projections for small bodies

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fragment of paper
M.E. Fleis, M.M. Borisov, M.V. Alexandrovich, P. Stooke, K.B. Shingareva CARTOGRAPHIC PROJECTIONS FOR SMALL BODIES OF THE SOLAR SYSTEM (XXIII International Cartographic Conference)

Formulae of two conformal projections (Transverse Cylindrical and Transverse Azimuthal) for "upturned" ellipsoid of revolution with revolution axis of ellipse matching with its major axis (not minor as usual) and being perpendicular to revolution axis of the body. Formulae are obtained without approximations and are true even for eccentricity close to 1.

Symbols

Projection constants
- semimajor axis of ellipse (for Eros asteroid we assume it to be 33000/2 m)
- semiminor axis of ellipse (for Eros asteroid we assume it to be 13000/2 m)
- eccentricity of ellipse
- longitude of the center of projection: or
- radius of the Equator of transverse graticule on Azimuthal projection under condition of no distortion in the center of projection
- auxiliary values
- minimal value of angular distance from the center of projection (we assume 0.0016 )
- maximal value of angular distance from the center of projection in the neighbourhood of point opposite to the center of projection (we assume 3.14 )

Input variables
- planetocentric latitude in normal aspect
- longitude west of zero meridian

Intermediate values
- longitude east of zero meridian
- longitude east of the center of projection
- angular distance from the center of projection
- azimuth as an angle measured clockwise from the north
- latitude at "upturned" ellipsoid of revolution
- radius of latitude circle on Azimuthal projection (transverse graticule) divided by
- radius of latitude circle on Azimuthal projection (transverse graticule) divided by

Output variables:
- rectangular coordinates ( - horizontal, - vertical)

Calculation of constant parameters:


(for Azimuthal projection)
(for Azimuthal projection)

Calculation of rectangular coordinates:



Azimuthal projection



To exclude the possibility of division by zero and breaking the domain of the functions we use three variants of coordinates' calculation depending on value.
1.



2.







where
,

3.
If , then , if also
1) let
2) let
3) let
4) let





where
,

Cylindrical projection







If then , if also
1) let
2) let


If , then , if also

1) let
2) let
3) let
4) let



To exclude the possibility of breaking the domain of the function we use three variants of azimuth calculation depending on cosine value.
1. If azimuth is equal to
2. If azimuth is equal to zero.
3. if
if
If then azimuth is increased by




Program for calculation of rectangular coordinates of points sets on the projections

Azimuthal projection
Cylindrical projection
longitude from   to   step  
latitude from   to   step  
semimajor axis of ellipse
semiminor axis of ellipse
longitude of the center of projection