h-Geometry Model of the Earth Dr. Donaldas Zanevičius. American Research Institute for Policy Development (ARIPD) ARIPD American Association of International Researchers President of Lithuanian Association of Engineers
Humanity has been interested in the shape of the Earth since the dawn of times. Many scientific works have been devoted to that. In geodetic science, it is accepted that the Earth is shaped as an ellipse. Geometry of the Earth is also analysed in cartography for forming maps of the Earth. As we know, the mathematical model of an ordinary ellipse (rk) in polar coordinate system is b
r
k
2
1 e ( cos ( ) )
2
(1)
where e
2
a b
2
a
2
2
(2)
Its algebraic expression is 2
x
2
a
2
y
2
1
0
b
(3)
When solving geodetic tasks, we constantly have to deal with function (1) differentiation and integration. In this case, the derivative cannot be integrated since it is an elliptic integral that does not have an analytical expression. For this purpose, various approximations are used. Usually, they are in the form of infinite series. The integral of the elliptic mathematical model does not have an analytical expression as well, and the integral itself is attributed to a special group of integrals – elliptic integrals.
b
d
2
1 e ( cos ( ) )
2
(4) We recommend using the mathematical model of slightly altered ellipse (neoellipse) used in hgeometry, where angles are measured in h-parameters [1] instead of radians. As we know, the relationship between radian and h-parameter is set by formulas h atan 1 h (5) or h
tan ( )
1 tan ( )
(6)
Using the system of h-parameters, the mathematical model of neoellipse is expressed as b h ( 1 h) 2
r
2
2
h ( 1 h)
2
(7)
where
( 1 )
b a
(8)
or b (sph) (cph) 2
r
2
(9)
where h
sp h 2
h ( 1 h )
2
(10)
1h
cp h 2
h ( 1 h )
2
(11)
The expression of neoellipse in α parameter system is b
r
1 ( cos ( ) )
2
(12)
Its algebraic expression is 2
x
a
y
2
b
2
x y
2
0
(13)
It is not difficult to determine that the expression (7) is easy to differentiate. Unlike in the case of an ordinary ellipse, where the derivative cannot be integrated, these derivatives can be easily integrated. When calculating the area of neoellipse (7), the function (7) can be integrated analytically. Compare numerical value of an ordinary ellipse (1), (rk) and neoellipse (7), (r). Set that a 63781370 b 63567520 (14) Calculation results are given in Table 1. Table 1 α h rk r
0 0 6378137 6378137
0,3 0,236254 6376261 6376264
0,5 0,353296 6373203 6373209
1,0 0,608979 6362973 6362980
1,3 0,782708 6358275 6358277
1,57 0,999204 6356752 6356752
As can be seen in the Table, calculation results (rk and r) correspond quite well.
Literature. 1. Donaldas Zanevičius h – Geometry. Neo-sines in mechanics. 2008 Vilnius. 2. Harvard‘s Department of Astronomy, Smithsonian Astrophysical Observatory and NASA Astrophysics Data System http://adsabs.harvard.edu/abs/2010GeCar..36..160Z http://www.kosmose.lt/users/www/uploaded/15-0309%20Geodezija-h-2%20EN.pdf