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Technologies for Calculating Geodetic Coordinates Applying h–Geometry Functions (2) Donaldas Zanevičius Faustas Keršys Space Technology Research Center Šermukšnių g. 3 LT- 35113 Panevėžys Lithuania E mail dzanevicius@yahoo.com faustas@baltgina.lt Abstract. The article proposes to apply h-geometry functions sph and cph instead of classic geometry functions sin and cos for conversion of coordinates in mathematical models. As known, numeric values of functions sin and cos can be calculated only when functions are developed to infinite string. h-geometry functions sph and cph have algebraic analytic expressions. It enables system mathematical models to be presented in analytic expressions. This means, that computer calculation times may be shortened 3-7-fold. This should be very important for developers of anti-ballistic defence systems. Classic geometry functions do not allow to obtain such expressions. In such case, systems’ mathematical models can be calculated only by applying iteration methods. Key words: geodetic coordinates, coordinate conversion, geodetic latitude, geodetic longitude, ellipsoidal height, geodetic rectangular coordinates. Introduction Magazine GEODESY AND CARTOGRAPHY 201 36(4); 160-163 has published article of Donaldas Zanevičius and Faustas Keršys TECHNOLOGIES FOR CALCULATING GEODETIC COORDINATES APPLYING H–GEOMETRY FUNCTIONS. (The article was written in Lithuanian). This article was cited in various international publications, including Harvard‘s Department of Astronomy, Smithsonian Astrophysical Observatory and NASA Astrophysics Data System (http://adsabs.harvard.edu/abs/2010GeCar..36..160Z ) It means that h–geometry methods are interesting to astrophysicists. h–geometry ideas in international press first time were published in 2010. [1]-[5]. Wide application area of h–geometry methods was shown in different areas of science, technologies and practical activity. One of such areas is space and ballistics. In European Satellite Navigation Conference 2010, Munich[6] it was shown that 3-7fold shorter computer calculation time is used for calculation of pathway and coordinates of ballistic rockets when using h-geometry functions sph and cph instead of classic geometry functions. 1. Classical geometry. Circle trigonometry functions


Geodesy, as well as astronomy, appeared ca. three thousand years ago. To support these sciences, geometry and one of its branches – trigonometry were developed. In geodesy, as well as in astronomy, the main variable is an angle. Astronomers, naturally, angle formed between two lines have put into circle centre, then value of an angle was possible to measure using length of the arc cut by the two lines coming from circle centre. Such (classical) trigonometry hereafter is called circle trigonometry. Angle measuring value was named degree. Soon after, radian was introduced as an angle unit. Functions named sinα, cosα, tanα and others appeared. These functions have not analytic expressions, therefore long time (and sometimes nowadays) tables were used. In calculations, circle trigonometry functions sinα, cosα, tanα and others are presented in the form of infinite strings. For example: 3

x

x−

sin( x)

3!

5

+

2

cos ( x)

x

1−

2! 3

tan( x)

x+

x

5!

− .........

(1)

4

+

x

4!

− .......

(2)

5

x 2 ⋅x π + + .......... x < 3 5 2

(3) When expressions in such form as (1) are included in an equation system or in a single equation, of course it may be calculated numerically, but such equation can not be solved analytically any more. Let’s take some examples. In tasks of space navigation, conversion of coordinate values is often required in case of transition from one coordinate system to another. Typical tasks are following: 1 Convert given geodetic coordinates B,L,H to geocentric rectangular coordinates X,Y,Z. 2 Convert given geocentric rectangular coordinates X,Y,Z to geodetic coordinates B,L,H.

Task 1. Remember how it is done now, when classical geometry functions sinα and cosα are used as a basis for a mathematical model. Mathematical models, when classic geometry functions sinα and cosα are applied, are well known. X ( N + H) ⋅ cos ( B) ⋅ cos ( L) (4) Y ( N + H) ⋅ cos ( B) ⋅ sin ( L) (5)

(

)

 1 − e2 ⋅ N + H ⋅ sin ( B)

(6) Angles B, as well as angles L, H are given in degrees, minutes and seconds. Therefore, at the beginning, angle values must be expressed in degrees and their decimal parts. Then angle value must be converted to radians. Value N in formulas (4)-(6) is calculated as follows: Z

a

N 2

1 − e ⋅ ( sin ( B) )

where

2

(7)


e

2

a −b

2

a

2

2

(8)

a, b and H numerical values are given. Let’s take an example. Given a B

6378137 o

b

1

6356752 11

55 ⋅ 19 ⋅ 6.73561

H

92.477

L

21 ⋅ 49 ⋅ 56.29320

o

1

11

(9)

or in radians (10) Using formulas (1)-(8) we obtain (calculated using Mathcad software) following: B

0.96549062

L

X Y Z

0.381045582

6

3.376643447× 10

(11)

6

1.352769851× 10

(12)

6

5.221718353× 10

(13)

Task 2. Mathematical model to solve the second task is obtained from mathematical model (4)-(7). p H Z

2

2

p

−N

X +Y cos ( B)

(

( N + H) ⋅ cos ( B)

(15)

)

 1 − e ⋅ N + H ⋅ sin ( B)

tan ( B) tan ( L)

2

Z p

⋅  1 − e ⋅ 2

(14)

  N + H N

(16) −1

(17)

Y

(18) In order to calculate ellipsoid height H(15), we must know geodetic latitude B, and in contrary, in order to calculate geodetic latitude B(17), we must know ellipsoid height H. Therefore many methods were developed for these calculations [7 ]. But they all are based on approximate or iterative methods. X

2. h–geometry. Non-circle trigonometry. Task 1. At the beginning, let’s write down initial equation system (4), (5), (6), but now we shall use h-geometry functions X ( N + H) ⋅ cphB ⋅ cphL (19) Y ( N + H) ⋅ cphB ⋅ sphL (20) Z

where

(

)

 1 − e2 ⋅ N + H ⋅ sphB

(21)


a

N 2

1 − e ⋅ ( sphB )

2

(22)

Where functions sph and cph are defined as follows hB

sphB 2

hB + ( 1 − hB)

2

(23)

1 − hB

cphB 2

hB + ( 1 − hB)

2

(24)

hL

sphL 2

hL + ( 1 − hL)

2

(25)

1 − hL

cphL 2

hL + ( 1 − hL)

2

(26)

In order to compare calculation results using one and another parametric system, we shall use formulas defining relation between these two system parameters. hB hL

tan ( B) 1 + tan ( B)

(27)

tan ( L) 1 + tan ( L)

(28)

hB, hL obtained using (27),(28): hB

0.591032524

hL

0.286033333

(29) (30)

Given B, L, H, a, b values as in classic. Using (19)-(21) we obtain: 6

3.376643446× 10

X

6

1.352769849× 10

Y

(31) (32)

6

5.221718354× 10

(33) As we can see, calculations on the basis of h-geometry (31), (32), (33) match calculations applying classical trigonometry functions (11), (12), (13). Z

Task 2 Values X,Y,Z are known. We need to find angle values measured by h-parameters hB, hL, hH and matching angle values B, L and H, when they are measured by degrees or radians. Let’s square and sum equations (4) and (5). Taking into account that 2

( sphL ) + ( cphL)

2

1

(34)

and assuming that 2

2

X +Y

we shall obtain

kh

(35)


(N2 + 2⋅N⋅H + H2)⋅(cphB) 2

kh

(36)

From here (36), we shall obtain two algebraic equations 3

2

A1⋅M + B1⋅M + C1 ⋅M + D1 2

A2⋅hb − B2⋅hb + C2

0

(37) (38)

0

where  1 − hb    hb 

M

E3 + 

C1

−3 ⋅Z⋅E3 a

2

1−

K 2

a

2

a −b 2

B2

2

3 ⋅E3 − E3

D1

a 2

1 − 3 ⋅E3 −

B1

a

2

E3

E3 + 1 − M

A2

3 ⋅Z

A1

2

C2

1

Equations (37), (38) are solved easily. After introduction of numerical values, we shall obtain hb

(39)

0.591033

As we see, this matches the value (29). It means that model (37),(38) is calculating correctly. From (22) we obtain 6

6.393 × 10

N

(40)

From (19),(20) we obtain 2

2

X +Y

H

cphB

−N

(41)

After introduction of known values, we shall obtain H

(42)

92.477

We know that tpx

y

x

x

1−x

atpy

y 1+ y

(43)

For our task Y

tphL

X

(44)

From here Y

hL

Y

X

1+

Y

X+Y

X

(45)

After introduction of known values, we shall obtain hL

0.286033

Relation between h-parameters and radians are defined as

(46)


L

 hL    1 − hL 

atan

(47)

In our case, with (47) we shall obtain (48) As we see, results are fully matching. Difference is such that instead of transcendental equation system (14) – (18) which may be calculated only by applying iteration methods, h–geometry mathematical models give system of two algebraic equations (37), (38) which is simply calculated. L

0.381046

References. 1. Donaldas Zanevičius Mathematics without sin α, cos α (When Angle α is Being Measured in Degrees) and π. International Journal of Applied Science and Technology Vol.2 No.6 June 2012 http://www.ijastnet.com/journals/Vol_2_No_6_June_2012/6.pdf 2. Donaldas Zanevičius Mathematics without π and sinα, cosα(when angle αis being measured in degrees). PDF]ISC-2011 - ISCA : International E-Publication www.isca.co.in/E-Souvinor%202011.pdf 3. Donaldas Zanevičius h–Geometry. Neo-sines in mechanics. 2008 Vilnius. 4. Donaldas Zanevičius zbMATH https://zbmath.org/?q=an:1161.70001 5. Donaldas Zanevičius h–Geometry.Neo-sines in space mechanics. 2010 Vilnius. 6. Faustas Keršys. Space Technology Research Centr. h–geometry functions sph and cph. Navigation mathematics. European Satellite Navigation Conference. 18-19 October 2010. Munich. ( conference.galileo-masters.eu/…/faustas_kersys.pdf ) 7. Marcin Lingas, Piotr Banasik Conversion between Cartesian and Geodetic coordinates on a rotational ellipsoid by solving a system of nonlinear equations. Geodesy and Cartography. Vol.60, No2, 2011, pp.145-159. Krakow,Poland.


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