<content> <p>Plot CreateMesh γ</p> </content>

0efc651d-08bf-4086-8c1c-3ddd653c6402 182 11 10 true true 0 decimal

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Transformation from Cartesian into geodetic coordinates

u v f5 ax u sin * v cos * ay u sin * v sin * b u cos * 31 mat :

γ 0.8232 - 0.4194 - 0.3827 0.5677 0.6187 - 0.543 0.009 0.6643 0.7474 33 mat :

Поворот на угол ψ вокруг оси OZ ψ Ω3 ψ cos ψ sin - 0 ψ sin ψ cos 000133 mat :

ax 6378.1370 : ay ax : b 6356.7523142 : 13 mat

sur f5 π 2 / - π 2 / 0 π 1515 CreateMesh :

sur1 f5 0 π - 2 / π - 2 / π 1515 CreateMesh :

aAγ A a stack γ * eval :

t 0 π 2 / 0.2 + (0.2 range : g0 ax t cos * vectorize ay t sin * vectorize 0 augment : gL ax t cos * vectorize 0 b t sin * vectorize augment : gP 0 ay t cos * vectorize b t sin * vectorize augment : B 000 ax 0023 mat gL stack : 51 line

Оси координат XYZ k OX 10013 mat (γ * k col eval : k OY 01013 mat (γ * k col eval : k OZ 00113 mat (γ * k col eval : 41 sys XYZ 00 1.1 ax * 1 OX * 1.1 ax * 2 OX * 00 1.1 ax * 1 OY * 1.1 ax * 2 OY * 0000 1.2 ax * 1 OZ * 1.2 ax * 2 OZ * 0082 mat : TextXYZ 00.10 brown 1.1 ax * 1 OX * 1.1 ax * 2 OX * X 12 brown 1.1 ax * 1 OY * 1.1 ax * 2 OY * y 13 brown 1.2 ax * 1 OZ * 1.2 ax * 2 OZ * Z 13 brown 45 mat : 31 sys

Plot3 g0 γ * 1 col g0 γ * 2 col augment B λ# - Ω3 * γ * 1 col B λ# - Ω3 * γ * 2 col augment XYZ A γ * (1 el A γ * (2 el A 12 15 mat a γ * (1 el a γ * (2 el a 13 15 mat A γ * (1 el A γ * (2 el . 15 green 15 mat gL γ * 1 col gL γ * 2 col augment TextXYZ a γ * (1 el a γ * (2 el . 18 green 15 mat gP γ * 1 col gP γ * 2 col augment aAγ 1 col aAγ 2 col augment sur1 γ * 1 col sur1 γ * 2 col augment 171 sys :

Convert Between Cartesian Coordinates and Geodetic Coordinates
often found in geodesy and astronomy, for example,when
determining the position by GPS.

The program for calculating Cartesian coordinates by geodetic coordinates
has the form (λ-longitude,φ-latitude,h-geodetic height):

(WGS 84) Earth’s geometrical parameters (km) ax 6378.137 : ay ax : b 6356.7520314245 : ex 1 b 2^ax 2^/ - sqrt eval : ee 1 ay 2^ax 2^/ - sqrt eval : 16 mat x f x 1 el (2^ax 2^/ x 2 el (2^ay 2^/ + x 3 el (2^b 2^/ + 1 - : 21 sys

Geodetic(φ,λ,h) to cartesian(x,y,z)  conversion λ φ h XYZ v ax 1 ex 2^φ sin (2^* - ee 2^φ cos (2^* λ sin (2^* - sqrt / eval : x v h + (φ cos * λ cos * eval : y v 1 ee 2^- (* h + (φ cos * λ sin * eval : z v 1 ex 2^- (* h + (φ sin * eval : x y z 13 mat 51 line :

Тhe inverse transformation is as follows:

Cartesian to geodetic coordinates conversion a λ 11 ee 2^- / a 2 el a 1 el / * atan : a φ 1 ee 2^- 1 ex 2^- / a 3 el 1 ee 2^- (a 1 el (2^* a 2 el (2^+ sqrt / * atan : 21 sys

where a is the orthogonal projection of point A onto the surface of the Earth's ellipsoid.

The code for calculating the orthogonal projection is borrowed from Program

https://en.smath.com/forum/yaf_postsm69369_Draghilev-method-revisited.aspx#post69369

Distance and projection of a point onto a surface

sur Pr c 13 range : g c el x f x c el diff : eN g g norme / : k 1 sur rows range : 14 mat d k el A sur k row - norme : a0 sur d augment 4 csort 1 row c el transpose eval : 12 mat x F x f eN 2 el A 3 el x 3 el - (* eN 3 el A 2 el x 2 el - (* - eN 3 el A 1 el x 1 el - (* eN 1 el A 3 el x 3 el - (* - 31 mat : a a0 transpose 1011 -^x F al_nleqsolve transpose eval : 12 mat a A f sign A a - norme * km * 12 mat 41 line :

1 (

Accuracy was defined as the difference between the initial (known) geodetic coordinates and the transformed coordinates

εh h# h - :

εφ φ# a φ - :

ελ λ# a λ - :

Ex.1 (GPS satellite)

h# 20300 km * :

λ# 30 ° * :

φ# 55 ° * :

Convert to XYZ:

X 13259.018058 km * :

y 7655.0976448 km * :

z 21830.169714 km * :

Restoration of geodetic coordinates:

A X y z augment km / :

a h 12 mat sur Pr :

h km 2.03 104^*

Plot3

a λ ° 29.999999999

a φ ° 54.999999999

εh mm 0.11005

εφ ° 1.06565 109 -^*

ελ ° 5.71703 1010 -^*

Ex.2 (100000 km)

h# 100000 km * :

λ# 40 ° * :

φ# 40 ° * :

Convert to XYZ:

X 62430.440421 km * :

y 52385.359531 km * :

z 68356.746253 km * :

Restoration of geodetic coordinates:

A X y z augment km / :

a h 12 mat sur Pr :

h km 1105^*

Plot3

a λ ° 40.000000000

a φ ° 40.000000000

εh mm 0.1618 -

εφ ° 1.70342 1010 -^* -

ελ ° 2.67291 1010 -^*

Ex.3.(Superdeep Borehole)

h# 3000 - (km * :

λ# 40 ° * :

φ# 35 ° * :

Convert to XYZ:

X 2124.2188597 km * :

y 1782.4312617 km * :

z 1917.1373296 km * :

Restoration of geodetic coordinates:

A X y z augment km / :

a h 12 mat sur Pr :

h km 3000 -

Plot3

a λ ° 40.000000000

a φ ° 34.999999999

εh mm 0.00815 -

εφ ° 6.85343 1010 -^*

ελ ° 2.2304 1010 -^*