Distance calculation on the sphere of given radius .

1. How can you improve the accuracy of the calculation of distances between two points on the route ?

2. How to calculate the maximum length between points, in preparation for a flight with maintaining loxodromic An exchange rate on the instrument ?

3. How to calculate the expected error in calculating the length of the route from the formulas of spherical trigonometry ?

4. How to calculate the spherical latitude, ensuring minimum error in the calculation of angles on a sphere with R = 6367.5km ?

5. With what accuracy can be measured and calculated TA and S? [ 7].

The radius R of the sphere, which approximates the ellipsoid is chosen to satisfy certain conditions in terms of accuracy . For example, in navigation in order to achieve the necessary precision for accurate calculations of angles and distances from the formulas of spherical trigonometry. The essence of these methods in the application of these formulas to calculate the latitude and radius of the spherical areas that would provide the least distortion of angles and lengths across its surface.

Minimum error calculating angles occur at equiangular image of a spheroid on a sphere of radius 6367.572 km . The magnitude of the error length computation does not exceed 0.17%. For example, if the actual length of the section on a spheroid 500km distance calculation error on the sphere is ± 0.85km .

Minimum length computation errors occur at equidistant along the meridians image of a spheroid. On a sphere of radius 6372.908 km . The error in determining the lengths equal to 0.08 %, and the angles of 5,7'cos² j. For example, at a length of 500 km of the route on j = 30 º error in the calculation of the distance from the formulas of spherical trigonometry (ie on the field ) is ± 0.4 km , and the error in the calculation of the angles will be 4.3 `≈ 0.07 º, as well . maximum at the equator - 5,7 '≈ 0.1º.

Formulas for calculating the optimum spherical latitude (j) and the sphere radius (R), ensuring the required accuracy of calculating angles and distances between the points on the sphere geodetic coordinates given example, the parameters common terrestrial ellipsoid WGS-84: a = 6378137m, e² = 0.00669437999 .

1. To provide conditions for conformality field using geodetic coordinates on the common terrestrial ellipsoid

Translation geodetic latitude in spherical, is as follows:

In j = В – 11'31"×sin2B,

Sphere radius R = a (1 - k_R e²) R;

R = 6378137×(1 – 0,25×0,00669437999) ≈ 6367,5 km

2. In order to ensure the conditions for minimum error in the calculation of distances on the field on geodetic coordinates of the points

j = В – 8'38"×sin2B.

Sphere radius R = 6378137×(1 – 0,125×0,00669437999) = 6372,8 km