assume that the spheroid, an ellipsoid of revolution, has an equatorial radius formula_2 and polar semi-axis formula_3. define the flattening formula_4, the eccentricity formula_5, and the second eccentricity formula_6. consider two points: formula_7 at (geographic) latitude formula_8 and longitude formula_9 and formula_10 at latitude formula_11 and longitude formula_12. the connecting great ellipse (from formula_7 to formula_10) has length formula_15 and has azimuths formula_16 and formula_17 at the two endpoints.