Vincenty's Great Circle Central Angle Formula

Great Circle Central Angle Equation, originally by eng. Fixed issue with denominator multiplying cos(φ₂) by itself instead of by cos(φ₁) and added absolute values to Δλ.

The Great Circle Central Angle equation uses three dimensional geometry to calculate the central angle of a great circle arc defined by a pair of latitude/longitude pairs.  This calculation is based on the special case of the Vincenty formula for an ellipsoid with equal major and minor axes. 

EXECUTING THIS EQUATION

Enter the latitudes and longitudes of two points on a sphere:

• Latitude 1 (`phi_1`) is the latitude of the first point
• Longitude 1 (`lambda_1`) is the longitude of the first point
• Latitude 2 (`phi_2`) is the latitude of the second point
• Longitude 2 (`lambda_2`) is the longitude of the second point

You can choose your own units for the angle output

DEFINITION OF A GREAT CIRCLE

The great-circle¹  is the shortest distance between two points on the surface of a sphere. Through any two points on a sphere which are not directly opposite each other, there is a unique great circle. 

Between two points which are directly opposite each other, called antipodal points, there are infinitely many great circles. All great circle arcs between antipodal points have the same length, i.e. half the circumference of the circle.

The Earth's shape can be approximated as nearly spherical, so great-circle distance formulas give the approximate distance between points on the surface of the Earth.

A great circle arc can be drawn between any two points on the earth's surface.

1. ^ orthodromic distance