Great Circle Central Angle (Vincenty)

The Great Circle Central Angle (Vincenty) calculator 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. 

INSTRUCTIONS: Enter the following:

• (`phi_1`) Latitude of First Point
• (`phi_2`) Latitude of Second Point
• (λ₁) Longitude of First Point
• (λ₂) Longitude of Second Point

Central Angle of a Great Circle Arc (θ): The calculator returns the angle in degrees.  However, this can be automatically converted to compatible units (e.g. radians) via the pull-down menu.

The Math / Science

The formula used in this calculator for the Great Circle Central Angle is:

`theta = "arctan"(\frac{sqrt( (cos phi2  * sin(Deltalambda) )^2  + ( cos phi1 * sin phi2    - sin phi1 *cos phi2 *cos (Deltalambda))^2 ) } { sin phi1 * sin phi2 + cos phi2*cos phi2*cos (Deltalambda) })`

where:

• `theta` = Central Angle of the great circle arc
• `phi_1` =  Latitude of First Point
• `phi_2` = Latitude of Second Point
• λ1 = Longitude of First Point
• λ2=  Longitude of Second Point

and where:

• Δλ = |λ1 - λ2|

DEFINITION OF A 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 

Great Circle Calculators

• Compute decimal degree angles from degrees, minutes and seconds,CLICK HERE.
• Compute the time to travel between two latitudes and longitudes
• Compute the Distance Between two Points on a Sphere (Great Circle Arc)
• Compute the Great Circle Arc Central Angle
• Great Circle Arc Angle (Vincenty's)