Arc Distance Formula by James Q. Jacobs

ARC_CALC_3 — Spherical Trigonometry Calculator by James Q. Jacobs

Finding the shortest distance between two points on the earth given latitude and longitude

Download ARC_CALC_3
This small program will do the calculations (shown below) in an Excel spreadsheet. You only need to input the coordinates.

The program supports input of three sites and calculates the three arc distances, the area of the spherical triangle and the bearings between sites.

Arc Distance Formula (below)

Most spreadsheet programs should be able to import this file format. The graphic below illustrates the spreadsheet, before corrections on July 24, 2002. Previous downloads have "A to C" and "B to C" labels swapped for arcs and bearings. Please let me know if there are other bugs in the applet.

Epoch_2000 Temporal Epoch Calculator is a similar Excel spreadsheet. It calculates
the temporal changes in astronomic constants, just enter a date.

RECENT

ArchaeoBlog

Ancient Earthworks of the
Eastern Woodlands Photo Galleries

REFERENCES
Astronomy Formulas	Astronomical Constants	Cosmographic Values	Geodesy Page
ESSAYS
The Àryabhatiya of Àryabhata The oldest exact astronomic constant?		Mesoamerican Archaeoastronomy
Archaeogeodesy Pages Archaeogeodesy can be defined as that area of study encompassing prehistoric and ancient place determination, point positioning, astronomy, geodynamic phenomena, measure and representation of the earth, and navigation (on land or water).

NEW: Google Earth™ Ancient Monuments Placemarks

Graphic of previous version of the Arc_Calc Excel spreadsheet.

Spherical triangle with parts labeled. Spherical Trigonometry
Arc Distance Formulas

Note: a and b are distinct from a (alpha) and b (beta).

1. Find distances a and b in degrees from the pole.

2. Find angle P by arithmetic comparison of longitudes. (If angle P is greater than 180 degrees subtract angle P from 360 degrees.)

Subtract result from 180 degrees to find angle y.

3. Solve for 1/2 ( a - b ) and 1/2 ( a + b ) as follows:

tan 1/2 ( a - b ) = - { [ sin 1/2 ( a - b ) ] / [ sin 1/2 ( a + b ) ] } tan 1/2 y
tan 1/2 ( a + b ) = - { [ cos 1/2 ( a - b ) ] / [ cos 1/2 ( a + b ) ] } tan 1/2 y

4. Find c as follows:

tan 1/2 c = { [ sin 1/2 ( a + b ) ] x [ tan 1/2 ( a - b ) ] } / sin 1/2 ( a - b )

5. Find angles A and B as follows:

A = 180 - [ ( 1/2 a + b ) + ( 1/2 a - b ) ]
B = 180 - [ ( 1/2 a + b ) - ( 1/2 a - b ) ]

Download ARC_CALC_3