# H. Rajpoot's formula to compute the minimum distance or great circle distance be

p={color:#000;}.

Harish Chandra Rajpoot Aug, 2016

M.M.M. University of Technology, Gorakhpur-273010 (UP), India

We know that the length of minor great circle arc joining any two arbitrary points on a sphere of finite radius is the minimum distance between those points. Here we are interested in finding out the minimum distance or great circle distance between any two arbitrary points on a spherical surface of finite radius (like globe) for the given values of latitudes & longitudes.

Let there any two arbitrary points & on the surface of sphere of radius & centre at the point O. The angles of latitude are measured from the equator plane & the angles of longitude are measured from a reference plane OPQ in the anticlockwise direction. Here, we are to find out the length of great circle arc AB joining the given points A & B. Draw the great circle arcs passing through the points A & B which intersect each other at the peak (pole) point P & intersect the equatorial line orthogonally at the points D & C respectively. (As shown by the dotted arcs PD & PC in the figure 1)

Join the points A, B, C & D by the dotted straight lines through the interior of sphere to get a plane quadrilateral ABCD.

N

[*Figure 1: The dotted great circle arcs passing through two given points A & B, intersecting each other at the peak (pole) point P, meet the equatorial line orthogonally at the points D & C respectively on a spherical surface of finite radius *] ow, the angle between the great circle arcs AD & DC, subtending the angles & respectively at the centre O of the sphere, meeting each other at the common end point D, is , hence the (plane) angle between the chords AD & DC of great circle arcs AD & DC is given by HCR’s cosine formula as follows
p<>{color:#000;}.

Now, setting the corresponding values in the above formula, we get

Similarly, setting the values of angles & & in the formula, the (plane) angle between the chords BC & CD of great circle arcs AD & DC, meeting each other orthogonally at the common end point C, is given as

Now, join the points A, B, C & D to the centre O of the sphere by the dotted straight lines (As shown in the figure 1 above).

In isosceles ,

Similarly, in isosceles ,

Similarly, in isosceles ,

Now, join the vertices B & D in the plane quadrilateral ABCD to get (As shown in the figure 2). Applying cosine rule in as follows

S

Figure 2: In plane quadrilateral ABCD, the sides AD, BC, CD & the angles & are known

etting all the corresponding values in above expression, we get
p<>{color:#000;}.

Let . Now applying sine rule in plane (see figure 2 above) as follows

Setting all the values (take the value of BD from (3) ), we get

Since hence taking cosine both the sides, we get

Setting the value of from eq(1),

Now, using cosine rule in as follows

Setting all the corresponding values in above expression, we get

Setting the value of from eq(4), we get

Where, the value of constant C is given as

Now, in isosceles (see figure 1 above)

Setting the value of from eq(5),

Since, great circle arc AB is the minimum distance between two given points A & B on the sphere hence

[
**]Where, C is HCR’s constant or distal constant given as

NOTE: It is obvious that the value of constant C depends on the difference of angles of longitude rather than the individual values of measured from a reference plane (like prime meridian for the globe) hence if the difference of angles of longitude is then setting in the expression of distal constant C, we get

NOTE: It’s worth noticing that the above formula of HCR’s constant C has symmetrical terms i.e. if are interchanged, the formula remains unchanged & hence the value of C is unchanged. It also implies that if the locations of two points for given values of latitude & longitude is interchanged, the distance between them does not change at all.

Since the equator plane divides the sphere into two equal hemispheres hence the above formula is applicable to find out the minimum distance between any two arbitrary points lying on any of two hemispheres. So for the convenience, the equator plane of the sphere should be taken in such a way that the given points lie on one of the two hemispheres resulting from division of sphere by the reference equator plane.

Case 1: If both the given points lie on the equator of the sphere then substituting , we get

Hence, the minimum distance between the points lying on the equator of the sphere of radius

The above result shows that the minimum distance between the points lying on the equator of the sphere depends only on the difference of longitudes of two given points & the radius of the sphere. This can easily be proved by using simple geometry.

If both the given points lie diametrically opposite on the equator of the sphere then substituting in above expression, the minimum distance between such points

Case 2: If both the given points lie on a great circle arc normal to the equator of the sphere then substituting in the formula of distal constant C, we get

Hence, the minimum distance between two points lying on a great circle arc normal to the equator of the sphere of radius

Consider any two arbitrary points A & B having respective angles of latitude & the difference of angles of longitude on a sphere of radius 25 cm. Now substituting the corresponding values, the distal constant C is given as follows

Hence, the minimum distance between the given points A & B

The above result also shows that the points A & B divide the perimeter of the great circle in two great circles arcs (one is minor arc AB of length & other is major arc AB of length ) into a ratio

Conclusion: It can be concluded that this formula gives the correct values of the great circle distance because there is no approximation in the formula. This is an analytic formula to compute the minimum distance between any two arbitrary points on a sphere which is equally applicable in global positioning system. This formula is extremely useful to calculate the geographical distance between any two points on the globe for the given latitudes & longitudes. This is a highly precision formula which gives the correct values for all the distances on the tiny sphere as well as the large sphere like giant planet if the calculations are made precisely.

Note: Above articles had been derived & illustrated by Mr H.C. Rajpoot (B Tech, Mechanical Engineering)

M.M.M. University of Technology, Gorakhpur-273010 (UP) India Aug, 2016