Journal of Theoretics


VECTOR ADDITION FORMULA ON A SPHERICAL SURFACE

Md. Shah Alam
Department of Physics
Shahjalal University of Science and Technology
Sylhet, Bangladesh
salam@sust.edu

ABSTRACT
Vector addition formula on a plane Euclidean surface is well known. In this paper we propose a vector addition formula on a spherical surface. Motion on a sphere (for example, the motion of an aircraft along the earth’s surface) can be studied using this formula.

KEYWORDS:  vector addition formula, spherical surface.

INTRODUCTION
We know that the surface of the earth is spherical. A line on the earth’s surface reduces to a straight line when its length is very small compared to the radius of the earth, but its curvature is manifest when its length is large. Therefore, if we draw a triangle on the earth’s surface it will be a plane triangle when its sides are small and it will be a spherical triangle when its sides are large. Our known vector addition formula is applicable on the plane triangle, but not on the spherical triangle. In this paper we define vectors on a spherical surface and their addition formula which can be applied on the spherical triangle.

DEFINITION OF VECTORS ON A SPHERICAL SURFACE
Let us consider two points A and B on the surface of a sphere. We draw a great circle on the surface of the sphere through the points A and B. AB and BA are two vectors on the spherical surface. The direction of AB vector is from A to B along the surface of the sphere and the direction of the vector BA is from B to A along the surface of the sphere. The magnitudes of these two vectors are same, that is, the tangent of the angle subtended at the center of the sphere by the curve AB.

GENERATION OF VECTOR ADDITION FORMULA ON A SPHERICAL SURFACE
Mushfiq Ahmad in his study of Lorentz Transformation has defined a formula of addition of vectors:1 

A B = (A + B + iAxB/c) / (1+A.B/c2) .

Here represents new addition:

Let c = ic' and c' = 1 then

A B = (A + B + AxB) / (1 - A.B) .              (1)

This is the vector addition formula on the spherical surface.

Where A, B etc. are the tangents of the angles subtended at the center of the sphere.  From this formula we can explained the following physical cases.

(i) When the angle between two vectors on the spherical surface is 0° then their addition gives a curved line on the spherical surface.

When angle between vectors A and B is 0° then using equation (1) we get

A B = (A + B) / (1 - A.B) = D (say)

or

D = (tanA + tanB) / (1-tanA tanB) = tan(A + B) .

(ii) When angle between the vectors A and B is 0° and angle between the vectors B and D     (AB = D) is 180° then it represents a circle:

For this case using equation (1) we can write

A = D (-B)

Here, D = tan180°, B = tan90°, and 

A = (D - B - DxB) / (1 + D.B)

or

A2 = (D2 - 2D.B +B2 - D2B2 + (D.B)2) / (1 + 2D.B + (D.B)2)

=(tan2180°+2tan180°tan90°+tan290°-tan2180°tan290°+(tan180°tan90° )2)/(1-2tan180°tan90°)

= tan2 90° 

A = tan 90° .

Which is exactly same as shown in the fig.1.

(iii) When the sides of a equilateral spherical triangle are very small then it behaves like a plane triangle.

Using equation (1) we get





Neglecting higher powers of D we get,

or, cos d = ˝ = cos 60° 

d = 60°

Which is the requirement of equilateral plane triangle.

The proposed vector addition formula is applicable on the spherical triangle is shown below.

EXAMPLES


(i) Let us consider a sphere as shown in fig.(2). 

A, B and D are three vectors on the surface of the sphere which form a spherical triangle and a, b, and d are angles of this spherical triangle.  In fig. 1 A = B = tan45°, d = 90° we want to find D = ?. 

Here d = 90°

A.B = AB cos(180°-d) = - ABcos d = - cos 90°

Using eqn.(1) we get

D = (A + B + AxB) / (1-A.B)

D2 = A2 + B2 + 2A.B + A2B2 - (A.B)2 / (1 - 2A.B + (A.B)2)



Using traditional method, we get, D = 60° where A, B, D etc. are angles subtended at the center of the sphere.2

(ii) Let, A = B = tan 45°, d = 60°

A.B = AB cos ( 180° - d ) = - cos d = - cos 60° = -(1/2 )

(iii) Let, A = B = tan30° and d = 90°, D = ?

A.B = AB cos (180°-d) = - cos d = - cos 90° = 0




ADVANTAGES
Using this formula we can explain rotation on a sphere clearly. In this representation A, B etc. are tangents of the angles subtended at the center. Traditionally, one represents them by the angles themselves subtended at the center. Our method provides a direct vector algebra which traditional method does not.3 

CONCLUSION
We have been able to obtain a generalized formula of addition of vectors which in one limit (when radius of the sphere ) reduces to addition of vectors on a plane (Euclidean) surface, and in another limit, AxB = 0 reduces to addition of tangents along a circle. 


ACKNOWLEDGEMENT
I am grateful to Mushfiq Ahmad, Dept. of Physics, Rajshahi University, Bangladesh and Prof. Habibul Ahsan of the Deparment of Physics, Shahjalal University of Science and Technology, Sylhet, Bangladesh for their help and advice.

REFERENCES
[1] Ahmad, Mushfiq (Dept. of Physics, R.U. Bangladesh). Private Communication. n.d.
[2] Green, Robin M. Green. Spherical Astronomy. Conn.: Cambridge UP, 1985. 
[3] Punmia, Dr. B. C. Surveying. 9th ed. Vol. 3. New Delhi: Laxmi Publications (P) Ltd.,1995. 

 

 

Journal Home Page

© Journal of Theoretics, Inc. 2001  (Note: all submissions become the property of the Journal)