Journal of Theoretics Vol.3-4

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1 Journal of Theoretics Vol.3-4 APPLICATIONS OF VECTOR ADDITION FORMULA ON A SPHERICAL SURFACE Md. Shah Alam Deptartment of Physics Shahjalal University of Science and Technology Sylhet, Bangladesh salam@sust.edu ABSTRACT Rotation can be represented as a vector, but one which is different from natural vector. P.E. Lewis and J.P. Ward derived a vector addition formula of rotations from Rodrigue s formula. 1 It was observed that this addition formula is not exactly correct with the justification of spherical triangular cosine formula. In this paper, we applied the vector addition formula on a spherical surface, [which is exactly consistent with the spherical triangular cosine formula.] KEYWORDS: vector addition formula, spherical surface. INTRODUCTION Vector addition formula for rotations derived from Rodrigue s formula is as follows: 1 Ω + Ω - Ω Ω Ω =... (1) 1 - Ω. Ω Ω θ Ω Ω Figure(1) where Ω and Ω are rotation vectors, and Ω is their addition, which is also a rotation vector. The direction of these vectors is that of a right-handed screw advancing in the positive direction along a coordinate axis. Magnitudes of these vectors are the tangent of the half of the angles subtended at the center of the sphere by the rotations. 1

2 If θ 1,θ 2 and θ 3 are the angles subtended at the center of the sphere by the rotations Ω, Ω and Ω respectively, then Ω = tan(θ 1 /2), Ω = tan(θ 2 /2) and Ω = tan(θ 3 /2). In fig. (1), θ is the angle between the sides Ω and Ω of the spherical triangle formed by the rotations Ω, Ω and Ω. Again, spherical triangular cosine formula is as follows: 2 A b θ c a Figure (2) cos a = cos b cos c + sin b sin c cos A (2) where a, b and c are the three sides of the spherical triangle, and A is the angle between the sides b and c. The magnitudes of a, b and c are the angles subtended at the center themselves. If θ 1, θ 2 and θ 3 are the angles subtended at the center of the sphere by the sides b, c and respectively, then b = θ 1, c = θ 2 and a = θ 3. In fig. (2), θ is the angle between the sides b and c then A = θ. Putting the values of b,c,a and A in equation (2) we get cosθ 3 = cosθ 1 cosθ 2 + sinθ 1 sinθ 2 cosθ (3) JUSTIFICATION OF VECTOR ADDITION FORMULA OF ROTATIONS SPHERICAL TRIANGULAR COSINE FORMULA Problem (1) : Determine a single rotation to the two successive rotations: WITH R 1 : about the x- axis through an angle of 90 R 2 : about the z- axis through an angle of 45 Solution of the problem (1) using Rodrigue s formula: Using Rodrigue s formula i.e. equation (1) the result obtained is [1] 98.4, which is physically incorrect. Solution of the problem (1) using spherical triangular cosine formula : 2

3 Equation (3) is: cosθ 3 = cosθ 1 cosθ 2 + sinθ 1 sinθ 2 cosθ Here, θ 1 = 90, θ 2 = 45 and cosθ 3 = cos 90 cos 45 + sin 90 sin 45 cos 90 = = 0 = cos 90 θ 3 = 90 Physically this value is correct. Therefore the result obtained using equation (1) differ from that obtained by using equation (3). Similarly we calculated the results of different cases using these two equations and the results obtained are shown in the Table 1. Table 1 Examples (i) θ 1 = 60, θ 2 = 60 (ii) θ 1 = 80, θ 2 = 80 (iii) θ 1 = 60, θ 2 = 60 (iv) θ 1 = 45, θ 2 = 45 (v) θ 1 = 30, θ 2 = 30 (vi) θ 1 = 70, θ 2 = 70 (vii) θ 1 = 30, θ 2 = 30 (viii) θ 1 = 30, θ 2 = 30 (ix) θ 1 = 45, θ 2 = 45 θ = 45, θ 3 =? (x) θ 1 = 45, θ 2 = 45 Results obtained using the vector addition formula of rotations derived from Rodrigue s formula Results obtained using spherical triangular cosine formula

4 From the above table we can conclude that the vector addition formula of rotations derived from Rodrigue s formula is not consistent with the spherical triangular cosine formula. Solution of the problem (1) using vector addition formula on a spherical surface: A and B are two vectors on the surface of the sphere whose radius is unit and D is their addition vector shown in figure (3). A θ B P (π-θ) Q D R Figure 3 According to vector addition Formula on the spherical surface [3] we can write: A + B + A B D = (4) 1 A.B where A and B are two rotation vectors and D is their addition. The magnitudes of these two vectors are the tangent of angles subtended at the center by the rotations. The direction of A vector is from Q to P along the surface of the sphere and the direction of B vector is from P to R along the surface of the sphere. If θ 1, θ 2 and θ 3 are angles subtended at the center of the sphere by the rotations A, B and D respectively, then A = tanθ 1, B = tanθ 2 and D = tanθ 3. In figure (3), on the spherical surface QPR = θ therefore the angle between the vectors A and B is equal to (π θ). Solution of the problem (1) using equation (4): Here θ = 90, A.B = AB cos(180 - θ) = - ABcosθ = - cos 90 Equation (4) is: D = A + B + A B 1 A.B 4

5 D 2 = A 2 + B 2 + 2A.B + A 2 B 2 (A.B) 2 1 2A.B + (A.B) 2 (5) Here, θ 1 = 90, θ 2 = 45 A = tanθ 1 = tan90 =, B = tanθ 2 = tan45 = 1 A.B = ABcos(180 - θ) = - ABcosθ [From fig. (3) angle between the rotation vectors A and B is (180 - θ)] Putting these values of A, B and A.B in equation (5) we get: D 2 = tan 2 θ 1 + tan 2 θ 2 + tan 2 θ 1 tan 2 θ 2 = = or, D = = tan90 or, D = tanθ 3 = tan90 θ 3 = 90 Therefore, the results obtained by using vector addition formula on a spherical surface and spherical triangular formula are exactly the same. We used the vector addition formula on a spherical surface and the spherical triangular cosine formula for the different cases, and the obtained results are shown in the following Table 2. Table 2 Examples (i) θ 1 = 60, θ 2 = 60 (ii) θ 1 = 80, θ 2 = 80 (iii) θ 1 = 60, θ 2 = 60 (iv) θ 1 = 45, θ 2 = 45 (v) θ 1 = 30, θ 2 = 30 Results obtained using vector addition Formula of rotations Results obtained using spherical triangular cosine formula

6 (vi) θ 1 = 70, θ 2 = 70 (vii) θ 1 = 30, θ 2 = 30 (viii) θ 1 = 30, θ 2 = 30 (ix) θ 1 = 45, θ 2 = 45 θ = 45, θ 3 =? (x) θ 1 = 45, θ 2 = Therefore we see that the vector addition formula on a spherical surface is directly consistent with the spherical triangular cosine formula. CONCLUSION Rotations can be represented as vectors. The vector addition formula on a spherical surface is the appropriate formula for the rotations on a spherical surface. Using the vector addition formula on a spherical surface we can explain rotations on the surface of a sphere clearly. Using spherical triangular cosine formula the same task could be done, but the vector addition formula on a spherical surface provides a direct vector algebra, which the spherical triangular cosine formula does not. 4 ACKNOWLEDGEMENTS I am grateful to Mushfiq Ahmad of the Department of Physics, Rajshahi University, Rajshahi, Bangladesh and Professor Habibul Ahsan of the Department of Physics, Shahjalal University of Science and Technology, Sylhet, Bangladesh for their help and advice. REFERENCES [1] Lewis, P. E., and J. P.Ward. Vector Analysis for Engineers and Scientists. Reading, Mass.: Addison-Wesley, [2] Green, Robin M. Spherical Astronomy. Cambridge, Mass.: Cambridge UP, [3} Alam, Md. Shah. "Vector Addition Formula on a Spherical Surface." Journal of Theoretics. 3-3 (2001) [4] Punmia, Dr. B.C. Surveying. Vol. 3. 9th ed. New Dehli, India: Laxmi Publications (P) Ltd., Journal Home Page mail@journaloftheoretics.com Journal of Theoretics, Inc (Note: all submissions become the property of the Journal) 6

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