ABSTRACT
The elements of the S matrix can be expressed in terms of the coordinates of the vertices of the triangle given by Equation 16.52 and repeated here for convenience: S j k e = 1 4 A b j b k + c j c k . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315115931/4707fdd3-f34e-4bb8-b9c6-7500461af290/content/eqn16_51a.jpg"/> They can also be expressed in terms of the angles of the vertices given by Equation 16.53 and repeated here for convenience: S e = 1 2 cot θ 2 + cot θ 3 − cot θ 3 − cot θ 2 − cot θ 3 cot θ 1 + cot θ 3 − cot θ 1 − cot θ 2 − cot θ 1 cot θ 1 + cot θ 2 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315115931/4707fdd3-f34e-4bb8-b9c6-7500461af290/content/eqn16_53a.jpg"/> It is further shown that Equation 16.53 can be written as S e = ∑ i = 1 3 cot θ i Q i , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315115931/4707fdd3-f34e-4bb8-b9c6-7500461af290/content/eqn16_107a.jpg"/> where [Q i ] are given by Equations 16.108 through 16.110. In this appendix, we will show that Equation 16.53 is obtained from Equation 16.51 based on the geometry and associated trigonometric identities [1]. Figure 16B.1 shows b’s and c’s from their definitions in terms of coordinates given by Equations 16.39b, 16.39c, 16.42b, 16.42c, 16.44b, and 16.44c. In the notation of cyclic values, we have b i = y i + 1 − y i + 2 , i = 1 , 2 , 3 , 4 , 5 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315115931/4707fdd3-f34e-4bb8-b9c6-7500461af290/content/eqn16b_1.jpg"/> c i = x i + 2 − x i + 1 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315115931/4707fdd3-f34e-4bb8-b9c6-7500461af290/content/eqn16b_2.jpg"/> i being cyclic values of 1, 2, 3, i = 4 is same as i = 1, and i = 5 is same as i = 2.
