X 237 the general theory of the logarithmic 538 19

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[X : 237] THE GENERAL THEORY OF THE LOGARITHMIC, 538 19. The Stereographic and Mercator’s Projections. The points of a unit sphere whose centre is the origin are projected from the south pole (whose coordinates are 0, 0, - 1) on to the tangent plane at the north pole. The coordi- nates of a point on the sphere are ξ , η , ζ , and Cartesian axes OX , OY are taken on the tangent plane, parallel to the axes of ξ and η . Show that the coordinates of the projection of the point are x = 2 ξ/ (1 + ζ ) , y = 2 η/ (1 + ζ ) , and that x + iy = 2 tan 1 2 θ Cis φ , where φ is the longitude (measured from the plane η = 0) and θ the north polar distance of the point on the sphere. This projection gives a map of the sphere on the tangent plane, generally known as the Stereographic Projection . If now we introduce a new complex variable Z = X + iY = - i log 1 2 z = - i log 1 2 ( x + iy ) so that X = φ , Y = log cot 1 2 θ , we obtain another map in the plane of Z , usually called Mercator’s Projection . In this map parallels of latitude and longitude are represented by straight lines parallel to the axes of X and Y respectively. 20. Discuss the transformation given by the equation z = Log Z - a Z - b , showing that the straight lines for which x and y are constant correspond to two orthogonal systems of coaxal circles in the Z -plane. 21. Discuss the transformation z = Log Z - a + Z - b b - a , showing that the straight lines for which x and y are constant correspond to sets of confocal ellipses and hyperbolas whose foci are the points Z = a and Z = b . [We have Z - a + Z - b = b - a exp( x + iy ) , Z - a - Z - b = b - a exp( - x - iy ); and it will be found that | Z - a | + | Z - b | = | b - a | cosh 2 x, | Z - a | - | Z - b | = | b - a | cos 2 y. ]
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[X : 237] EXPONENTIAL, AND CIRCULAR FUNCTIONS 539 22. The transformation z = Z i . If z = Z i , where the imaginary power has its principal value, we have exp(log r + ) = z = exp( i log Z ) = exp( i log R - Θ) , so that log r = - Θ, θ = log R + 2 , where k is an integer. As all values of k give the same point z , we shall suppose that k = 0, so that log r = - Θ , θ = log R. (1) The whole plane of Z is covered when R varies through all positive values and Θ from - π to π : then r has the range exp( - π ) to exp π and θ ranges through all real values. Thus the Z -plane corresponds to the ring bounded by the circles r = exp( - π ), r = exp π ; but this ring is covered infinitely often. If however θ is allowed to vary only between - π and π , so that the ring is covered only once, then R can vary only from exp( - π ) to exp π , so that the variation of Z is restricted to a ring similar in all respects to that within which z varies. Each ring, moreover, must be regarded as having a barrier along the negative real axis which z (or Z ) must not cross, as its amplitude must not transgress the limits - π and π . We thus obtain a correspondence between two rings, given by the pair of equations z = Z i , Z = z - i , where each power has its principal value. To circles whose centre is the origin in one plane correspond straight lines through the origin in the other.
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