Stereographic projection from the north pole of the unit sphere sends a point with…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch Manifolds, charts and tangent spaces

Problem

Stereographic projection from the north pole of the unit sphere sends a point with Cartesian coordinates (x,y,z)(x,y,z), x2+y2+z2=1x^2+y^2+z^2=1, to the plane point (u,v)=(x,y)/(1−z)(u,v) = (x,y)/(1-z). Write x=sin⁡θcos⁡ϕx=\sin\theta\cos\phi, y=sin⁡θsin⁡ϕy=\sin\theta\sin\phi, z=cos⁡θz=\cos\theta, with θ\theta the colatitude measured from the north pole (θ=0\theta=0). Consider the circle of latitude at colatitude θ0=2π/3\theta_0 = 2\pi/3 (i.e. 120∘120^\circ, a circle below the equator). ((a)) Show that the image of this circle under the projection is itself a circle in the (u,v)(u,v)-plane centred at the origin, and find its radius. ((b)) State, with a one-line reason, whether this radius should be less than, equal to, or greater than 11 (the radius of the image of the equator), and check that your answer in part (a) is consistent with this.

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