Stereographic projection from the north pole of the unit sphere is

PHYSICS · Subjective · 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 (0,0,1)(0,0,1) of the unit sphere is (u,v)=(x,y)/(1−z)(u,v) = (x,y)/(1-z). A meridian at fixed longitude ϕ0=π/6\phi_0 = \pi/6 is the curve (x,y,z)=(sin⁡θcos⁡ϕ0, sin⁡θsin⁡ϕ0, cos⁡θ),θ∈(0,π],(x, y, z) = (\sin\theta\cos\phi_0,\ \sin\theta\sin\phi_0,\ \cos\theta), \qquad \theta \in (0, \pi], where θ\theta is colatitude (the north pole, θ=0\theta = 0, is excluded since it is the projection point). Find the image of this meridian under the stereographic chart and describe it geometrically.

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