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Stereographic Projection

1. Let z and z' be points in C with corresponding points on the unit sphere Z and Z' by stereographic projection. Let N be the north pole N(0,0,1).
a) Show that z and z' are diametrically opposite on the unit sphere iff z(z bar)'=-1
ps. here z bar means conjugate of z
b) Show that the triangles Nz'z and NZZ' are similar. The order of the vertices is important and is as given. Use this to derive the formula for the euclidean distance in R^3 d(Z,Z') = (2|z-z'|)/(sqrt(1+|z|^2)x sqrt(1+|z'|^2))
c) Show that the stereographic projection preserves angles by looking at two lines l1 and l2 through the point z in the complex plane and their images of the Riemann sphere, which are two arcs thru the north pole. Compare the angle between l1 and l2 with the angle of the arcs at N and the image Z of z under the projection.

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Stereographic projections

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1. Let z and z' be points in C with corresponding points on the unit sphere Z and Z' by stereographic projection. Let N be the north pole N(0,0,1).
a) Show that z and z' are diametrically opposite on the unit sphere iff z(z bar)'=-1
ps. here z bar means conjugate of z
b) Show that the triangles Nz'z and NZZ' are similar. The order of the vertices is important and is as given. Use this to derive the formula for the euclidean distance in R^3 d(Z,Z') = (2|z-z'|)/(sqrt(1+|z|^2)x sqrt(1+|z'|^2))
c) Show that the stereographic projection preserves angles by looking at two lines l1 and l2 through the point z in the complex plane and their images of the Riemann sphere, which are two arcs thru the north pole. Compare the angle between l1 and l2 with the angle of the arcs at N and the image Z of z under the projection.

First let's review the basics of stereographic projections.

The Riemann sphere, S, is the unit sphere centered at the origin of , and we call the north pole .

We have the complex plane embedded in as a horizontal plane through the origin. We create the "extended" complex plane by adding a further element to which we'll call the "point at infinity".

There is a one to one correspondence ...

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Stereographic projection are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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