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Solution to 2.6
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Joined: 12 Nov 2008, 02:55
Posts: 12
Location: Japan
Solution to 2.6
In Penrose's solution to the geometric problem whereby a stereographic projection sends circles to circles and that the map is conformal, I cannot see geometrically speaking that the angle at the top of the circle (near N) and marked with the dot is shown to be truly equivalent to the other angles marked with a dot. The justification appears to be missing. If anyone could help me see the simple geometry I am apparently not realizing it would be sincerely appreciated.
Thanks
Tim

12 Nov 2008, 06:13

Joined: 12 Mar 2008, 10:57
Posts: 69
Location: India
Re: Solution to 2.6
Tim,
Welcome to the forum.I would suggest you post such topics in the exercise discussion section.
The concept of angle on a curved surface is a subtle one.It actually refers to the angle in the tangent plane at the given point.So draw a tangent plane at the point on the sphere where the great circles meet.Hope you got it.

16 Nov 2008, 15:44

Joined: 12 Nov 2008, 02:55
Posts: 12
Location: Japan
Thanks
Sameed Zahoor
Thanks for your help it has helped & most appreciated
Cordially
Tim

18 Nov 2008, 02:49
Supporter

Joined: 07 Jun 2008, 08:21
Posts: 235
Re: Solution to 2.6
Tim
In the solution to 2.6 on the Road to Reality website the two angles marked with a dot which are on the circumference of the circle are equal because:
ALL angles at the circumference of a circle subtended by the same chord are equal.

So if you choose any point on the circumference of the circle and draw lines from that point to the end points of the chord, then the angle at the circumference will be equal to the dot angle.

Try and prove this. If you can't I can post a proof to you.

The other two dot angles are equal because the two lines with arrows are parallel

18 Nov 2008, 17:44
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