3: Let n = (0,0,1) and let o: S\ {n} →R? be the stereographic projection given by ø(r, y, z) = ( 2). %3D (a) Let u = (2,2, 1) and v = (1, –2, 2). Find the image of the spherical line segment [u, v] under o. (b) Let u = (1,0,0), v = (0, 1,0) and w = (0, 1, 1). Find the area of the image of the spherical triangle [u, v, w] under ó.
3: Let n = (0,0,1) and let o: S\ {n} →R? be the stereographic projection given by ø(r, y, z) = ( 2). %3D (a) Let u = (2,2, 1) and v = (1, –2, 2). Find the image of the spherical line segment [u, v] under o. (b) Let u = (1,0,0), v = (0, 1,0) and w = (0, 1, 1). Find the area of the image of the spherical triangle [u, v, w] under ó.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![3: Let n =
(0,0, 1) and let ø: s \ {n} →R? be the stereographic projection given by ø(r, y, z) = ( 2).
(a) Let u = (2,2, 1) and v = (1, –2, 2). Find the image of the spherical line segment [u, v] under ø.
(b) Let u =
(1,0,0), v = (0, 1,0) and w =
(0, 1, 1). Find the area of the image of the spherical triangle
[u, v, w] under ø.
(c) Identify R2 with C so o(x, y) =+i. Show that o-( ) = (x,-y,-2) for all (x, y, z) € S²\{±n}.
We remark that this shows that -10 is the rotation R1,0,0),m
4: Given a point u e S² and a (spherical) line L in S?, we define the (spherical) distance from u to L to be](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F11543e2e-0d2a-4bb4-9823-b362f3bac988%2F9e775300-5b7b-493d-b6ca-304a8119b233%2F98ocofe_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3: Let n =
(0,0, 1) and let ø: s \ {n} →R? be the stereographic projection given by ø(r, y, z) = ( 2).
(a) Let u = (2,2, 1) and v = (1, –2, 2). Find the image of the spherical line segment [u, v] under ø.
(b) Let u =
(1,0,0), v = (0, 1,0) and w =
(0, 1, 1). Find the area of the image of the spherical triangle
[u, v, w] under ø.
(c) Identify R2 with C so o(x, y) =+i. Show that o-( ) = (x,-y,-2) for all (x, y, z) € S²\{±n}.
We remark that this shows that -10 is the rotation R1,0,0),m
4: Given a point u e S² and a (spherical) line L in S?, we define the (spherical) distance from u to L to be
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