7 - 4. Rotation of the sphere by 180◦ about the X -axis...

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Unformatted text preview: 4. Rotation of the sphere by 180◦ about the X -axis has the effect φ : (X, Y, Z ) → (X, −Y, −Z ). Recalling that the projection from the sphere to the plane is given by π : (X, Y, Z ) → Thus Y X +i . 1−Z 1−Z X −Y +i , 1+Z 1+Z π ◦ φ(X, Y, Z ) = π (X, −Y, −Z ) = and 1 = π (X, Y, Z ) = = = = = Which gives π ◦ φ(X, Y, Z ) = X 1− Z (1 − Z ) X + iY (1 − Z )(X − iY ) X2 + Y 2 (1 − Z )(X − iY ) 1 − Z2 X − iY 1+Z X −Y +i . 1+Z 1+Z 1 . π (X, Y, Z ) 1 z 1 Y + i 1− Z So rotation by 180◦ about the X -axis corresponds to z → in C. 7 ...
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This note was uploaded on 07/27/2010 for the course MATH Math2009 taught by Professor Koskesh during the Spring '09 term at SUNY Empire State.

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