t03sol - The University of Sydney Math3061 Geometry and...

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Unformatted text preview: The University of Sydney Math3061 Geometry and Topology Web page: www.maths.usyd.edu.au/u/UG/SM/MATH3061/ 2009 Tutorial 3 1. Given that is the line through (1 , 0) with direction u = bracketleftBig 2 1 bracketrightBig , find the image of the point ( x, y ) under the glide reflection , u . Solution. We note, to begin with, that Q = (1 , 0) is a point on and that u = 1 5 bracketleftbigg 1 2 bracketrightbigg is a unit vector perpendicular to . Let P = ( x, y ) and ( P ) = ( x , y ). Then PQ = bracketleftbigg 1 x y bracketrightbigg and P ( P ) = bracketleftbigg x x y y bracketrightbigg . We have P ( P ) = bracketleftbigg x x y y bracketrightbigg = 2( PQ. u ) u = 2 parenleftbigg (1 x ) 2 y 5 parenrightbigg 1 5 bracketleftbigg 1 2 bracketrightbigg = 2 5 ( x 2 y 1) bracketleftbigg 1 2 bracketrightbigg = bracketleftbigg ( 2 x + 4 y + 2) / 5 (4 x 8 y 4) / 5 bracketrightbigg Therefore x = x + ( 2 x + 4 y + 2) / 5 = (3 x + 4 y + 2) / 5 and y = y + (4 x 8 y 4) / 5 = (4 x 3 y 4) / 5 . Thus ( P ) = ((3 x + 4 y + 2) / 5 , (4 x 3 y 4) / 5) . This completes the reflection part of the glide reflection. Now u = bracketleftbigg 2 1 bracketrightbigg . Thus , u ( P ) = u ( P ) = u ( 3 x +4 y +2 5 , 4 x 3 y 4 5 ) = (2 + 3 x +4 y +2 5 , 1 + 4 x 3 y 4 5 ) = ( 3 x +4 y +12 5 , 4 x 3 y +1 5 ) . 2 2. If the three lines , m and n are neither all parallel nor concurrent, then m n is a glide reflection. Find the line b and the vector u such that m n = b, u in the following circumstances. ( i ) , m and n are the lines x = 1, y = 0 and x + y = 2. ( ii ) , m and n are the lines y = x , y = 0 and x + y = 1....
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t03sol - The University of Sydney Math3061 Geometry and...

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