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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 4 1. Show that if PQR is a triangle and is an affine transformation which maps P to Q , Q to R and R to P then ( i ) 3 = the identity ( ii ) is an isometry if and only if the triangle PQR is equilateral. Solution. ( i ) Since PQR is a triangle (that is, P, Q, R are non-collinear), there is a unique affine transformation which maps P to P , Q to Q and R to R . Now 3 ( P ) = 2 ( Q ) = ( R ) = P 3 ( Q ) = 2 ( R ) = ( P ) = Q 3 ( R ) = 2 ( P ) = ( Q ) = R and, also, ( P ) = P, ( Q ) = Q, ( R ) = R. Therefore 3 = . ( ii ) If is an isometry then PQR is equilateral, since d ( P,Q ) = d ( ( P ) , ( Q )) = d ( Q,R ) and d ( Q,R ) = d ( ( Q ) , ( R )) = d ( R,P ) . Conversely, suppose PQR is equilateral. Consider the rotation about the centre of the triangle, through angle 2 / 3. Now maps P to Q , Q to R and R to P . However, there is only one affine transformation which maps P to Q , Q to R and R to P , since P, Q, R are non-collinear. So = and we see that is an isometry. 2. Which of the following maps of the plane to itself are affine transformations? Which are isometries? Classify those that are isometries. ( i ) ( x,y ) = (2 x 2 + 3 y + 3 , 4 y + 1) ( ii ) ( x,y ) = ( y + 4 ,x 7) ( iii ) ( x,y ) = (2 x + 4 y 7 ,x + 2 y + 1) ( iv ) ( x,y ) = ( y, 3 x + 4) 2 Solution. Only (ii) and (iv) are affine transformations. Indeed, (i) and (iii) are not even transformations, since in (i), ( 1 , 0) = (1 , 0) and so is not injective, and in (iii), the function maps the entire Euclidean plane onto the line y = 1 2 ( x +9), so...
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