t02 - ℓ and m such that σ ℓ σ m is the rotation about...

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The University of Sydney Math3061 Geometry and Topology Web page: www.maths.usyd.edu.au/u/UG/SM/MATH3061/ 2009 Tutorial 2 1. Consider the set S consisting of the four transformations α , β , γ , ι given by α ( x, y ) = ( x, - y ) , β ( x, y ) = ( - x, y ) , γ ( x, y ) = ( - x, - y ) , ι ( x, y ) = ( x, y ) . ( i ) Describe α , β , γ , ι geometrically. ( ii ) Show that S is a group. 2. Suppose that and m are lines, and that n = σ m ( ). Show that and n are parallel if and only if either is parallel to m or is perpendicular to m . 3. Suppose that α is an isometry, m is a line and Q is a point ( m refers to the Frst part of the question and Q refers to the second part). Show that: ( i ) ασ m α 1 is a re±ection with axis α ( m ), ( ii ) αη Q α 1 is the halfturn about α ( Q ), ( iii ) if β is any rotation then αβα 1 is a rotation. 4. ²ind the image X = ρ ( X ) of the the given point X under the following rota- tions ρ . ( i ) X = (3 , 4) , ρ = ρ O,π/ 3 ( ii ) X = (3 , 4) , ρ = ρ Q,π/ 3 where Q = (1 , 2). 5. ( i ) ²ind equations for a pair of lines
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Unformatted text preview: ℓ and m such that σ ℓ σ m is the rotation about (1 , 1) with angle π 3 . ( ii ) Describe the isometry σ ℓ σ m when ℓ is the line x + 2 y = 0 and m is the line 2 x-y = 0. 6. ( i ) ²ind equations for a pair of lines ℓ and m such that σ ℓ σ m is the translation τ u where u = b 1 √ 3 B . ( ii ) Describe the isometry σ ℓ σ m when ℓ is the line x + y = 1 and m is the line x + y = 3. 7. By expressing each of α and β suitably as a composite of two re±ections, show the following. ( i ) If α and β are rotations then either αβ is a rotation or αβ is a translation. ( ii ) If α is a rotation and β is a translation then αβ is a rotation. 8. Show that the set of all translations and rotations is a group....
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This note was uploaded on 10/17/2011 for the course MATH 3061 taught by Professor ? during the Three '11 term at University of Sydney.

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