t03 - 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 2 1 , find the image 1. Given that ℓ is the line through (1, 0) with direction u = of the point (x, y ) under the glide reflectionγℓ,u . 2. If the three lines ℓ, m and n are neither all parallel nor concurrent, then σℓ σm σn is a glide reflection. Find the line b and the vector u such that σℓ σm σn = γb,u in the following circumstances. (i ) ℓ, m and n are the lines x = 1, y = 0 and x + y = 2. (ii ) ℓ, m and n are the lines y = x, y = 0 and x + y = 1. 3. (i ) Suppose that N is a point on line ℓ. Find the set of all isometries which map ℓ onto the line through N perpendicular to ℓ. (ii ) Is this set a group ? 4. If α is an affine transformation, then its derivative α∗ maps vectors to vectors. Find α∗ (u) when (i ) α is a translation and u is any vector 1 (ii ) α is the rotation ρ0,θ and u = 0 (iii ) α is a half-turn and u is any vector (iv ) α is a reflection and (a) u is parallel to the axis, or (b) u is perpendicular to the axis. 5. (i ) Find the derivative α∗ of the affine transformation α, given that α∗ 3 1 and = 4 1 α∗ 5 −1 . = 6 1 (ii ) Write down a formula for α, given the extra information that α(0, 0) = (3, 5). 6. Find the formula for the unique affine transformation α with derivative 4 1 6 2 which maps (1, 1) to (3, 5). 7. Find a formula for the affine transformation α which maps (1, 1) to (3, 8), (3, 8) to (2, 5) and (2, 5) to (1, 1). ...
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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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