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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 , 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
(ii ) α is the rotation ρ0,θ and u =
(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
1 α∗ 5
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
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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