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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 5 1. When φ is a collineation, a point C is called a centre for φ if φ maps every line through C onto itself; and a line c is called an axis for φ if φ maps every point of c to itself. Show that each of the following has a centre and an axis, when considered as a collineation of the real projective plane: ( i ) Any translation ( ii ) Any halfturn ( iii ) Any reflection. Solution. ( i ) For the identity, every point is a centre and every line is an axis. Now consider an arbitrary nontrivial translation τ = τ −→ PQ . Recall that τ maps any line ℓ to a line parallel to ℓ . So τ fixes every point at infinity, ie the line at infinity, ℓ ∞ , is an axis for τ . Also, τ maps any line parallel to PQ onto itself. So the point at infinity on the line PQ is a centre for τ . ( ii ) The halfturn η P with centre P maps each point at infinity to itself, since each point at infinity is the point of intersection of a family of parallel lines in E , which are mapped to one another by η P . b b b b b P ℓ η P ( ℓ ) b ℓ ∞ Thus ℓ ∞ is an axis for η P . The point P is a centre for η P , since every ordinary point on each line through P is mapped to another ordinary point on the same line, while each point at infinity is fixed. ( iii ) The reflection σ ℓ with axis ℓ fixes every point (including the point at infinity) on ℓ . So ℓ is an axis for σ ℓ . Also, σ ℓ maps every line perpendicular to ℓ onto itself. So σ ℓ has as centre the point at infinity on the lines 2 perpendicular to ℓ . m ℓ σ ℓ ( m ) n 1 n 2 ℓ ∞ b b 2. Suppose that φ is a collineation which has a centre C and an axis c , and that φ maps P to P ′ , where P is a point distinct from C and not on c . Find a construction for φ ( Q ), where Q is any point not on the line CP , given C , c , P and P ′ . Solution....
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 Three '11
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 Geometry, Topology, Conic section, ηp

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