t05 - ( iv ) ±ind all the ²xed points of φ . 4. ±or...

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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 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 reFection. 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 . ±ind a construction for φ ( Q ), where Q is any point not on the line CP , given C , c , P and P . 3. Suppose that φ is the unique collineation which maps (1 : 0 : 0) to (0 : 1 : 0), (0 : 1 : 0) to (0 : 0 : 1), (0 : 0 : 1) to (1 : 1 : 1), and (1 : 1 : 1) to (1 : 0 : 0). ( i ) ±ind a matrix for φ . ( ii ) ±ind the line φ ( ). ( iii ) ±ind the image under φ of the line x + y + z = 0.
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Unformatted text preview: ( iv ) ±ind all the ²xed points of φ . 4. ±or each of the following conics, ²nd an associated symmetric matrix and show that the conic is nonsingular. ±ind all points where ℓ ∞ meets the conic and say whether the conic is an ellipse, a hyperbola or a parabola. ( i ) x 2 + y 2-z 2 + 2 xy-2 yz + 4 zx = 0 ( ii ) x 2-y 2 + 2 xy-2 yz + 4 zx = 0 ( iii ) x 2 + 4 y 2-2 xy + 2 z 2 + 2 yz-6 zx = 0 5. ±or all real λ, let C ( λ ) denote the conic x 2 + 2 λxy + y 2-17 10 xz-17 10 yz-2 λz 2 = 0 . Show that ( i ) C ( λ ) passes through each of the points (4 : 1 : 2) and (1 : 4 : 2), for all λ . ( ii ) C ( λ ) is a hyperbola if | λ | > 1. ( iii ) C ( λ ) is an ellipse if | λ | < 1. ( iv ) C ( λ ) is a parabola if λ =-1. ( v ) C ( λ ) is a singular conic if λ = 1. ( vi ) C ( λ ) is a circle if λ = 0....
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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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