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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 is a collineation, a point C is called a centre for if maps every line through C onto itself; and a line c is called an axis for if 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 non-trivial translation = PQ . Recall that maps any line to a line parallel to . So fixes every point at infinity, ie the line at infinity, , is an axis for . Also, maps any line parallel to PQ onto itself. So the point at infinity on the line PQ is a centre for . ( ii ) The halfturn 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 P . b b b b b P P ( ) b Thus is an axis for P . The point P is a centre for 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 with axis fixes every point (including the point at infinity) on . So is an axis for . Also, maps every line perpendicular to onto itself. So has as centre the point at infinity on the lines 2 perpendicular to . m ( m ) n 1 n 2 b b 2. Suppose that is a collineation which has a centre C and an axis c , and that maps P to P , where P is a point distinct from C and not on c . Find a construction for ( Q ), where Q is any point not on the line CP , given C , c , P and P . Solution....
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