chap12 MATH477 - PROJECTIVE GEOMETRY b3 course 2003 Nigel...

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Unformatted text preview: PROJECTIVE GEOMETRY b3 course 2003 Nigel Hitchin hitchin@maths.ox.ac.uk 1 1 Introduction This is a course on projective geometry. Probably your idea of geometry in the past has been based on triangles in the plane, Pythagoras’ Theorem, or something more analytic like three-dimensional geometry using dot products and vector products. In either scenario this is usually called Euclidean geometry and it involves notions like distance, length, angles, areas and so forth. So what’s wrong with it? Why do we need something different? Here are a few reasons: • Projective geometry started life over 500 years ago in the study of perspective drawing: the distance between two points on the artist’s canvas does not rep- resent the true distance between the objects they represent so that Euclidean distance is not the right concept. The techniques of projective geometry, in particular homogeneous coordinates, provide the technical underpinning for perspective drawing and in particular for the modern version of the Renaissance artist, who produces the computer graphics we see every day on the web. • Even in Euclidean geometry, not all questions are best attacked by using dis- tances and angles. Problems about intersections of lines and planes, for example are not really metric. Centuries ago, projective geometry used to be called “de- 2 scriptive geometry” and this imparts some of the flavour of the subject. This doesn’t mean it is any less quantitative though, as we shall see. • The Euclidean space of two or three dimensions in which we usually envisage geometry taking place has some failings. In some respects it is incomplete and asymmetric, and projective geometry can counteract that. For example, we know that through any two points in the plane there passes a unique straight line. But we can’t say that any two straight lines in the plane intersect in a unique point, because we have to deal with parallel lines. Projective geometry evens things out – it adds to the Euclidean plane extra points at infinity, where parallel lines intersect. With these new points incorporated, a lot of geometrical objects become more unified. The different types of conic sections – ellipses, hyperbolas and parabolas – all become the same when we throw in the extra points. • It may be that we are only interested in the points of good old R 2 and R 3 but there are always other spaces related to these which don’t have the structure of a vector space – the space of lines for example. We need to have a geometrical and analytical approach to these. In the real world, it is necessary to deal with such spaces. The CT scanners used in hospitals essentially convert a series of readings from a subset of the space of straight lines in R 3 into a density distribution....
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chap12 MATH477 - PROJECTIVE GEOMETRY b3 course 2003 Nigel...

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