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Vectors in Projective Geometry

Asked by Mark Pimm-Smith on August 28, 1997:
How do you define and operate with vectors in projective geometry?
(Note: the interested reader will find a brief introduction to projective geometry in response to another question on this web site).

Vectors in projective geometry--and in any kind of geometry on a curved space, for that matter--arise as tangent vectors. The best example to think of is the surface of a sphere. At any point on the sphere there is a tangent plane, and this tangent plane is a "vector space": a set of vectors that can be added and scaled as desired. Vector techniques can thus be used on the sphere. For example, given two intersecting curves on the sphere, one can measure the angle between them by looking at their tangent vectors and finding the angle between these vectors (with the dot product).

The same thing is true for the "projective plane" in projective geometry, and also for any of a wide class of mathematical objects called manifolds. Roughly speaking, a manifold is any set of points which can be covered by "patches", each of which can be flattened into something that looks like part of ordinary Euclidean space of some dimension. A sphere is an example of a manifold; the top and bottom hemispheres can be flattened into disks, even though the entire sphere itself cannot be. The projective plane is also an example of a manifold, as are most other smooth shapes you can think of.

On each manifold, there is a tangent space associated to every point. When the manifold is embedded in ordinary Euclidean space (as a sphere is), the tangent space is easy to visualize (in the case of a sphere, it's just a plane sitting inside ordinary space). But even in more abstract cases when things aren't as easy to visualize, the tangent space still exists, and vector techniques can be used to say a lot about the geometry of the manifold (such as defining the angle between two curves, even defining the length of a curve as the limiting value of successive approximations by short tangent vectors). All of this comes about through the techniques of calculus, which can relate tangent vectors of curves (which are essentially a kind of derivative) back to the curve itself.

I realize the above description is very sketchy (it's also a little inaccurate, for a knowledgeable reader will see that I have left out some important conditions and technical details). This is a huge subject, known as differential geometry, and it is hard to give a comprehensive introduction to it in such a short space. The interested reader may want to consult the book Calculus on Manifolds by Michael Spivak, which gives a reasonably elementary introduction to the subject (but it still requires some multivariable and advanced calculus as a prerequisite).

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