# Download PDF by Chazelle B., Goodman J.E., Pollack R. (eds.): Advances in Discrete and Computational Geometry By Chazelle B., Goodman J.E., Pollack R. (eds.)

ISBN-10: 0821806742

ISBN-13: 9780821806746

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Extra resources for Advances in Discrete and Computational Geometry

Example text

This intersection point is of course exactly the representative of the form (x : y : 1) and the lines in the xy-plane have homogeneous coordinates (x : y : 0). Now, if is a line in the plane z = 1, then the union of all the 1-dimensional spaces intersecting the plane in that line is the intersection of the plane with a 2-dimensional subspace (cf. Fig. 20). Hence it is natural to say that a line in the projective plane is the set of 1-dimensional subspaces that lie in a given 2-dimensional subspace.

Ak ], −[a1 ], −[a2 ], . . , −[ak ] is the set of zeroes of h counted with multiplicity. We may also choose a sequence [b1 ], [b2 ], . . , [bk ] which does the same thing for the poles (and these sequences have the same length as the number of poles and zeroes are the same). Now put g(z) := ℘ (z) − ℘ (ai ) . i ℘ (z) − ℘ (bi ) i This elliptic function has the same pole and zero sets (counted, of course, with multiplicities) as h, as the poles at  from the different factors cancel, which means that h(z)/g(z) has neither zeroes nor poles and is hence a constant which finishes the proof.

Figure 24 gives an example where one starts with one point and then takes multiples of it. Note that the inverse of a point (x, y) ∈ X is given by (x, −y). 7 1 5 3 8 2 9 4 Figure 24. A point and some of its multiples. 50 5 The group structure on an elliptic curve We have not actually proved that the binary operation thus defined gives us a commutative group. For those elliptic curves over the complex numbers that actually come from doubly periodic functions, it is clear as we have seen that the binary operation corresponds to addition of complex numbers modulo a lattice.