Catálogo de publicaciones - libros

An interesting surface was discussed by H. F. Minding in the 1830s in papers he published in Crelle’s [159] although these papers were then largely forgotten for 30 years; this was a surface of constant negative curvature with the property that geodesics between points are not necessarily unique. Minding’s surface is formed by rotating a tractrix about its vertical axis. A tractrix — the name is due to Huyghens — is the curve of the obstinate dog. A point on a line ℓ is attached to a point by a curve of fixed length. , the dog, is dragged behind , the owner, who walks along ℓ, and the path of is called the tractrix. It is conventional for the walk to start with perpendicular to ℓ. Minding’s surface, which is sometimes called the pseudosphere, is formed by rotating the tractrix about ℓ; the point then generates a circle of singular points (see Figure 11.1).

This is an unusual chapter, one of three, that offers advice on how to tackle a piece of assessed work. An example of suitable assessment questions will be found at the end of this chapter, along with yet more advice.

In 1827 Möbius published his [162] or . The word “barycentre” means centre of gravity, but the book is not about mechanics but geometry. It is most concerned with, and is best remembered for, introducing a new system of coordinates, the barycentric coordinates.

An algebraic plane curve is, by definition, the locus in the plane corresponding to a polynomial equation of some degree, : (,) = 0. For example + + = 0 represents a quartic. The equation may be written in homogeneous coordinates [] by setting = ′/′, = ′/′ and multiplying through by the lowest power of ′ that produces a polynomial equation (if you wish you may then remove the primes) when the curve is considered to lie in the projective plane. The example above becomes + + = 0 in homogeneous form.

This chapter provides a summary of the basic facts about what are called singular points on a curve, and an account of how they resolve the duality paradox. The next chapter gives the mathematical theory behind these facts.

We are interested in the local behaviour of a curve, that is to say what small pieces of them look like. Either they will have one branch passing through a point, or more than one. If only one, then the curve will have a tangent there. To study the nature of a curve near a particular point we may always suppose that we have moved the point to the origin.

In all this welter of original work on the geometry of curves, one matter has remained stubbornly unclear, although to a modern mathematician the need for it is painfully apparent. Is the subject the algebraic geometry of real curves in the real projective plane, or has everything migrated to the complex projective plane? In a sense, the question is anachronistic. It does not arise for Poncelet because, as we saw, he had his own way of talking about imaginary points and resisted very strongly Cauchy’s suggestion of making everything algebraic. But Plücker and Hesse were avowedly algebraic, yet they do not seem to have openly confronted the issue. At one stage, Plücker even outlined a way in which complex coordinates could be written out of the theory in favour of certain symmetry consideration (a topic there is not room to explore here). Generally speaking, he and Hesse seem to have pursued a policy of quiet acceptance: intersection points of one curve with another, tangents to a curve, and all manner of objects may be complex if the algebra forces them to be so — but one will not explore too closely. This is a curious position. A cubic curve, let us say, was thought of as a real object in the real plane with nine inflection points, six of which at least were necessarily imaginary. A cubic and a quartic curve meet in 12 points, of which in any given case quite a number might be complex. But a curve was not made up of complex points — points with complex coordinates — or if it was one did not enquire too closely how this could be so.

Riemann was the archetype of the shy mathematician, not much drawn to topics other than mathematics, physics and philosophy, devout in his religion, conventional in his tastes, close to his family and awkward outside them. As a child, he was taught by his father, a pastor, and then for some years at school before going to Göttingen University. There he had initially intended to study theology, in accordance with his father’s wishes — Göttingen was the only university in Riemann’s native Hanover with strong links to the Hanover church — but his remarkable ability at mathematics led him to switch subjects. He was always inclined to the conceptual side of things, rather than the computational or algorithmic. His written German is scholarly and old-fashioned — Victorian, one might say — and his Latin (required for academic purposes) is no easier.

Let us consider a surface in space, given by an equation of the form () = 0. We shall assume we have a map from a region of the plane, with coordinates () onto part of the surface, and that we can differentiate this map as often as we like. We assume that at each point of the surface the directions u-increasing and -increasing are distinct.

What Beltrami did in his , as the extracts given in Chapter 19 will have suggested, was to take the usual metric for a map of the sphere on the plane and modify it so that it was defined only within the unit disc, but the space mapped onto the interior of that disc had the following properties.