stoutfellow (![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png)
stoutfellow) wrote2006-01-14 04:08 pm
stoutfellow) wrote2006-01-14 04:08 pm
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DG, Part 0: Are We Not Geometers?
Before I begin with differential geometry proper, I'd like to make a few comments on the use of coordinates. Coordinate geometry was developed independently and more-or-less simultaneously by René Descartes and Pierre de Fermat in the 17th century, but they developed it for different purposes, and that difference has consequences that are worth glancing at.
The great virtue of coordinate geometry is that it provides a connection between algebra and geometry, allowing problems in the one to be recast as problems in the other, so that the very different techniques of the two fields can both be brought to bear. Now, Fermat was at heart an algebraist; he was interested in numbers. Descartes was (in this respect) a geometer; his interest was in shapes and configurations. For Fermat, the geometry was a tool for solving algebraic questions; he had no need to make it any more complicated than strictly necessary. For Descartes, the geometry was central; it was the algebra, the numerical calculations, that were the tool.
Under Fermat's approach, the geometry is adapted to the algebra. A set system of coordinates is imposed on the plane (or on 3-space) ahead of time. Its axes are automatically assumed to be perpendicular, mostly for simplicity's sake. An algebraic function or relation is then graphed (for example) with respect to those axes, and the properties of the resulting shape are exploited to derive algebraic information. (It is ironic that, although we speak of "Cartesian" coordinates, the way they are taught in school more closely resembles Fermat's vision!)
Descartes' approach is different. The geometry is prior; if there is to be a coordinate system, it should be adapted to the shapes and configurations under study. The axes need not be perpendicular, if the situation dictates otherwise. (Indeed, in the configuration Descartes is studying when he introduces coordinates, the axes are almost guaranteed not to be perpendicular.) The center of coordinates and the axes should be chosen so as to make computations simpler.
Let me clarify with an example. An ellipse in the plane comes equipped with various special points and lines: a pair of foci, a center, a major and a minor axis, and two directrices. If you want to study the ellipse, the most sensible way to choose coordinates is to put the origin at the center of the ellipse, the x-axis along the major axis of the ellipse, and the y-axis along its minor axis. This makes the equation of the ellipse exceptionally simple: it has the form x2/a2+y2/b2=1, where a and b are the lengths of the two semiaxes of the ellipse, and the locations of the foci and directrices can be quickly determined. If the coordinate system had been prearranged, the equation would take the form Ax2+Bxy+Cy2+Dx+Ey+F=0, where the constants A,B,C,D,E,F satisfy certain conditions, and identification of the various special points and lines would be much more complicated.
There are two points, then, that must be kept in mind in studying geometry. First, the coordinate system should always be adapted to the problem. It may be that, after studying the situation, you will have to change to another coordinate system; in fact, as your attention shifts from aspect to aspect of the problem, it may be necessary to do this repeatedly. (This is not as daunting as it sounds; to some extent, even this can be systematized, and that's one of the topics I plan to address.) Second, note that in the first version of the equation of the ellipse, the coefficients a,b can be easily interpreted geometrically: they have clear meaning. In the second version, the coefficients A,B,C,D,E,F are artifacts of the choice of coordinates, and do not have any real geometric meaning. In studying a geometric problem, you should look for quantities which do have such meaning; one good indicator is coordinate-invariance - in other words, if the expression defining a quantity doesn't change when you change coordinate systems, there's a good chance that it actually has geometric significance.
I'm speaking rather vaguely and generally, I realize, but we'll get down to specific examples shortly.
The great virtue of coordinate geometry is that it provides a connection between algebra and geometry, allowing problems in the one to be recast as problems in the other, so that the very different techniques of the two fields can both be brought to bear. Now, Fermat was at heart an algebraist; he was interested in numbers. Descartes was (in this respect) a geometer; his interest was in shapes and configurations. For Fermat, the geometry was a tool for solving algebraic questions; he had no need to make it any more complicated than strictly necessary. For Descartes, the geometry was central; it was the algebra, the numerical calculations, that were the tool.
Under Fermat's approach, the geometry is adapted to the algebra. A set system of coordinates is imposed on the plane (or on 3-space) ahead of time. Its axes are automatically assumed to be perpendicular, mostly for simplicity's sake. An algebraic function or relation is then graphed (for example) with respect to those axes, and the properties of the resulting shape are exploited to derive algebraic information. (It is ironic that, although we speak of "Cartesian" coordinates, the way they are taught in school more closely resembles Fermat's vision!)
Descartes' approach is different. The geometry is prior; if there is to be a coordinate system, it should be adapted to the shapes and configurations under study. The axes need not be perpendicular, if the situation dictates otherwise. (Indeed, in the configuration Descartes is studying when he introduces coordinates, the axes are almost guaranteed not to be perpendicular.) The center of coordinates and the axes should be chosen so as to make computations simpler.
Let me clarify with an example. An ellipse in the plane comes equipped with various special points and lines: a pair of foci, a center, a major and a minor axis, and two directrices. If you want to study the ellipse, the most sensible way to choose coordinates is to put the origin at the center of the ellipse, the x-axis along the major axis of the ellipse, and the y-axis along its minor axis. This makes the equation of the ellipse exceptionally simple: it has the form x2/a2+y2/b2=1, where a and b are the lengths of the two semiaxes of the ellipse, and the locations of the foci and directrices can be quickly determined. If the coordinate system had been prearranged, the equation would take the form Ax2+Bxy+Cy2+Dx+Ey+F=0, where the constants A,B,C,D,E,F satisfy certain conditions, and identification of the various special points and lines would be much more complicated.
There are two points, then, that must be kept in mind in studying geometry. First, the coordinate system should always be adapted to the problem. It may be that, after studying the situation, you will have to change to another coordinate system; in fact, as your attention shifts from aspect to aspect of the problem, it may be necessary to do this repeatedly. (This is not as daunting as it sounds; to some extent, even this can be systematized, and that's one of the topics I plan to address.) Second, note that in the first version of the equation of the ellipse, the coefficients a,b can be easily interpreted geometrically: they have clear meaning. In the second version, the coefficients A,B,C,D,E,F are artifacts of the choice of coordinates, and do not have any real geometric meaning. In studying a geometric problem, you should look for quantities which do have such meaning; one good indicator is coordinate-invariance - in other words, if the expression defining a quantity doesn't change when you change coordinate systems, there's a good chance that it actually has geometric significance.
I'm speaking rather vaguely and generally, I realize, but we'll get down to specific examples shortly.