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Since 1637 Rene Descartes’ two principal lines have evolved into the more familiar array of straight and/or curved lines, the grids on which we display the geometric analogues of algebraic equations. The chief application of this analytic (or algebraic) geometry comes to us through Isaac Newton’s dynamic geometry, which we now call mathematical physics. In that example the curves that we draw on our coordinate grids represent laws of nature: an ellipse, for example, represents the law of gravity by displaying the path that a small, light body follows due to the gravitational attraction of a much heavier body at one of the foci.
Though Descartes (and even Newton) never used it in their works, we refer to the grid comprising sets of parallel straight lines, the lines of each set crossing the lines of the others at a right angle, as a Cartesian grid. We use one set of mutually parallel straight lines for each dimension of the space (usually an abstract space) in which we want to work out our analytic geometry: the six-dimensional phase space that we use in some branches of physics provides a good example of such an abstract space. But the Cartesian grid of squares, cubes, or their higher dimensional analogues doesn’t always provide the best choice for describing a physical situation. In the example given above of an elliptical orbit, we would better use a grid of plane polar coordinates.
In the plane we establish the chessboard grid of Cartesian coordinates by drawing an horizontal straight line that we designate our x-axis with the positive x-direction extending to our right. We draw a second straight line crossing the x-axis at a right angle and designate it the y-axis. We determine the positive y-direction by imagining standing where the two lines cross, which point we call the origin of our grid, with the positive x-direction extending to our right and then saying that the positive y-direction extends away from us in the forward direction. We then fill in the grid by drawing evenly spaced straight lines parallel to each of the axes. On that grid we can locate any point in the plane by listing its coordinates [x,y], in which x represents the distance of the point from the y-axis and y represents the distance of the point from the x-axis.
Plane polar coordinates appear in a grid that consists of evenly spaced concentric circles with straight lines radiating away from their common center, the point that we designate the origin of the grid. We measure radial distance (r) along those straight lines from the origin. We designate one of those lines the grid’s pole and measure angular distance (θ) from it, positive angles in the counterclockwise direction and negative angles in the clockwise direction. Thus we locate any point in the plane by specifying the coordinate pair [r,θ].
To construct a geometric figure on a coordinate grid we set aside our Euclidean compass and straightedge and use an equation that expresses one coordinate’s values as a function of the other coordinate’s values. With that equation we calculate the coordinate pairs representing the locations of the points that comprise the figure. Thus we use Descartes’ insight to bring the concept of measurement into geometry in a way that lets us unite geometry with algebra. So, if we put the primary focus of our ellipse (its center of gravity if it represents an orbit) onto the origin of a Cartesian grid, then we describe it by calculating the y-coordinate corresponding to any x-coordinate through the equation
In that equation a represents the length of the semi-major axis of the ellipse (half the long axis of symmetry), b represents the length of the semi-minor axis (half the length of the short axis of symmetry), and e represents the ellipse’s eccentricity, which we can calculate from the equation
Note that if we let b=a, then e=0 and Equation 1 becomes the equation describing a circle centered on the origin of our Cartesian grid.
On a polar grid we again put the primary focus of our ellipse on the origin and put the point that we would identify as the periapsis (the perihelion of a planetary orbit) onto the grid’s pole. In this case we describe the ellipse by calculating the r-coordinate corresponding to any givenθ-coordinate through the equation
By taking arbitrary values for the angle, calculating the corresponding radial distance, and then plot those points on the grid we create a pointillist image of the ellipse. If we create a sufficiently dense set of points we can create the illusion of a continuous line, an Impressionist’s figure that coincides with a true ellipse. On that basis we can say that Equation 1 or Equation 3 give us something like the Platonic Form of an ellipse.
It may happen that two researchers studying the ellipse and its dynamic properties may each use one or the other of those representations. Miscommunication thus becomes inevitable, as if, while discussing the new analytic geometry Mr. Descartes spoke only French and Mr. Newton spoke only English (we ignore for the moment the facts that both men actually spoke fluent Latin and that Descartes died when Newton was only eight years old). As in that example, we need a means of translating one set of data into the equivalent set of data in the other system.
Pick a point in the plane and draw a straight line from it to the origin of the grid. For this exercise we imagine that we have superimposed our grids, one upon the other, with their origins occupying the same point and with the pole of the polar grid coinciding with the positive x-axis of the Cartesian grid. We can now conceive our straight line from the origin ([0,0] in both grids) to the point [x,y] or [r,θ] as the hypotenuse of a right triangle whose sides have the lengths x (the distance of the point from the y-axis) and y (the distance of the point from the x-axis). With a hypotenuse of length r and a primary vertex angle of θ, that triangle epitomizes the definitions of the basic trigonometric functions, so we have
Now imagine that our two researchers study events a minuscule distance apart. They will need to use differentials and to translate between the two sets of differentials that they obtain from their measurements, so we differentiate Equations 4 and 5 to get
Those equations look like the transformation equations that we would use between two Cartesian grids that have a common origin and an angle ofθ between their positive x-axes and they look that way for a good reason: at the differential scale all orthogonal coordinates coincide with Cartesian grids in the regions around the points where we make the translations. Recall that in the differential calculus a segment of any smooth, continuous curve becomes indistinguishable from a straight line in the limit as the length of the curve approaches the infinitesimal. That fact allows us to inter-relate our grids through the Pythagorean theorem in the form
in which gij represents the elements of the metric tensor describing the space that the coordinates qi and qj measure and N represents the number of dimensions in that space. Mathematicians and physicists call Equation 8 the metric equation. Note that in orthogonal coordinate systems (those in which all of the coordinate lines cross each other at right angles) gij=0 whenever ij. In that case we have
in which hi represents the scale factor in the line element
in which êi represents the unit vector pointing in the positive qi-th direction and ds2=dr•dr. We can then express the scale factor as
because Equation 10 gives us an exact differential.
In our analytic geometry we often want to transform between coordinate grids our descriptions of features other than lines. We may want to transform an area or a volume or some entity of even higher dimensionality.
We can describe a minuscule element of area dσ in terms of line elements, dr1 and dr2, that emanate from the same point and lie on two of its sides. The rhomboid thus defined has an area equal to the product of the lengths of dr1 and dr2 and of the sine of the angle between them. But that calculation coincides with the outer (or cross) product of the vectors dr1 and dr2:
in which the êi represent the unit vectors of the coordinate grid. In Cartesian coordinates hx=hy=hz=1, so we can write for a square area defined by the elemental vectors dx and dy
More generally we have
And in the most general case of all the determinant would have no zero terms, indicating that the element of area does not lie entirely parallel to one plane in the given space.
We can also describe a minuscule element of volume dτ as a rhombohedron whose edges we define by the three vectors dr1, dr2, and dr3 that emerge from the same point. The volume of that figure equals the area dr2×dr3 multiplied by the length of dr1 and the cosine of the angle between dr1 and dr2×dr3; which means, we have the inner (or dot) product
in which detJ represents the Jacobian determinant
At any point where detJ=0, that point represents a singular point of the coordinate transformation. Because we have detJ as a determinant, its vanishing as a point implies a lack of invertability of the coordinates at that point. That means that at the singular point some of the coordinates’ values remain undetermined (for example, the angular coordinatesθ and φ at the origin of the polar coordinate grid): we call those ignorable coordinates. Among the coordinates qi at the singular point let those coordinates for k+1≤i≤m be ignorable with respect to Cartesian coordinates xj: that means that any function of qi will be independent of the ignorable coordinates. So we must have as true to mathematics, in light of Equation 15,
In a two-dimensional space, for example, in which we use Equations 4 and 5 to convert polar coordinates to Cartesian coordinates, we have
which gives us
which vanishes at the origin. In this case q2=θ, the angular coordinate, represents the ignorable coordinate at the origin, so k=1 and we have
so we have Equation 17 as
Thus we obtain the information we need to convert between Cartesian and plane polar coordinates.
But this analysis sets us up to go beyond simple coordinate conversions, even those of Special Relativity. The metric tensor described above plays an important role in General Relativity, even coming into play as a dynamic object in its own right. In examining this material, then, we have begun to prepare ourselves to confront the most difficult and most profound theory in modern physics.
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