Rational Parameterizations of the Circle
In chapter 2 we offered the rational parameterization
of the circle x2 + y2 = 5 and here we disclose its derivation.
First, a reminder of one definition and the statement of another:
• A rational point in the plane is one for which both coordinates are rational numbers.
• A rational line in the plane is one whose equation can be written as ax + by + c = 0, where a, b, c are rational numbers.
We can set up a one-to-one correspondence between the rational points on a circle and the rational points on a rational line in the following way, referring to figure B.1:
• Draw the circle.
• Given there is one, pick any rational point P on the circle to act as a fixed origin.
• Draw a convenient rational line L.
• For every other rational point Q on the circle draw the line PQR, where R is the point of intersection of the line with L.
• The one-to-one correspondence is Q ↔ R.
To establish the one-to-one correspondence, for any rational point Q on the circle, since P is a rational point, the line PQ is a rational line. It is a trivial fact that if two rational lines intersect, they do so in a rational point and this ensures that R, which is the intersection of PQ and L, is a rational point. Conversely, take any rational point R on L and draw the rational line PR. The intersection of this line and our circle will yield a quadratic equation (in x, say, and so of the form ax2 + bx + c = 0) with rational coefficients and with one of its roots the (rational) x coordinate of P. Since the sum of the two roots = −b/a, which is rational, the other root must be rational and we have the x coordinate and therefore y coordinate of the rational point Q on the circle. The one-to-one correspondence is thereby established. For the intersection of the two lines to be assured we must ensure that they are not parallel and for convenience we will take L to be the line perpendicular to OP and passing through O; also, the point P itself is associated with ‘the point at infinity’ of L.
Figure B.1.
Figure B.2.
With the general idea dealt with, we will first see how the method generates the expected algebraic parameterization of the standard unit circle x2 + y2 = 1.
Take P = (−1, 0) and L the y-axis, as shown in figure B.2. Our line PQR has equation y = t(1 + x), making R = (0, t) and the x coordinate of Q given by the equation
1 − x2 = [t(1 + x)]2 = t2(1 + x)2
cancelling by the known factor of 1 + x results in x = (1 − t2)/(1 + t2) and substituting back into the y equation results in y = 2t/(1 + t2) and so we have the standard rational parameterization
Figure B.3.
For x2 + y2 = 5, referring to figure B.3, we will take P = (1, 2), then L has equation y = −
x. Our line PQR has equation y − 2 = t(x − 1) and this makes
and the x coordinate of Q given by the equation 5 − x2 = [t(x − 1) + 2]2, which becomes the quadratic equation
(1 + t2)x2 + 2t(2 − t)x + (t2 − 4t − 1) = 0.
We know that x = 1 is a root and that the sum of the two roots is −(2t(2 − t))/(1 + t2) and therefore the x coordinate of Q = −(2t(2 − t))/(1 + t2) − 1 = (t2 − 4t − 1)/(1 + t2) and substitution gives its y coordinate as y = (2 − 2t − 2t2)/(1 + t2).
So, if there is a single rational point P on the circle, the procedure generates the full infinity of them but fails with the likes of x2 + y2 = 3 since P simply does not exist. The method applies to conic sections in general.
