This entry is based on the prologue for the book Elliptic Tales: Curves, Counting, and Number Theory (challenging but quite fun reading on the morning Tube commute!):

is the familiar equation for the unit circle and in the prologue the authors show how a straight line with a rational slope intersects a circle at two point which is rational i.e, of the form then the second point is also rational and that all such lines trace out the full set of rational points on the circle.

But then the book goes further –

We say that circle has a “rational parametrization”. This means that the coordinates of every point on the circle are given by a couple of functions that are themselves quotients of polynomials in one variable. The rational parametrization for we have found is the following:

So this is what I wanted to check… it surely isn’t claiming that all the points on the circle are rational, is it? Merely that the above – if (which corresponds to the slope of our line through the two points) is rational, generates the full set of rational points on the circle. Because if is not rational then the second point will not be either? Is that right?

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Not having read the book, I can only speculate on the intent of the quote, but I think they mean that the rational functions given generate the entire circle, not just the rational points on it. The “rational” in “rational parametrization” may refer to a rational function (quotient of polynomials) rather than a function of a rational argument. The specified function maps all rational values of m to all rational points on the circle and all irrational values of m to all irrational points on the circle.

You never let me down! Your explanation makes complete sense, and answers what was confusing me.