3. Postdoctoral Program - MSRI

For example, for a circle with radius r < 1 centered at z0 there exists a bijection
between the circle and the line as shown in figure 1. The rationality of the distance
between the points on the the codomain follows from the identity
� �
�
�
1 1 �
� − �
z w �
= �z − w�
�z� · �w� .
2.3 Finding Rational Distance Sets on a Parabola
Consider the parabola y = ax 2 + bx + c where a, b, c, ∈ Q.
We provide a geometric intuition of the problem. One major note for this con-
struction is that the points are at rational distance, but they do not have to be
rational. We present a method for constructing a rational distance set of three points
on a parabola: First, we choose some point on the parabola and construct a circle
of rational radius around it. We then construct a second concentric circle of rational
radius, as shown in figure 2.
In figure 2, we see that the line segments P1P2 and P1P3 are rational. All we
require now is that P2P3 be rational. This segment is not necessarily rational. We
can, however, make it rational. If we allow the contruction to move along the parabola,
we notice that the length of segment P2P3 changes. We see from this construction
that the length of segment P2P3 is a continuous function on an interval. Therefore,
we can find some point in the parabola such that the segment P2P3 is rational.
First, we will introduce minor results for clarification.
By equation 1, the distance dij between two rational points Pi and Pj on a
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