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Rational Distance Sets on Conic Sections

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Rational Distance Sets on Conic Sections Rational Distance Sets on Conic Sections Megan D. Ly Shawn E. Tsosie Lyda P. Urresta Loyola Marymount UMASS Amherst Union College July 2010 Abstract Leonhard Euler noted that there exists an infinite set of rational points on the unit circle such that the pairwise distance of any two is also rational; the same statement is nearly always true for lines and other circles. In 2004, Garikai Campbell considered the question of a rational distance set consisting of four points on a parabola. We introduce new ideas to discuss a rational distance set of four points on a hyperbola. We will also discuss the issues with generalizing to a rational distance set of five points on an arbitrary conic section. 1 Introduction A rational distance set is a set whose elements are points in R2 which have pairwise rational distance. Finding these sets is a difficult problem, and the search for rational distance sets with rational coordinates is even more difficult. The search for rational distance sets has a long history. Leonard Euler noted that there exists an infinite set of rational points on the unit circle such that the pairwise distance of any two is also rational. In 1945 Stanislaw Ulam posed a question about a rational distance set: is there an everywhere dense rational distance set in the plane [2]? Paul Erd¨osalso considered the problem and he conjectured that the only irreducible algebraic curves which have an infinite rational distance set are the line and the circle [4]. This was later proven to be true by Jozsef Solymosi and Frank de Zeeuw [4]. First, we provide the formal definition of a rational distance set: 2 Definition 1.1. A rational distance set S is a set of elements Pi = (xi; yi) 2 R , 1 ≤ i ≤ n, such that for all Pi;Pj 2 S, jjPi − Pjjj is a rational number. We are primarily concerned with finding rational distance sets of rational points. p 2 2 The distance between two points on a graph is (xi − xj) + (yi − yj) . It will prove useful to rewrite this formula as s 2 yi − yj jxi − xjj 1 + (1) xi − xj Now, consider the following lemma: 1 Lemma 1.2. The rational solutions to the equation α2 = 1 + β2 (2) m2 + 1 m2 − 1 can be parameterized by α = ; β = , for m 2 ; m 6= 0. 2m 2m Q For the proof, refer to Campbell [1, Lemma 2.1]. 2 Main results 2.1 Line Theorem 2.1. For any line L: ax + by + c = 0, for a; b; c 2 Q, there exists a dense rational distance set with rational coordinates if and only if there exists m 2 Q such that a2 + b2 = m2. Proof. Suppose L has a dense rational distance set with rational coordinates. Then a c we can express the line L as y = − b x − b . Consider two points on L,(xi; yi) and (xj; yj). Using Lemma 1.2, we can write the distance between these two points as r a2 jx − x j 1 + : (3) i j b a2 2 2 2 2 2 The distance is rational only if 1 + b2 = n for n 2 Q. That is, if a + b = b n . Hence, we may set m = bn. a c Suppose a2 +b2 = m2 for m 2 Q. Then line L can be written as y = − x− . The bq b a 2 distance between two rational points on L,(xi; yi) and (xj; yj), is jxi − xjj 1 + b . 2 2 2 a2 m2 Note that from Lemma 1.2 we can write a + b = m as 1 + b2 = b2 . Hence, we see q a2 m 1 + b2 = b , which is rational. 2.2 Circle To show that there exists a rational distance set of rational points on a circle, we identify the cartesian plane R2 with the complex plane C. Theorem 2.2. Let r 2 Q with r 6= 0 and let C : kz − z0k = r be a circle in the complex plane with z0 = x+iy; x; y 2 Q. Then, there exists a dense rational distance set on C consisting of rational points. 2 1 Sketch of Proof. Consider the vertical line L: Re z = 2r . Let f : C ! C be the M¨obiustransformation defined by (z − r)z − 1 f(z) = 0 : z Then f maps the line L into the circle C. The following figure shows this for a circle with radius r < 1 centered at z0. A dense rational distance set of rational points on L is given by 1 s2 − 1 S = z = 1 + i s 2 ; s 6= 0 : 2r 2s Q Notice that Lemma 1.2 implies that each z 2 S has rational norm kzk. The codomain f(S) = ff(z) z 2 Sg will then be a dense set of points on C with rational coordinates. Finally notice that f is a translation composed with the map z 7! −1=z. Then, the rationality of the distance between the points on the codomain follows from the identity 1 1 kz − wk − = : z w kzk · kwk 2.3 Rational Distance Set of Three Points on a Parabola Consider the parabola y = ax2 + bx + c where a; b; c 2 Q. We provide a geometric intuition of the problem. An important note for this construction is that while the points are at rational distance, the points themselves do not have to be rational. We present a method for constructing a rational distance set of three points on a parabola. Note that we can find a rational distance set using 3 (a) Two concentric circles with (b) Moving the construction rational radii are centered at along the parabola, we note some point on the parabola. that the length of segment P2P3 changes Figure 1: While P1P2 and P1P3 are rational, we cannot guarantee that P2P3 is also rational. Since the length of P2P3 changes, the segment P2P3 must have a rational length at some point on the parabola. this method on any conic section. First, we choose some point on the parabola and construct two circles of rational radii around it, as shown in Figure 1. In Figure 1, 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 construction 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 where the segment P2P3 is rational. Now we turn to the more difficult problem of finding rational distance sets of rational points on the parabola. By equation 1, the distance dij between two rational points Pi and Pj on a parabola is q 2 dij = jxi − xjj 1 + (axi + axj + b) : 2 mij −1 1 m2−1 By Lemma 1.2, axi + axj + b = . We let g(m) = ( − b), which leads us to 2mij a 2m the following theorem: 2 Theorem 2.3. Given a parabola y = ax +bx+c, let S = fP1;P2;P3g. S is a rational distance set of rational points if and only if 1 ≤ i < j ≤ 3 we can find a nonzero 4 rational value mij such that g(m ) + g(m ) − g(m ) x = 12 13 23 1 2 g(m ) − g(m ) + g(m ) x = 12 13 23 2 2 −g(m ) + g(m ) + g(m ) x = 12 13 23 3 2 Proof. For each 1 ≤ i < j ≤ 3 where xi + xj = g(mij), we have three equations: x1 + x2 = g(m12) x1 + x3 = g(m13) x2 + x3 = g(m23) This is equivalent to the following row reduced augmented matrix: 0 g(m12)+g(m13)−g(m23) 1 1 0 0 2 B 0 1 0 g(m12)−g(m13)+g(m23) C @ 2 A −g(m12)+g(m13)+g(m23) 0 0 1 2 which gives us our desired values. 2.4 Rational Distance Set of Four Points on a Parabola We can follow a similar argument to find a rational distance set of four rational points on a parabola. 2 Theorem 2.4. Given a parabola y = ax + bx + c, let S = fP1;P2;P3;P4g. S is a rational distance set of rational points if and only if there are rational values mij, 1 ≤ i < j ≤ 4 such that 1 x = (g(m ) + g(m ) − g(m )) 1 2 12 13 23 1 x = (g(m ) − g(m ) + g(m )) 2 2 12 13 23 1 x = (−g(m ) + g(m ) + g(m )) 3 2 12 13 23 1 x = (−g(m ) − g(m ) + g(m ) + 2g(m )) 4 2 12 13 23 14 and g(m13) + g(m24) = g(m23) + g(m14) = g(m12) + g(m34) 5 Proof. For each 1 ≤ i < j ≤ 4 where xi + xj = g(mij), we have six equations: x1 + x2 = g(m12) x1 + x3 = g(m13) x1 + x4 = g(m14) x2 + x3 = g(m23) x2 + x4 = g(m24) x3 + x4 = g(m34) This is equivalent to the following row reduced augmented matrix: 0 g(m12)+g(m13)−g(m23) 1 1 0 0 0 2 B 0 1 0 0 g(m12)−g(m13)+g(m23) C B 2 C B −g(m12)+g(m13)+g(m23) C B 0 0 1 0 2 C B −g(m12)−g(m13)+g(m23)+2g(m14) C B 0 0 0 1 C B 2 C @ 0 0 0 0 g(m13) + g(m24) − g(m23) − g(m14) A 0 0 0 0 g(m12) + g(m34) − g(m23) − g(m14) which gives us our desired equations.
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