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«AN ANALOGUE OF PTOLEMY’S THEOREM AND ITS CONVERSE IN HYPERBOLIC GEOMETRY JOSEPH EARL VALENTINE Vol. 34, No. 3 July 1970 PACIFIC JOURNAL OF ...»

-- [ Page 1 ] --

Pacific Journal of

Mathematics

AN ANALOGUE OF PTOLEMY’S THEOREM AND ITS

CONVERSE IN HYPERBOLIC GEOMETRY

JOSEPH EARL VALENTINE

Vol. 34, No. 3 July 1970

PACIFIC JOURNAL OF MATHEMATICS

Vol. 34, No. 3, 1970

AN ANALOGUE OF PTOLEMY'S THEOREM AND ITS

CONVERSE IN HYPERBOLIC GEOMETRY

JOSEPH E. VALENTINE

The purpose of this paper is to give a complete answer to the question: what relations between the mutual distances of n (n ^ 3) points in the hyperbolic plane are necessary and sufficient to insure that those points lie on a line, circle, horocycle, or one branch of an equidistant curve, respectively ?

In 1912 Kubota [6] proved an analogue of Ptolemy's Theorem in the hyperbolic plane and recently Kurnick and Volenic [7] obtained another analogue. In 1947 Haantjes [4] gave a proof of the hyperbolic analogue of the ptolemaic inequality, and in [5] he developed techniques which give a new proof of Ptolemy's Theorem and its converse in the euclidean plane. In the latter paper it is further stated that these techniques give proofs of an analogue of Ptolemy's Theorem and its converse in the hyperbolic plane. However, Haantjes' analogue of the converse of Ptolemy's Theorem is false, since Kubota [6] has shown that the determinant | smh\PiPs/2) |, where i, j = 1, 2, 3, 4, vanishes for four points P19 P2, P3, P 4 on a horocycle in the hyperbolic plane. So far as the author knows, Haantjes is the only person who has mentioned an analogue of the converse of Ptolemy's Theorem in the hyperbolic plane.

Relations between the mutual distances of three points are obtained which are necessary and sufficient to insure that those points determine a line, circle, horocycle, or equidistant curve, respectively.

It will be recalled that Ptolemy's Theorem and its converse for the euclidean plane may be stated as follows.

(Ptolemy). Four points P19 P2, P3, P 4 of the euclidean

THEOREM

plane lie on a circle or line if and only if the determinant C(Plf P2, P3, P4) = I PiPj21 vanishes, where PiPά = P^ denotes the distance of the points Piy P3, (i, j = 1, 2, 3, 4).

The analogous theorem which will be proved in this paper is the following.

THEOREM. Four points Plf P2, P3, P 4 of the hyperbolic plane lie on a circle, line, horocycle, or one branch of an equidistant curve if and only if the determinant 818 JOSEPH E. VALENTINE I sinh (PiPj/2) \ vanishes (i, j = 1, 2, 3, 4).

Moreover, the techniques used in this paper provide an easy extension of this analogue of Ptolemy's Theorem and its converse to give necessary and sufficient conditions for n, (n ^ 4), points to lie on a line, circle, horocycle, or equidistant curve.

2* Preliminary remarks* It is well known [1, pp. 273-274] that the determinants A4(P,, P2, P3, P4) - ( cosh (PtPj) I, (i, j = 1, 2, 3, 4) and A5(Ply P2, P3, P4, P5) = I cosh (PtPj) I, (i, j = 1, 2, 3, 4, 5) vanish for each quadruple and each quintuple of points in the hyperbolic plane. Moreover, A,(Pίy P2, P3) = | cosh (P^) | ^ 0, (i, i = 1,2, 3) for each triple of points and vanishes if and only if the triple is collinear. These determinants play an important part in this paper.

Poincare's circular model of the hyperbolic plane will be used.

3* Analogue of Ptolemy's Theorem and its converse* As was indicated in the § 2, we will use Poincare's model of the hyperbolic plane. In order to prove the analogue of Ptolemy's Theorem and its converse, the "cross ratio" of four points is defined and shown to be an inversive invariant. Since lines, circles, horocycles, and equidistant curves are all equivalent under the group generated by hyperbolic inversions, this reduces Ptolemy's Theorem and its converse to that of considering four points on a line.

DEFINITION 3.1. If A, B, C, D are four distinct points, then the

cross ratio {AB, CD) is defined by:

{AB, CD) = [sinh AC/2 sinh BD/2] / [sinh AD/2 sinh BC/2].

–  –  –

Substituting the values of (*, *), together with the same identities when x9 X are replaced by y, Y, respectively, in (*) and solving the

new equation for sinh X ' Y'/2, we obtain:

sinh X ' Γ'/2 - k[(l-x2)/(x2 - k2)]1'2 [(l-y2)/(y2 - k2)]112 sinh 1 7 / 2. (r) Application of (τ) to all pairs of any four points A, J5, C, D yields,

–  –  –

Proo/. Since Λ4(P1? P 2, P 3, P4) = 0, it follows that the determinant H obtained from Λ4(P1? P2, Pz, P 4 ), by bordering it with a first row and a first column with a common element — 1, and the rest of the elements in the first column all ones and the rest of the elements in the first row all zeroes, also vanishes.

The result of subtracting the first column of this bordered determinant from the second, third, fourth, and fifth columns, respectively, and making use of the fact that cosh(β) — 1 = 2 sinh2(#/2) is

–  –  –

If the four points contain a nonlinear triple, assume the labeling so that Λ3(P2, P 3, P4) Φ 0. Letting [1, 2] denote the cofactor of the element in the first row and second column, an expansion theorem from determinants yields

–  –  –





quently, the rank of the determinant in (1) is three. Therefore, K(P1,P2,Ps,Pi) = 0.

If P, Q, R, S is a quadruple of distinct points, then COROLLARY.

the three products, sinh (PQ/2) sinh(i2S/2), sinh (PR/2) sinh (QS/2), sinh (PS/2) sinh (QR/2), of the hyperbolic sines of half the "opposite" distances satisfy the triangle inequality.

–  –  –

REMARK. The above corollary is the hyperbolic analogue of the ptolemaic inequality. It has been shown [3] that the ptolemaic inequality itself is valid in the hyperbolic plane.

–  –  –

THEOREM 3.3.

// P, Q, R, S are distinct points and if PR//QS then {PS, QR} + {PQ, RS} = 1.

Proof. Theorem 3.2 and the above remarks show the validity of the theorem in case the cycle is a line. Suppose, then, that the four points lie on a horocycle, circle or equidistant curve. There exists an inversion which maps such a cycle onto a line. Thus, if P', Sr, Qr,Rf ANALOGUE OF PTOLEMY'S THEOREM 821 are the inverse points of P, Q, R, S then P'R'HQ'S' and {P'S'9 Q'R'} + {P'Q', R'S'} = 1. Since cross ratio is an inversive invariant, {PS, QR} + {PQ, RS} = 1.

THEOREM 3.4.

// P, Q, iϋ, £ are four distinct points such that {PS, QR} + {PQ, RS} = 1 then PR//QS.

Proof. Three points of the hyperbolic plane lie on a line, circle, horocycle, or equidistant curve. We first suppose three of the points, say, Q, R, S are collinear. Since Q, R, S are collinear, Λ3 (Q, R, S) — 0, K(Q, R,S)^0 and with the notation of Theorem 3.2, I(Q, R, S) = 0.

By hypothesis K (P, Q, R, S) = 0, and I(P, Q, R, S) = 0. It now follows that the rank of I (P, Q, R, S) is three. Consequently, the rank of Λ4 (P, Q, R, S) is two. Therefore, Λ3 (P, Q, S) = 0 and P, Q, S are collinear. It now follows that P, Q, i?, S are collinear and PR//QS.

If Q, R, S lie on any cycle, this cycle may be mapped onto a line by an inversion. If P',ζ', R', S' are inverse points of P, Q, i2, S then by the above, P'R'UQ'S'. If the line of P', Q', i2', S' is mapped back onto the cycle of Q, R, S by the same inversion, it follows that PR//QS.

The following theorem, which is a hyperbolic analogue of Ptolemy's Theorem and its converse, has now been obtained.

THEOREM 3.5.

Four distinct points P19 P 2, P3, P 4 of the hyperbolic plane lie on a line, circle, horocycle, or one branch of an equidistant curve if and only if the determinant K{Pt, P2, P3, P4) = I sinWiPiPj/Z) | vanishes (i, j - 1, 2, 3, 4).

In order to obtain the generalization of this analogue of Ptolemy's Theorem and its converse, we need the following lemma.

LEMMA 3.6.

If P19 P 2, P3, P 4, P 5 are five points then the determinant K{Pt, P2, P3, P4, P5) = I smh^PiPj'β) \ vanishes (i, j^ = 1,2,3,4,5).

Proof. Since the rank of A5(PX, P2, P3, P4, P5) is less than or equal to three, it follows that the determinant, H, obtained from Λ5 (P19 P2, P3, P4, P5) by bordering it with a first column and a first row with common element —1, and the rest of the elements of the first column all ones and the rest of the element in the first row all zeroes has rank less than or equal to four. The determinant, I, obtained from H by substracting the first column from the second, third, 822 JOSEPH E. VALENTINE fourth, fifth and sixth columns, respectively, and then substituting 2 sinh2(#/2) for cosh (x) — 1 also has rank less than or equal to four.

Hence K(Pl9 P2, P3, P4, P5) = 0.

3.7. If P19 P 2, «, P W are w pairwίse distinct points

THEOREM

of the hyperbolic plane, (n ^ 4), ί/^ew necessary and sufficient condition that P19 P 2,, P Λ ϊie ow a iwβ, circle, horocycle, or one branch of an equidistant curve is that the matrix

–  –  –

Proof. Suppose P 1? P2,, P w are pair wise distinct points of the hyperbolic plane which lie on a line, circle, horocycle, or one branch of an equidistant curve. Then the leading principal minor of order three of K (P19 P2,, Pn) is nonzero, while the determinants obtained from this principal minor by adjoining one row and one column or two rows and two columns vanish by Theorem 3.3 and Lemma 3.6, respectively. It follows that the rank of K(P19P2, •, Pn) is three.

Conversely, if the rank of K(P19, Pn) is three then every principal minor of K (P19 P2, «,P Λ ) of order four is zero. It follows from Theorem 3.5 that each quadruple of the points and hence the n points lie on a line, circle, horocycle, or one branch of an equidistant curve.

4* Circles, horocycles and equidistant curves* It is noted, in the hyperbolic analogue of Ptolemy's Theorem and its converse, that the relation satisfied by the mutual distances of four points is the same for four points on a line, circle, horocycle, and one branch of an equidistant curve. Since three points are collinear if and only if one of the distances determined by the three points is equal to the sum of the other two, the characterization of three collinear points is immediate. The purpose of this section is to find metric conditions which are necessary and sufficient to insure that three points lie on a circle, horocycle, or one branch of an equidistant curve, respectively.

In this section the vertices of a triangle will be denoted A, B, C and α, 6, c will denote the lengths of the sides opposite the vertices A, B, C respectively.

–  –  –

Proof. Clearly, A, B, C are points on a circle if and only if a point T exists such that TA = TB = TC = R and 2R ^ max {a, δ, c) by the triangle inequality. Since A* (A, B, C, T) = 0, it follows, by factoring coshi? from the last row and column of A* (A, B, C, T), that a (A, B, C) = 0.

Conversely, suppose A, 5, C are vertices of a triangle and a real number R exists such that α (A, J5, C) = 0 and 2R ^ max {α, 6, c}.

Then

–  –  –

T w o cases are to be considered.

Case 1. The distance AB is less than 2R.

Since the function AX, (X on the perpendicular bisector I, of the segment joining A and B), is continuous, monotone increasing, and is not bounded above as X recedes along either half-line of I determined by the midpoint of the segment joining A and B, points S, S' of I r exist such that AS = AS' = R. Also, BS = AS = BS. It follows that CS Φ CS', for otherwise C would lie on the line joining A and B, contrary to the fact that A, B, C are noncollinear points.

Denote by A (A, B, C, S; x) the function obtained from A4 (A, B, C, S) by replacing CS by x. This function is not identically zero, since the coefficient of cosh2# in the development of the determinant is — A2(A, B) Φ 0. It follows that the function vanishes for at most two values of x. Since AS = AS', BS = BS' and A4(A, B, C, S) = Λ4 (A, B, C, S') = 0, CS are CS' are the distinct roots of A(A,B,C,S; x) = 0. From (2), R is also a root of A (A, B, C, S x) = 0. Therefore, CS = R or CSr = R and A, B, C lie on a circle with circumradius R.

Case 2. The distance AB is equal to 2R.

–  –  –

from Λ4 (A, B, C, S) by replacing CS by x. By an expansion theorem for determinants Λ(A, B, C, S;x) = [Λ3(A, B, C)Λ3(A, 5, S) - [coshα;]2]/Λ2(A, B), where [coshcc] denotes the cofactor of the element coshα; in the determinant Λ(A, B, C, S; x). Hence, since Λ3(A, B, S) = 0, the equation Λ(A, B, C, S; x) = 0 has only one root. From (2), R is a root of this equation, while Λ4 (A, B, C, S) = 0 by the Preliminary Remarks.

Therefore, CS = R and A, _B, C lie on a circle with circumradius R.

–  –  –

As a result of Theorems 4.1 and 4.2, the following theorem is immediate.

THEOREM 4.3.

If A, B, C are vertices of a triangle, then A, B, C lie on one branch of an equidistant curve if and only if for each real number R ^ 1/2 max {a, b,c},a (A, B, C) Φ 0 Φ β {A, B, C).

The author wishes to thank Professor L.M. Blumenthal for the suggestion of the problem and for his helpful criticism. The author also wishes to thank H. S. M. Coxeter for his suggestion to use Poincare's model. The proofs of some of the theorems given here are due to his helpful suggestions. Moreover, Theorem 3.1 is due to him.

REFERENCES

1. L. M. Blumenthal, Theory and applications of distance geometry, The Clarendon Press, Oxford, 1953.

2. H. S. M. Coxeter and S. L. Greitzer, Geometry revisited, Random House Inc., New York, 1967.

3. R. W. Freese, Ptolemaic Metric Spaces, doctoral dissertation, University of Missouri, Columbia, 1961.

4. J. Haantjes, A characteristic local property of Geodesies in certain metric spaces, Proc. Akad. Wetensch., Amsterdam, 5O (1947), 496-508.

5., De Stelling von Ptolemeus, Simon Stevin 2 9 (1952), 25-31



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