The Inscribed Angle Theorem for the Hyperbola Jack Williams

University of Cambridge [email protected]

April 22, 2021

Abstract

The inscribed angle theorem, a famous result about the angle subtended by a chord within a circle, is well known and commonly taught in school curricula. In this paper, we present a generalisation of this result (and other related circle theorems) to the rectangular hyperbola. The notion of angle is replaced by pseudo-angle, deﬁned via the Minkowski inner product. Indeed, in Minkowski space, the unit hyperbola is the set of points a unit metric distance from the origin, analogous to the Euclidean unit circle. While this is a result of pure geometrical interest, the connection to Minkowski space allows an interpretation in terms of special relativity where, in the limit c → ∞, it leads to a familiar result from non-relativistic dynamics. This non-relativistic result can be interpreted as an inscribed angle theorem for the parabola, which we show can also be obtained from the Euclidean inscribed angle theorem by taking the limit of a family of ellipses ananlogous to the non-relativistic limit c → ∞. This simple result could be used as a pedagogical example to consolidate understanding of pseudo-angles in non-Euclidean spaces or to demonstrate the power of analytic continuation.

1 Introduction cles. The inscribed angle theorem implies that the proper time between these two events depends only on the relative velocity of the two By replacing t with it, many results and tech- particles and not on the relative velocity between niques familiar in the Euclidean setting can be the particles and the observer. This is the rela- applied in physically meaningful Lorentzian set- tivistic analogue of a similar fact familiar from tings [1]. Making use of these Wick rotations, non-relativistic kinematics and, of course, the non- we obtain a generalisation of the inscribed angle relativistic result can be obtained from the rela- theorem to rectangular hyperbolae in Minkowski tivistic one by taking the limit c → ∞. space. The hyperbola itself is analogous to a Eu- clidean circle and is related to it by a Wick rotation Since the trajectory of a non-relativistic observer of the plane coordinates. The Euclidean angle is moving with constant acceleration is a parabola, replaced by the Minkowski pseudo-angle and the this limit leads to an inscribed angle theorem proof of the hyperbolic inscribed angle theorem for the parabola. Since the parabola is the sin- arXiv:2104.10518v1 [math.HO] 19 Apr 2021 is similar to a proof of the Euclidean result, but gular conic between families of hyperbolae and with trigonometric functions replaced by hyper- ellipses, it is natural that the inscribed angle theo- bolic trigonometric functions. rem should arise by taking this limit of a family While this is a result of pure geometrical inter- of hyperbolae. We show that the same result can est, the connection to Minkowski space allows an also be obtained from the Euclidean inscribed interpretation in terms of special relativity. Ob- angle theorem by taking a limit of a family of serving that hyperbolae in Minkowski space occur ellipses. naturally as the trajectory of observers moving This result is accessible to early undergraduates with constant proper acceleration, we provide and could be presented as a pedagogical example a physical interpretation of the inscribed angle to build intuition around the analogy between theorem in terms of an observer who ejects two Euclidean angles and rapidity in special relativity, particles while moving along this trajectory. The or Euclidean rotations and Lorentz transforma- observer subsequently collides with both parti- tions. It also demonstrates the power of analytic

1 The Inscribed Angle Theorem for the Hyperbola

2 continuation. R1,1 2( ) = (x·y) vectors in via cosh θ x2y2 whenever x and y are either both spacelike or both timelike [3]. 2 Inscribed Angle Theorem A similar construction exists for the pseudo-angle between a spacelike and a timelike vector. on the Unit Hyperbola In the case that x and y are both timelike, this pseudo-angle is, in fact, the relative rapidity of 2.1 Inscribed angle theorem in 2D Eu- observers A and B moving with velocities x and clidean space y. Indeed, if B moves with velocity vector parallel to (1, v) in the rest frame of A, then In 2D Euclidean space, the inscribed angle theorem states that given any three points P0, P1, P2 on the unit circle, the lines P P and P P inter- (x · y)2 1 0 1 0 2 cosh2(θ) = = sect at an acute angle independent of the position x2y2 1 − v2 of P0 [2]. Further, if the points are represented =⇒ v = ± tanh(θ). (4) as Pi = (cos(θi), sin(θi)), where 0 ≤ θi < 2π, then the acute angle between P0P1 and P0P2 is 1 1 2 |θ1 − θ2| (or π − 2 |θ1 − θ2|, whichever is acute). This follows because the gradients of the chords 2.3 Inscribed angle theorem in 2D P P are 0 i Minkowski space θ +θ θ −θ 2 cos 0 i sin 0 i sin θ0 − sin θi 2 2 = θ +θ θ −θ In 2D Minkowski space, the analogue of the unit cos θ0 − cos θi 0 i 0 i 2 sin 2 sin 2 2 2 circle is the hyperbola −x0 + x1 = 1, consisting of θ + θ points a unit Minkowski distance from the origin. = cot 0 i . (1) 2 Let P0, P1, P2 be any three points on the branch of this hyperbola with x1 > 0 with coordinates Given that the angle between them satisﬁes Pi = (sinh(θi), cosh(θi)). Observe that the chords P P must be timelike. (x · y)2 0 i cos2(θ) = , (2) x2y2

where x and y are tangent to these chords, we P ﬁnd 0

θ +θ θ +θ 2 1+cot 0 1 cot 0 2 2 2 2 cos (θ) = 2 θ0+θ1 2 θ0+θ1 1+cot 2 1+cot 2 − 2 θ1 θ2 P = cos . (3) 1 2

P It follows immediately that θ is independent of 2 θ0 and depends only the the difference between θ1 and θ2.

2.2 Pseudo-angles in 2D Minkowski Figure 1: Inscribed angle theorem in Minkowski space space It can be checked directly that the inner product 1,1 In a similar way to the Euclidean case, we find on R given by x · y = −x0y0 + x1y1 satisfies a θ0+θi form of the Cauchy-Schwarz inequality: for any that the gradient of the line P0Pi is coth 2 non-null vectors x and y, (x · y)2 ≥ x2y2. Using and hence that the pseudo-angle between these this, we can define the pseudo-angle between two chords is given by

2 The Inscribed Angle Theorem for the Hyperbola

θ0+θ1 θ0+θ1 coth 2 and tanh 2 respectively. Their θ +θ θ +θ 2 tangent vectors are orthogonal since coth 0 1 coth 0 2 −1 2 2 2 cosh (θ) = θ +θ θ +θ coth2 0 1 −1 coth2 0 1 −1 θ0+θ1 θ0+θ1 2 2 x · y = coth 2 tanh 2 − 1 = 0. (7) θ − θ = cosh2 1 2 . (5) 2 2.5 Relationship between the inscribed It follows again that θ does not depend on the angle theorem and isometries position of P . 0 In the traditional statement of the inscribed angle theorem in Euclidean space, the points P1 and 2.4 Analogues of other circle theorems P2 are fixed and the point P0 varies on the cir- in Minkowski space cle. However, to obtain an alternative viewpoint one keeps P fixed and rotates the two lines P P There are similar analogues of other familiar Eu- 0 0 1 and P P together, keeping their relative angle clidean results. For example, on the circle, the 0 2 fixed. After the rotation, the new points of inter- angle P P P is half the angle P OP [2]. This 1 0 2 1 2 section, P0 and P0, of these lines with the circle holds true in Minkowski space since the pseudo- 1 2 are separated by the same relative angle as the angle between the spacelike chords OP1 and OP2 original points P1 and P2. In fact, the rotation is given by 0 0 taking P1 and P2 to P1 and P2 is an isometry of (x · y)2 the Euclidean plane. cosh2(θ) = x2y2 2 = (cosh (θ1) cosh (θ2) − sinh (θ1) sinh (θ2)) P 2 0 = cosh (θ1 − θ2) . (6)

P1’ θ1−θ2 and is hence twice 2 , the pseudo-angle between P0P1 and P0P2. P2’

’ P1

P 1 Figure 3: Alternative interpretation of Euclidean inscribed angle theorem

Likewise, in Minkowski space one can rotate the lines P0P1 and P0P2 together about P0, keeping their relative pseudo-angle ﬁxed. The pseudo- angular distance between the new points of in- Figure 2: Thales’ theorem in Minkowski space 0 0 tersection, P1 and P2, of these lines with the hyperbola remains θ1−θ2 . Since isometries of Furthermore, the analogue of Thales’ theorem, 2 Minkowski space of the form which states that the angle subtended by a di- ameter is a right angle [2], is the result that if x0 cosh φ sinh φ x0 P0 = −P is the point in Minkowski space ob- 7→ (8) 1 1 x1 sinh φ cosh φ x1 tained from P1 by a reﬂection in the origin, then 0 the chords P0P1 and P0P1 are orthogonal. This map points (sinh(θi), cosh(θi)) to (sinh(θi + 0 holds because the gradients of P0P1 and P0P1 are φ), cosh(θi + φ)), there is an isometry of

3 The Inscribed Angle Theorem for the Hyperbola

0 0 Minkowski space mapping P1 and P2 back to P1 Since the position of the observer at proper and P2. This isometry relates the two perspectives time τ is (sinh(k(τ − τ0)), cosh(k(τ − τ0))), the on the inscribed angle theorem. inscribed angle theorem (viewed from the per- spective in which P0 is ﬁxed and the two lines rotate) implies that the proper time separating P 0 these two collisions is independent of the velocity with which A was ejected by the observer. It depends only on the relative velocity between A and B.

P1’

P 1 3.2 Connection with non-relativistic P’ 2 kinematics

P 2 This is an analogue of another familiar fact from non-relativistic kinematics. An observer with constant acceleration follows a trajectory of the form 1 2 x = 2 at + ut + x0. If, at time 0, two particles are Figure 4: Alternative interpretation of hyperbolic inscribed ejected with velocities α and β, then these particles angle theorem follow trajectories x = αt + x0 and x = βt + x0. The difference between the times at which the 2 observer collides with these particles is a (α − β), 3 Relativistic interpretation which depends only on the relative velocity of the particles. 3.1 Constant proper acceleration A corresponding inscribed angle theorem for the parabola arises from this non-relativistic re- An observer moving with constant unit proper sult, where the pseudo-angle between two lines acceleration in Minkowski space follows a rect- P0P1 and P0P2 is proportional to the difference in 2 2 angular hyperbola of the form −x0 + x1 = 1 [4]. the t-coordinates of P1 and P2 and is independent Suppose this observer ejects two particles, A and of P0. The inscribed angle theorem states that B with ﬁxed relative rapidity (i.e. separated by this angle measure is constant as P0 varies on the a ﬁxed pseudo-angle), in the same direction as parabola. the observer’s acceleration but with greater velocity. Then at some future times, the observer will collide again with each of these objects.

Figure 6: Inscribed angle theorem for parabola

Naturally, this non-relativistic inscribed angle Figure 5: Kinematic interpretation of hyperbolic inscribed theorem for the parabola can be obtained from angle theorem the inscribed angle theorem for the hyperbola by

4 The Inscribed Angle Theorem for the Hyperbola

taking the limit c → ∞. In order to do this, it P0P1 and P0P2 is given by is helpful to reintroduce dimensionful quantities. 2 The trajectory of a relativistic observer moving 4 2 2 +c 2 a (θ0+θ1)(θ0+θ2) with constant proper acceleration a in the posi- cos (θ) = (13) 4 2 4 2 2 2 +c 2 2 +c tive x-direction and passing through the origin is a (θ0+θ1) a (θ0+θ2) given by the hyperbola 2 a2 2 which implies θ ∼ 2 (θ1 − θ2) , recovering the c2 aθ c aθ 4c x = a cosh c − 1 , t = a sinh c (9) result obtained earlier. In summary, the hyperbolic inscribed angle the- where θ is the observer’s proper time. In the orem is related to the circular inscribed angle non-relativistic limit, these become theorem by Wick rotation, while the parabolic in- 1 scribed angle theorem can be obtained from both x = aθ2, t = θ (10) 2 by taking a limit in which the hyperbola/circle degenerates into a parabola. and we recover the non-relativistic parabola. The 2 = gradient of a chord P0Pi is a(θ +θ ) where Pi 0 i 1 2 4 Acknowledgements θi, 2 aθi . 2 = − 2 2 + 2 Using the metric ds c dt dx to com- The author is grateful to Maciej Dunajski and pute inner products, the angle between the chords Gary Gibbons for useful discussions. P0P1 and P0P2 is given by

2 4 2 2 −c References 2 a (θ0+θ1)(θ0+θ2) cosh (θ) = (11) 4 2 4 2 2 2 −c 2 2 −c a (θ0+θ1) a (θ0+θ2) [1] G. C. Wick, “Properties of Bethe-Salpeter Wave Functions,” Phys. Rev. 96, 1124-1134 2 and so cosh (θ) → 1 as c → ∞. Using (1954) doi:10.1103/PhysRev.96.1124 2 2 2 a2 2 cosh (θ) ∼ 1 + θ , we ﬁnd θ ∼ 2 (θ1 − θ2) , 4c [2] T. L. Heath, “Euclid: The Thirteen Books of recovering the result that the angle between P0P1 The Elements,” Dover, 1956. and P0P2 is proportional to the difference in the t-coordinates of P1 and P2 and is independent of [3] G. L. Nabel, “The Geometry of Minkowski P0. Spacetime,” Springer, 2010.

3.3 The parabolic limit of the Euclidean [4] C. W. Misner, K. S. Thorne, J. A. Wheeler, “Gravitation,” W. H. Freeman, 1973. inscribed angle theorem The inscribed angle theorem for the parabola can also be obtained in a suitable limit from the Eu- clidean inscribed angle theorem. Consider the 2 − c2 + 2 2 = c4 ellipse x a c t a2 passing through the origin. In the limit c → ∞, this becomes the 1 2 parabola x = 2 at . It can be parametrised as

c2 aθ c aθ x = a 1 − cos c , t = a sin c (12) which is the limit c → ∞ takes the form (10). 2 The gradient of the chord P0Pi is where a(θ0+θi) 1 2 Pi = θi, 2 aθi . Using the metric ds2 = dx2 + c2dt2, in which the ellipse is a metric circle, the angle between