Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 36 (1995), No. 2, 235-241.

On the Derivation of the Non-Euclidean Angle of Parallelism Function

Dieter Ruoff

Dept. of Mathematics and Statistics, University of Regina Regina, Canada S4S 0A2

Abstract. This paper investigates the role that elementary area theory can play in deriving the formula for the non-Euclidean angle of parallelism function. It is shown how the pivotal feature of area theory, the connection between area and angle defect of a polygon, extends to a quantitative relation among the entities associated with a given angle of parallelism. The sought-after formula then is easily determined. The whole investigation is carried out in the framework laid out by Oskar Perron.

1. Introduction

The non-Euclidean angle of parallelism formula has been derived in quite a variety of ways, on the one hand because of its great importance for non-Euclidean geometry, and on the other hand because no single way of deducing the function has emerged as the most natural one. In 1944 Oskar Perron published a particularly elementary deduction of the angle of parallelism function in the Mathematische Annalen; in his approach the function is considered together with and calculated right after the introduction of the trigonometric functions (s. [5]). This procedure (which has also been integrated in Perron’s textbook of non-Euclidean geometry [6]) suffers however from the fact that long-winded and nonintuitive analytic passages make up a considerable part of it. As will be shown in this paper, it is possible to derive the angle of parallelism function in a direct, geometric way, and to do that still in the context of the trigonometric functions, and still in an entirely elementary way. The decisive connection in determining the sought-after formula will be that between area measure and angular defect of (proper and improper) polygons. In section 2 a few preliminary facts will be compiled, and in section 3 the actual derivation of the angle of parallelism function will be presented.

2. Abbreviations, Preliminaries

Let A, B be two distinct points. Then AB denotes the segment determined by them, |AB|

the length of the segment, g(AB) the straight line through A and B, \overrightarrow{(A)B} the halfline

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1995 Heldermann Verlag, Berlin

236 D. Ruoff: On the Derivation of the Non-Euclidean Angle of Parallelism Function

bounded by A, through B, and \overrightarrow{AB} = \{A\} ∪ \overrightarrow{(A)B} the related ray. By | <^{)} AOB| we mean the size of the angle <^{)} AOB and by |ΔABC| the area of the triangle ΔABC. The letters ω, τ, … are used for ends, improper points of concurrence of boundary parallel lines. The end symbols appear prominently in the abbreviations ΔABω, … for trilaterals, i.e. for simply asymptotic triangles with two proper vertices and an improper one. Our deduction of the angle of parallelism function depends especially on the following facts.

(i) For <^{)} AOB a fixed angle of size α, Y a running point on the halfline \overrightarrow{(O)B} and X its

normal projection on \overrightarrow{(O)A} (s. Fig. 1),

lim_{|OY |→0} \frac{|XY |}{|OY |} = sin α, lim_{|OY |→0} \frac{|OX|}{|OY |} = cos α.

(This result has been deduced by Perron; s. [5], p. 247ff., [6], p. 51ff.)

Fig. 1 Fig. 2

(ii) For a polygon the defect (nπ − 2π− angle sum in the case of an n-gon) constitutes a suitable area measure in the following sense. Any decomposition of the polygon into polygonal parts has the property that the sum of the defects of the parts is equal to the defect of the whole polygon. (For details see any of the books listed in the Bibliography.) Also for a trilateral the defect (π− sum of the proper angles) qualifies as a suitable area measure.

(iii) A trilateral has an area of less than ε if only its (finite) inner side is sufficiently small. In order to show that we construct out of a right trilateral ΔABω another right trilateral ΔA_{1}B_{1}ω of less than half the area (s. Fig. 2). Repeated arbitrarily this construction leads to a right trilateral ΔUV ω of area < ε/2. The area of the equiangular trilateral ΔUU^{∗}ω with symmetry line g(V ω) then measures < ε (s. Fig. 3), and the same is true for any other trilateral whose inner side is not larger than UU^{∗}.

Fig. 3 Fig. 4

(iv) Let again, as in (i), <^{)} AOB be a fixed angle of size α, Y a variable point on \overrightarrow{(O)B} and

X its normal projection on \overrightarrow{(O)A}. Then

lim_{|OX|→0} \frac{|ΔOXY |}{|OX|} = 0 and lim_{|XY |→0} \frac{|ΔOXY |}{|XY |} = 0.

Proof. Given ΔOX_{0}Y_{0}, ΔOX_{1}Y_{1}, where |OX_{1}| = 1/2|OX_{0}|, we construct a point Z_{0} such that ΔOX_{1}Y_{1}, ΔX_{0}X_{1}Y_{1} and ΔY_{1}Z_{0}X_{0} form three congruent triangles with mutually disjoint interior, and such that | <^{)} Y_{1}X_{0}Z_{0}| = | <^{)} OY_{1}X_{1}| = β_{1} (s. Fig. 4). Since | <^{)} OX_{0}Z_{0}| = | <^{)} X_{1}Y_{1}Z_{0}| = α + β_{1} and α + β_{1} < π/2 (see ΔOX_{1}Y_{1}) it follows that <^{)} OX_{0}Z_{0} < <^{)} OX_{0}Y_{0} and <^{)} X_{1}Y_{1}Z_{0} < <^{)} X_{1}Y_{1}Y_{0}. As a result Z_{0} lies in the interior of ΔOX_{0}Y_{0}, and the three triangles are part of a polygonal decomposition of ΔOX_{0}Y_{0}. Consequently |ΔOX_{1}Y_{1}| < 1/3|ΔOX_{0}Y_{0}| and

\frac{|ΔOX_{1}Y_{1}|}{|OX_{1}|} < \frac{\frac{1}{3}|ΔOX_{0}Y_{0}|}{\frac{1}{2}|OX_{0}|} = \frac{2}{3} \frac{|ΔOX_{0}Y_{0}|}{|OX_{0}|} .

Introducing additional triangles ΔOX_{2}Y_{2}, ΔOX_{3}Y_{3}, … which one after another fulfill |OX_{i+1}| = 1/2|OX_{i}| we can expand our result to

\frac{|ΔOX_{n}Y_{n}|}{|OX_{n}|} < (\frac{2}{3})^{n} \frac{|ΔOX_{0}Y_{0}|}{|OX_{0}|} . (n = 1, 2, 3, …)

Because |ΔOXY |, and for |OX| > 0 also |ΔOXY |/|OX| are continuous functions of |OX| this provides the proof for the first formula of our statement. By means of

lim_{|OX|→0} \frac{|OX|}{|XY |} = cot α

the second formula follows from the first one at once.

3. The Derivation of the Angle of Parallelism Function

Let <^{)} ωOτ be a right angle, defined by the proper point O and by the ends ω and τ, and

let X, Y, … be arbitrary points on \overrightarrow{(O)τ}. The rays \overrightarrow{Xω}, \overrightarrow{Y ω}, … are boundary parallel to

\overrightarrow{Oω} and the angles <^{)} OXω, <^{)} OY ω, … are the angles of parallelism associated with the segments OX, OY, … (s. Fig. 5). The function which relates the measures of these segments and angles is the angle of parallelism function φ = Π(x). If |OX| < |OY |, <^{)} OXω is an exterior angle of ΔXY ω, and consequently | <^{)} OXω| > | <^{)} XY ω| = | <^{)} OY ω|. This shows Π(x) is a decreasing function. Also, | <^{)} OXω|−| <^{)} OY ω| = π−| <^{)} Y Xω|−| <^{)} XY ω|, i.e. the difference of the angles of parallelism related to OX, OY is identical with the defect, and thus with the area measure of ΔXY ω. Because |ΔXY ω| → 0 for |XY | → 0 (see 2., (iii)) the function Π(x) must be continuous.

Fig. 5

We now consider three points A, B, C in this order on \overrightarrow{(O)τ} and assume B is the midpoint

of AC. From these points we draw the rays \overrightarrow{Aω}, \overrightarrow{Bω}, \overrightarrow{Cω}, as well as the rays \overrightarrow{Aω_{A}} and \overrightarrow{Cω_{C}} which are defined by the equation | <^{)} OAω_{A}| = | <^{)} OCω_{C}| = | <^{)} OBω| = φ, and by the stipulation that ω_{A} and ω_{C} lie on the same side of line g(Oτ) as ω (s. Fig. 6). We complete our figure by normally projecting the points A, C onto the points A′, C′ of g(Bω). The congruence of the ensuing triangles ΔAA′B, ΔCC′B entails that the segments A′A, C′C have equal length, and as a consequence that the related angles of parallelism <^{)} A′Aω, <^{)} C′Cω satisfy | <^{)} A′Aω| = | <^{)} C′Cω| = α. Note also | <^{)} A′AB| = | <^{)} C′CB| = β. Finally, defining | <^{)} ω_{A}Aω| = δ_{A}, | <^{)} ω_{C}Cω| = δ_{C}, we have that Π(OA) = φ + δ_{A}, Π(OB) = φ and Π(OC) = φ − δ_{C}, in agreement with the decreasing character of the angle of parallelism function. In addition, the relations

φ + β < \frac{π}{2}  ( in ΔAA′B),
φ + δ_{A} + α + β = π  ( at the point A) and
δ_{C} + α = φ + β  ( at the point C)

lead to δ_{A} + α > \frac{π}{2} , δ_{C} + α < \frac{π}{2} and thus to δ_{C} < δ_{A}. This means that the continuous, decreasing function φ = Π(x) is convex downwards at x = |OB| and has negative lefthand side and right-hand side derivatives there. These one-sided derivatives φ′_{−}, φ′_{+} satisfy (s. Fig. 6)

|φ′_{−}| = lim_{|AB|→0} \frac{|(φ + δ_{A}) − φ|}{|AB|} = lim_{|AB|→0} \frac{δ_{A}}{|AB|}

and

|φ′_{+}| = lim_{|CB|→0} \frac{|(φ − δ_{C}) − φ|}{|CB|} = lim_{|CB|→0} \frac{δ_{C}}{|CB|} .

Fig. 6

At this point one has to take the important fact into account that δ_{A} = π − (α + β + φ) is the defect and thus the area measure of the trilateral ΔABω, and δ_{C} = β + φ − α = π − [(α − β) + (π − φ)] is the defect and thus the area measure of the trilateral ΔCBω. Consequently

|φ′_{−}| = lim_{|AB|→0} \frac{|ΔABω|}{|AB|} and |φ′_{+}| = lim_{|CB|→0} \frac{|ΔCBω|}{|CB|} .

From the expansion

|φ′_{−}| = lim_{|AB|→0} \frac{|ΔABω|}{|AB|} = lim_{|AB|→0} \frac{|ΔABω|}{|AA′|} lim_{|AB|→0} \frac{|AA′|}{|AB|} = lim_{|AA′|→0} \frac{|ΔABω|}{|AA′|} sin φ

of the left-hand side derivative follows that the limit which appears last in the chain of equations exists and is positive. Because of ΔABω = ΔAA′ω + ΔAA′B it satisfies the equation

lim_{|AA′|→0} \frac{|ΔABω|}{|AA′|} = lim_{|AA′|→0} \frac{|ΔAA′ω|}{|AA′|} + lim_{|AA′|→0} \frac{|ΔAA′B|}{|AA′|}

in which the second limit on the right-hand side is 0 (s. 2., (iv)). The remaining first limit on the right-hand side applies to the areas of right trilaterals as a function of, and divided by their inner sides; it is independent of x = |OB| resp. φ and represents a general non-Euclidean constant κ. Thus for every argument x > 0

|φ′_{−}| = κ sin φ.

The right-hand side derivative is calculated analogously (except that ΔCBω = ΔCC′ω − ΔCC′B), and the result is likewise

|φ′_{+}| = κ sin φ.

This shows that φ has an all-over derivative which is determined by the differential equation

φ′ = −κ sin φ.

Its solution, the angle of parallelism function, has the formula

φ = Π(x) = 2 tan^{−1}(e^{−κx}) .

References

[1] Baldus, R.; Löbell, F.: Nichteuklidische Geometrie, Sammlung Göschen, Berlin, 1964.

[2] Gans, D.: An Introduction to non-Euclidean Geometry, Academic Press, Inc., Orlando etc., 1973.

[3] Klotzek, B.; Quaisser, E.: Nichteuklidische Geometrie, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978.

[4] Norden, A. P.: Elementare Einführung in die Lobatschewskische Geometrie, VEB Deutscher Verlag der Wissenschaften, Berlin, 1958.

[5] Perron, O.: Neuer Aufbau der nichteuklidischen (hyperbolischen) Trigonometrie, Math. Ann. 119 (1944), 247–265.

[6] Perron, O.: Nichteuklidische Elementargeometrie der Ebene, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1962.

[7] Simon, M.: Nichteuklidische Geometrie in elementarer Behandlung (bearbeitet und herausgegeben von K. Fladt), B. G. Teubner, Leipzig, 1925.

Received September 5, 1994; revised version February 6, 1995.