8. Hyperbolic triangles


 Ethelbert Walker
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1 8. Hyperoli tringles Note: This yer, I m not doing this mteril, prt from Pythgors theorem, in the letures (nd, s suh, the reminder isn t exminle). I ve left the mteril s Leture 8 so tht (i) nyody interested n red out hyperoli trigonometry, nd (ii) to sve me hving to renumer Letures 9 onwrds! 8.1 Rightngled tringles In Euliden geometry there re mny wellknown reltionships etween the sides nd the ngles of rightngled tringle. For exmple, Pythgors theorem gives reltionship etween the three sides. Here we study the orresponding results in hyperoli geometry. Throughout this setion, will e rightngled tringle. The internl ngles will e, β, π/2, with the opposite sides hving lengths,,. 8.2 Pythgors theorem In Euliden geometry, Pythgors theorem gives reltionship etween the three side lengths of rightngled tringle. Here we prove n nlogous result in hyperoli geometry. Theorem 8.1 (Pythgors theorem for hyperoli tringles) Let e rightngled tringle in H with internl ngles, β, π/2 nd opposing sides with lengths,,. Then osh = osh osh. (8.1) Remrk. If,, re ll very lrge then pproximtely we hve + log 2. Thus in hyperoli geometry (nd in ontrst with Euliden geometry), the length of the hypotenuse is not sustntilly shorter thn the sum of the lengths of the other two sides. Proof. Let e tringle stisfying the hypotheses of the theorem. By pplying Möius trnsformtion, we my ssume tht the vertex with internl ngle π/2 is t i nd tht the side of length lies long the imginry xis. It follows tht the side of length lies long the geodesi given y the semiirle entred t the origin with rdius 1. Therefore, the other verties of n e tken to e t ki for some k > 0 nd t s + it, where s + it lies on the irle entred t the origin nd of rdius 1. See Figure
2 ki i β s + it 0 Figure 8.1: Without loss of generlity, we n ssume tht hs verties t i, ki nd s + it Rell from Leture 5 tht for ny z, w H osh d(z, w) = 1 + z w 2 2 Im(z) Im(w). Applying this formul to the the three sides of we hve: s + i(t 1) 2 osh = 1 + = 1 + s2 + (t 1) 2 = 1 2t 2t t, (8.2) (k 1)2 osh = 1 + = 1 + k2 2k 2k, (8.3) osh = 1 + s + i(t k) 2 2tk = 1 + s2 + (t k) 2 2tk = 1 + k2 2tk, (8.4) where to otin (8.2) nd (8.4) we hve used the ft tht s 2 + t 2 = 1, s s + it lies on the unit irle. Comining (8.2), (8.3) nd (8.4) we see tht osh = osh osh, proving the theorem. 8.3 Two sides, one ngle For rightngled tringle in Euliden geometry there re wellknown reltionships etween n ngle nd ny of two of the sides, nmely sine = opposite / hypotenuse, osine = djent / hypotenuse nd tngent = opposite / djent. Here we determine similr reltionships in the se of hyperoli rightngled tringle. 2
3 Proposition 8.2 Let e rightngled tringle in H with internl ngles, β, π/2 nd opposing sides with lengths,,. Then (i) sin = sinh / sinh, (ii) os = tnh / tnh, (iii) tn = tnh / sinh. Proof. As in the proof of Theorem 8.1, we n pply Möius trnsformtion to nd ssume without loss in generlity tht the verties of re t i, ki nd s + it, where s + it lies in the unit irle entred t the origin nd the rightngle ours t i. ki i β s + it x 0 Figure 8.2: The point x is the entre of the semiirle orresponding to the geodesi through ki nd s + it The verties t ki nd s + it lie on unique geodesi. This geodesi is semiirle with entre x R. The (Euliden) stright line from x to ki is inlined t ngle from the rel xis. See Figure 8.2. The line from x to ki is rdius of this semiirle, s is the line from x to s + it. Clulting the lengths of these rdii, we see tht so tht k 2 + x 2 = (s + x) 2 + t 2 k 2 = 1 + 2sx, (8.5) using the ft tht s 2 + t 2 = 1. By onsidering the Euliden tringle with verties t x, ki, 0, we see tht tn = k x = 2ks k 2 1, (8.6) 3
4 where we hve sustituted for x from (8.5). Using the fts tht osh 2 sinh 2 = 1 nd tnh = sinh / osh it follows from (8.2) nd (8.3) tht sinh = k2 1, tnh = s. 2k Comining this with (8.6) we see tht tn = tnh sinh, proving sttement (iii) of the proposition. The other two sttements follow y using trig identities, reltionships etween sinh nd osh, nd the hyperoli version of Pythgors theorem. Exerise 8.1 Assuming tht tn = tnh / sinh, prove tht sin = sinh / sinh nd os = tnh / tnh. Exerise 8.2 We now hve reltionships involving: (i) three ngles (the GussBonnet theorem), (ii) three sides (Pythgors theorem) nd (iii) two sides, one ngle. Prove the following reltionships etween one side nd two ngles: osh = os ose β, osh = ot ot β. Wht re the Euliden nlogues of these identities? 8.4 Nonrightngled tringles: the sine rule Rell tht in Euliden geometry the sine rule tkes the following form. In tringle (not neessrily rightngled) with internl ngles, β nd γ nd side lengths, nd we hve sin = sin β = sin γ. The hyperoli version of this is the following. Proposition 8.3 Let e hyperoli tringle with internl ngles, β nd γ nd side lengths,,. Then sin sinh = sin β sinh = sin γ sinh. 4
5 Exerise 8.3 Prove Proposition 8.3 in the se when is ute (the otuse se is simple modifition of the rgument, nd is left for nyody interested...). (Hint: lel the verties A, B, C with ngle t vertex A, et. Drop perpendiulr from vertex B meeting the side [A, C] t, sy, D to otin two rightngled tringles ABD, BCD. Use Pythgors theorem nd Proposition 8.2 in oth of these tringles to otin n expression for sin.) 8.5 Nonrightngled tringles: osine rules The osine rule I Rell tht in Euliden geometry we hve the following osine rule. Consider tringle (not neessrily rightngled) with internl ngles, β nd γ nd sides of lengths, nd, with side opposite ngle, et. Then 2 = os γ. The orresponding hyperoli result is. Proposition 8.4 Let e hyperoli tringle with internl ngles, β nd γ nd side lengths,,. Then osh = osh osh sinh sinh os γ. Proof. See Anderson s ook The osine rule II The seond osine rule is the following. Proposition 8.5 Let e hyperoli tringle with internl ngles, β nd γ nd side lengths,,. Then osh = os os β + os γ. sin sin β Proof. See Anderson s ook. Remrk. The seond osine rule hs no nlogue in Euliden geometry. Oserve tht the seond osine rule implies the following: if we know the internl ngles, β, γ of hyperoli tringle, then we n lulte the lengths of its sides. In Euliden geometry, the ngles of tringle do not determine the lengths of the sides. 5
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