I was puzzeling with Area of hyperbolic triangle definition and could not figure it out, but then i thought there should be a (maybe solvable) simpler problem so here it is:

suppose:

  • an hyperbolic plane with a Gaussian curvature of -1
  • on this surface there is a triangle $\triangle ABC$
  • $\angle C$ is a right angle

Then given the lengths of sides a and b what is the area?

Off course we could do:

the area $ = \frac{\pi}{2} - \angle A -\angle B$ or

$ \frac{\pi}{2} - arctan (\frac{\tanh(a)}{\sinh(b)} )- arctan (\frac{\tanh(b)}{\sinh(a)}) $

(free after http://en.wikipedia.org/wiki/Hyperbolic_triangle )

But is there not a nicer formula that does not contain any or only one trigonometric function?

(I guess the hyperbolic functions are needed)

UPDATE 07/10/2015

following https://en.wikipedia.org/wiki/Inverse_trigonometric_functions#Arctangent_addition_formula

$$ \arctan u + \arctan v = \arctan \left( \frac{u+v}{1-uv} \right) $$

and

$$ \frac{\pi}{2} -arctan (\frac{u}{v}) = arctan (\frac{v}{u})$$

(complement tangent = cotangent)

I get to

$$ Area = arctan(\frac{ \sinh(a)\sinh(b) - \tanh(a)tanh(b)}{\sinh(a)\tanh(a)+ \sinh(b)\tanh(b)} )$$

Can this be simplified any further? (Do i overlook the obvious again , or even did i make a mistake on the way somewhere?)

  • 1
    Is $\tan(S/2)=\tanh(a/2)\tanh(b/2)$ nice enough? – Grigory M Oct 3 '15 at 19:15
  • You mean Area $ = 2 arctan ( \tanh ( a / 2 ) \tanh ( b / 2) ) $ ? It is a nice formula , can you show me the deriviation of it from where I started? (I did not manage that at all, i like to know and I would gladly accept it as an answer) – Willemien Oct 4 '15 at 7:35
up vote 2 down vote accepted
+50

Let's consider an arbitrary hyperbolic triangle with angles $\alpha$, $\beta$, $\gamma$ opposite respective sides $a$, $b$, $c$. The area ($T$) is given by $$T = \pi - \alpha - \beta - \gamma \tag{0}$$ so that, writing $X_2$ for $X/2$, $$\begin{align} \cos T_2 &= \cos\left(\frac{\pi}{2}-\alpha_2-\beta_2-\gamma_2\right) = \sin\left(\alpha_2+\beta_2+\gamma_2\right) \tag{1a}\\ &= \sin\alpha_2 \cos\beta_2\cos\gamma_2+\cos\alpha_2\sin\beta_2\cos\gamma_2 \tag{1b} \\[4pt] &+\cos\alpha_2\cos\beta_2\sin\gamma_2-\sin\alpha_2\sin\beta_2\sin\gamma_2 \end{align}$$

The hyperbolic Law of Cosines tells us, for instance, $$\cos\gamma = \frac{\cosh a \cosh b - \cosh c}{\sinh a \sinh b} \tag{2}$$ so that $$\begin{align} \cos^2\gamma_2 &= \frac{1+\cos\gamma}{2} = \frac{\cosh(a+b)-\cosh c}{2\sinh a\sinh b} = \frac{\sinh s \sinh s_c}{\sinh a\sinh b} \tag{3a} \\[4pt] \sin^2\gamma_2 &= \frac{1-\cos\gamma}{2} = \frac{\cosh c-\cosh(a-b)}{2\sinh a\sinh b} = \frac{\sinh s_a \sinh s_b}{\sinh a\sinh b} \tag{3b} \end{align}$$

Likewise, $$\cos^2\alpha_2 = \frac{\sinh s \sinh s_a}{\sinh b\sinh c} \qquad \sin^2\alpha_2 = \frac{\sinh s_b \sinh s_c}{\sinh b\sinh c} \tag{4}$$ $$\cos^2\beta_2 = \frac{\sinh s \sinh s_b}{\sinh c\sinh a} \qquad \sin^2\beta_2 = \frac{\sinh s_c \sinh s_a}{\sinh c\sinh a} \tag{5}$$ From these, we have $$\begin{align} \cos T_2 &= \frac{s_0 s_b s_c+s_0 s_c s_a+s_0 s_a s_b - s_a s_b s_c}{\sinh a\sinh b\sinh c} \tag{6a}\\[4pt] &= \frac{2\sinh a_2 \sinh b_2\sinh c_2\;(1+\cosh a + \cosh b + \cosh c)}{8\;\sinh a_2 \cosh a_2\;\sinh b_2 \cosh b_2\;\sinh c_2 \cosh c_2} \end{align} \tag{6b}$$

where the numerator of $(6b)$ is a bit of a slog. (See Addendum below.) Ultimately, we conclude

$$\begin{align} \cos T_2 &= \frac{1+\cosh a + \cosh b + \cosh c}{4\;\cosh a_2\cosh b_2\cosh c_2} \tag{7a}\\[4pt] \sin T_2 &= \frac{\sqrt{1-\cosh^2 a-\cosh^2 b - \cosh^2 c+ 2 \cosh a\cosh b\cosh c}}{4\cosh a_2 \cosh b_2 \cosh c_2} \tag{7b} \end{align}$$


In the particular case of a right triangle $\gamma = \pi/2$, we have $\cosh c = \cosh a \cosh b$, and the above formulas reduce thusly:

$$\begin{align} \cos T_2 &= \frac{(1+\cosh a)(1 + \cosh b)}{4\;\cosh a_2\cosh b_2\cosh c_2} = \frac{4\cosh^2 a_2 \cosh^2 b_2}{4\cosh a_2 \cosh b_2 \cosh c_2} = \frac{\cosh a_2 \cosh b_2}{\cosh c_2} \tag{8a}\\[4pt] \sin T_2 &= \frac{\sqrt{(\cosh^2 a-1)(\cosh^2 b-1)}}{4\cosh a_2 \cosh b_2 \cosh c_2} = \frac{\sinh a\sinh b}{4\cosh a_2 \cosh b_2 \cosh c_2} = \frac{\sinh a_2 \sinh b_2}{\cosh c_2} \tag{8b} \end{align}$$

so that

$$\tan T_2 = \tanh a_2 \tanh b_2 \tag{9}$$


Addendum. Here's a walkthrough of deriving the numerator of $(5)$. I did this mostly on-the-fly; the reader is encouraged to seek a much quicker path. (Expanding everything in terms of exponentials is tedious, but effective. Of course, a computer algebra system can verify the relation in an instant.)

$$\begin{align} &\phantom{+\;} \sinh s \sinh s_b \sinh s_c + \sinh s \sinh s_c \sinh s_a \tag{Aa}\\ &+ \sinh s \sinh s_a \sinh s_b - \sinh s_a \sinh s_b \sinh s_c \\[6pt] = &\phantom{+\;} \sinh s \sinh s_c \left( \sinh s_b + \sinh s_a \right) \tag{Ab}\\ &+ \sinh s_a \sinh s_b \left( \sinh s - \sinh s_c \right) \\[6pt] = &\phantom{+\;\;} \frac12 \left(\cosh(s\;+s_c)-\cosh(s\;-s_c)\right)\cdot 2 \sinh\frac{s_a+s_b}{2}\;\cosh\frac{s_a-s_b}{2} \tag{Ac} \\[4pt] &+ \frac12 \left(\cosh(s_a+s_b)-\cosh(s_a-s_b)\right) \cdot 2\cosh\frac{s\;+s_c}{2}\;\sinh\frac{s\;-s_c}{2} \\[6pt] = &\phantom{+\;} \left(\cosh(a+b)-\cosh c\right)\cdot \sinh c_2 \cosh(a_2-b_2) \tag{Ad} \\[4pt] &+ \left(\cosh c-\cosh(a-b)\right) \cdot \cosh(a_2+b_2)\;\sinh c_2 \\[6pt] = &\phantom{+\;}\sinh c_2 \left(\; \begin{array}{c} (\cosh a\cosh b - \cosh c)\;( \cosh(a_2-b_2) - \cosh(a_2+b_2)) \tag{Ae} \\[2pt] + \sinh a \sinh b\;( \cosh( a_2 - b_2 ) + \cosh(a_2 + b_2 ) ) \end{array} \;\right) \\[4pt] = &\phantom{+\;}\sinh c_2 \left(\; \begin{array}{c} -2 \sinh a_2 \sinh b_2 (\cosh a\cosh b - \cosh c) \\[2pt] + 2 \sinh a \sinh b \cosh a_2 \cosh b_2 \end{array} \;\right) \tag{Af} \\[4pt] = &\phantom{+\;}2 \sinh c_2 \left(\; \begin{array}{c} \sinh a_2 \sinh b_2 (\cosh c - \cosh a\cosh b ) \\[2pt] + 4 \sinh a_2 \sinh b_2 \cosh^2 a_2 \cosh^2 b_2 \end{array} \;\right) \tag{Ag} \\[4pt] = &\phantom{+\;}2 \sinh a_2 \sinh b_2 \sinh c_2 \left(\; \cosh c - \cosh a\cosh b + ( \cosh a + 1 )( \cosh b + 1 ) \;\right) \tag{Ah} \\[4pt] = &\phantom{+\;}2 \sinh a_2 \sinh b_2 \sinh c_2 \left(\; 1 + \cosh a + \cosh b + \cosh c \;\right) \tag{Ai} \end{align}$$

  • can you work out the bit on how you get to (cosh(a+b)-cosh(c))/(2 sinh a sinhb) and the next steps? – Willemien Oct 9 '15 at 9:38
  • Use the identity $$\cosh x-\cosh y \equiv 2\sinh(x_2+y_2)\sinh(x_2-y_2)$$ in the expressions for $\cos^2\gamma_2$ and $\sin^2\gamma_2$. Reducing the "$s$" expression is somewhat too involved for a comment; I'll put an addendum to the answer. – Blue Oct 9 '15 at 10:04
  • Thanks, can you also number the equations (makes commenting easier) – Willemien Oct 9 '15 at 12:26

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