Puzzling with curvature

Question

This is a post inspired by Gauss Bonnet theorem validation with hyperbolic circles

I was thinking also about curvature and how to measure it

And was puzzling with the following :

Suppose some critters land on a hyperbolic plane (a surface with a constant negative curvature) and they want to measure the curvature of it.

(critters is a nod to an old question about critters on a hyperbolic plane :)

I was thinking on how they could measure the curvature and I found one the following way:

The critters have brought their own measure m (think meters/ miles/yards /feet) and construct an isosceles right triangle with the top angle as the right angle

The legs of this triangle are both length $a m$ and the hypotenuse has length $b m$ ,

(In Euclidean geometry the base angles are $45^o$ and the length of the hypotenuse is always $bm = \sqrt {2} a m$ )

In hyperbolic geometry the base angles are $< 45^o$ and length of the hypotenuse is between $\sqrt {2} a m $ and $ 2 a m $

Even better from this construction you can estimate what the hyperbolic absolute lengths of $a m$ and $b m $ are.

$a m$ and $b m$ are also related by $\cosh(b m) = \cosh(a m) \cosh(a m) =\cosh^2(a m)$

So for every $\frac{b m}{a m} = \frac{b}{a } $ there is only one pair of absolute lengths $l_h = am$ and $h_h = b m$

And from this $m$ and the curvature can be estimated .

To make it all a bit clear in a table: ( I am still myself puzzeling with this )

 b/a   -> l     -> h
 1.43  0.372    0.532
 1.44  0.479    0.690
 1.45  0.567    0.822
 1.5   0.919    1,378
 1.6   1.523    2.438
 1.7   2.224    3.816

So for example: if the by de critters measured lengths are $a m= 2m$ and $bm = 3m$ then the lengths in absolute hyperbolic lengths are 0.919 and 1,378 meaning $m=\frac{0.919}{2}=\frac{1,378}{3}= 0.459$

But then:

What is the curvature ? Is it $ -m$ ,$ -m^2$, $\frac{1}{-m}$ or $\frac{1}{-m^{2}} $?

Is there a more direct way to calculate $l_h$ and $h_h$ from a/b? (Instead of estimating it by looking it up in a table)

I like this way of calculating curvature (it is quite simple just a simple triangle an measuring of lenghts ) are there an even more simple ones ?

Not sure what you mean by an equilateral right triangle in hyperbolic geometry (equilateral triangles have equal internal angles, all less than $\pi/3$...?), but curvature is "angular defect per unit area". Precisely, if $T$ is a geodesic triangle in a space of constant curvature $K$, then $$K = \frac{\text{sum of internal angles} - \pi}{\operatorname{Area}(T)}.$$ — Andrew D. Hwang, Aug 05 '17 at 11:18
Sorry mend isosceles triangle and the angle between them is right) — Willemien, Aug 05 '17 at 12:06
You say in other words an isosceles right hyperbolic triangle has invariant side lengths and $k_g$ curvatures of its sides by non-euclidean slidings arbitrarily on a pseudospherical surface... Is that correct? — Narasimham, Aug 24 '17 at 08:30
@narasimham not sure what you mean , my idea is that on a pseudo spherical surface you can calculate the curvature by constructing an isosceles right angled triangle and measuring the lengths of the legs and the hypotenuse — Willemien, Aug 24 '17 at 12:24
If geodesic curvature $\kappa_g$ is constant in Poincare models,they should be zero in $\mathbb R^3$ to apply here into GB theorem, Right? — Narasimham, Aug 24 '17 at 21:28
Not sure what you mean , in both poincare models of hyperbolic geometry the curvature is (by ???). -1 not even sure how a model would look if the curvature was something else but that is in models , a real hyperbolic plane can have any curvature (when you scale a pseudo sphere you do get a different curvature, only not sure how and so) — Willemien, Aug 25 '17 at 12:33
The geodesic curvature of all geodesic lines in Gauss Bonnet theorem is zero. What is there to measure in $\kappa_g$ curvature any further? — Narasimham, Aug 25 '17 at 20:20
Am afraid cannot understand. Suppose a calculated isosceles triangle of geodesic arcs/sides is drawn on a constant negative Gauss curvature surface in 3-space with $(a,b,c,A,B,C)$ given/calculated would that serve as an answer? — Narasimham, Aug 25 '17 at 20:52
No the ratio of the hypotenuse /legs is given/ measured ,this let's you approximate the absolute lengths (curvature = -1) and given the measured ( relative, wrong measure unit) length let's you calculate the curvature (with some unit) — Willemien, Aug 25 '17 at 21:00
See also https://math.stackexchange.com/q/2270994/88985 especially the first answer is the same idea in spherical geometry — Willemien, Aug 26 '17 at 04:57

score 0 · Answer 1 · edited Jun 12 '20 at 10:38

Did not fully understood what geodetic information the critters get, however, am attempting a basic answer hoping it would serve as background for possible discussion / insight with 3D hyperbolic triangles constructed by pseudospherical trigonometry measurements.

Given: $\,C= \pi/2,a=b$

To Find:

A relation between $(a,A)$ for hyperbolic isosceles right triangles to find other angles and sides.

Sinh and Cosh Laws in pseudospherical Trigonometry

(The term used from Roberto Bonola's Non-Euclidean Geometry text-book pp 137)

Let Gauss curvature be $ -1/T^2.$

By hyperbolic Law of (Sinhs)

$$ \frac{\sin A}{\sinh (a/T)}=\left(\frac{\sin B}{\sinh (b/T)}\right)=\frac{\sin C}{\sinh (c/T)}=\frac{1}{\sinh (a/T)}$$

$$ \frac{\sin A}{\sinh (a/T)}=\frac{1}{\sinh (c/T) } \tag1$$

From hyperbolic Law of Cosines (Coshs) triangle sides with a right angle can be generalized to "hyperbolic Pythagoras " theorem:

$$\cosh(c/T) = \cosh(a/T) \cosh(b/T) = \cosh^{2}(a/T) \tag2$$

Combine (1) and (2) and simplify to get a relation for isosceles hyperbolic right triangles as

$$\sin A \sqrt{ 1+ \cosh^{2} a/T }=1 \tag3$$

The graph of above trigonometric relation.. $ (a,A) $ plotted on $(x,y)$ axes respectively.. and representative 3D projection of triangles made on bisected sides $a=(1,2) $ is shown red and blue triangles. Also

$$\sin A \sqrt{ 1+ \cosh \, c/T }=1 \tag4$$

Eliminate $A$ from (3) and (4) side/hypotenuse ratio is found variable for right hyperbolic isosceles requirement but you are trying to fix it up .. so cannot understand how that could be done.

If $ a=1,c=\sqrt2 $ are given, dividing (3) and (4) numerical solution $ T \approx 7300,K=-1/T^2 \approx \frac{-1}{7300^2}$ the Gauss curvature is seen to be very low, near to a flat plane.

c=Sqrt[2.]; a=1. ;Plot[{Cosh[c/tt]/Cosh[a/tt]^2}-1., {tt,6500., 8000.}]

Numerical data

Clairaut's constant for all geodesics shown

$$r_i \sin \psi_i= 1/4 $$

$$ K=-1/T^2,\, (a,b,c)= (2,2,3.38021); \quad (a_{small},b_{small},c_{small})=(1,1,1.51337);\quad (A,B,C)= (0.249795,0.249795,\pi/2);\quad(for \ both) $$

When three arcs are plotted with these lengths and surface angles, a closed isosceles triangle forms checking accuracy of calculation.

An interesting take on this is about Integral curvature $\int\int K dA$ is same for all right isosceles triangles same $A$ when $a$ is partitioned in anyway.. as the pseudospherical deficit is same for the red and blue side triangles given above or for any other triangle constructed in a similar way.It evaluates to $ I.C.= (\pi/2-2A)$.

The set of hypotenuses are not parallel in any sense.

In response to your question in comments that I still cannot understand.. for all prescribed ratios of leg length $a$ to hypotenuse $c..$ by virtue of Equation (2) we can have only one trivial triangle solution $ a=c=0.$

In conclusion, ratio $c/a$ is a fixed number. In the last $\cosh$ graph $c/T,a/T$ relation is not a straight line through the origin. To me it appears the "ratio" of side/hypotenuse restriction is perhaps our hang-up from Euclidean trigonometry concept of angle definition. One has to abandon it .. large nonlinearities of non-Euclidean geometries demand it. Even in spherical trigonometry a construction in with different $c/a$ ratios produces different integral curvatures.

EDIT1:

The conclusion was not comprehensive enough.

We should be able to plot $K$ as a function of hypotenuse to side ratio $c/a$. The plot should include flat Euclidean special case as well as spherical geometry if possible. The following does precisely that. Dividing (3) by (4)

$$ \cosh (c \sqrt{|K|})/ \cosh ^2 (a \sqrt{|K|}) = 1 ,\quad \cos (c \sqrt{|K|})/ \cos ^2 (a \sqrt{|K|}) = 1 \tag5 $$

The hyperbolic triangle hypotenuse/side ratio$c/a$ should be $\sqrt 2 $ when $K \approx 0$ for flatter triangles and $c/a=2$ when the sides are deep and curved...

You cannot choose a ratio like 1.2 or 3.0 outside this interval.

That solves the puzzle perhaps more than earlier ..

The critters know nothing more than 1) the ratio between between the legs and the hypotenuse, 2) they have some unit of lenght / distance and how long the sides are in these units) — Willemien, Aug 28 '17 at 08:43
Maybe clearest can make it the ratio $c:a $ differs from the ratio $ c_{small} : a_{small} $ and from this difference the question follows. — Willemien, Aug 30 '17 at 03:16
My question is not about the integral curvature but about the total (Gaussian) curvature ( the product of the principal curvatures at a point) — Willemien, Aug 30 '17 at 12:47
Omg! By assumption it is the same for all the hyperbolic triangular patches $ K= -1/T^2, T=1 $ — Narasimham, Aug 30 '17 at 13:45
yes it is the same for the whole surface, but what is its value and how do you calcukate it ?(from a,b and their ratio) you can construct different speudo spheres with different gaussan curvature (like you can construct different spheres with different curvatures) atleast that is my presumption — Willemien, Aug 30 '17 at 19:26

Puzzling with curvature

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