A *circular triangle* is a closed simple curve in the Euclidean plane $\mathbb{R}^2$ that can be expressed as the union of three circular arcs. Data naturally associated with such a figure includes:


* the edge lengths $\ell\_i$,
* the edge curvatures $\kappa\_i$, and
* the interior angles $\theta\_i$


(for $i \in {1,2,3}$). In English, surprisingly little seems to have been written about these creatures. [Mathworld](http://mathworld.wolfram.com/CircularTriangle.html) provides some pointers, with perhaps the most useful reference being [Area of Common Overlap of Three Circles](http://www.dsto.defence.gov.au/publications/4815/DSTO-TN-0722.pdf). However, this investigation is a bit brute force/trigonometric/algorithmic, which makes it hard to get a clean picture of what's really going on in the "space of circular triangles." A few natural questions:


**Question 1:** *What's a (nice) way to characterize data $\theta\_i$, $\ell\_i$, $\kappa\_i$ that describes a valid circular triangle?*


For instance, when the curvatures $\kappa\_i$ are all zero (i.e., just a standard Euclidean triangle), the necessary and sufficient conditions are of course that the angles sum to $\pi$ and the lengths satisfy the triangle inequality. What about the general case? One can of course express, say, $\theta$ in terms of $\ell$ and $\kappa$, but the expressions get rather nasty. A related question is:


**Question 2:** *Can all circular triangles be expressed as the image of geodesic triangles from some "nice" space under some "nice" map?*


For instance, one can study the subset of circular triangles coming from spherical geometry (via stereographic projection) or hyperbolic geometry (via the Poincaré model). What about the rest?


If a global description is too nasty, I'd be willing to settle for a more local characterization:


**Question 3:** *Suppose $\theta\_i,\ell\_i,\kappa\_i$ describe a valid circular triangle. Generically, how can one vary this data so that the triangle remains valid? (Morally: what does the "tangent space" look like? What's its dimension?)*


Finally, and more generally:


**Question 4:** *Is there another name for this object? What else is known?*


Any pointers are appreciated. Thanks!



The space of circular triangles?
Actually the literature on circular triangles is enormous. Most of it is in German.
Here is a little sample:


A. Schonflies,
Ueber Kreisbogendreiecke und Kreisbogenvierecke,
Math. Ann., 44 (1894) 105--124.


F. Klein, Ueber die Nullstellen der hypergeometrischen Reihe,
Math. Ann., 37 (1890) 573--590.


F. Klein, Vorlesungen uber die hypergeometrische Funktion, 1933, reprint:
Springer-Verlag, Berlin-New York, 1981.


There is a complete classification and description of the moduli space.
The connection with hypergeometric function is the following.
Let $y^"+Py=0$ be a hypergeometric equation with real coefficients.
Let $y\_1,y\_2$ be two linearly independent solutions.
Then $f=y\_1/y\_2$ maps the upper half-plane conformally onto a spherical triangle
(region bounded by 3 arcs of circles). And every spherical triangle (region, surface)
is obtained in this way.


Of course, your definition of the spherical triangle is different: your triangle
is just a curve. But classification of triangles-curves is simpler that classification
of triangles-regions, and it can be found in those papers I mentioned. 


Notice also that fractional-linear transformations of the Riemann sphere act on
the set of circular triangles. By a fractional-linear transformation, every triangle
can be sent to a triangle with vertices $0,1,\infty$. Then two sides are straight, and 
one, $(0,1)$ is straight or circle. This leads to a simple classification. 


