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 provides some pointers, with perhaps the most useful reference being Area of Common Overlap of Three Circles. 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!

  • $\begingroup$ All these triangles are Moebius-equivalent to spherical triangles or hyperbolic triangles (in, say, unit disk model) or Euclidean triangles. This answers your question about "the rest" of triangles. $\endgroup$
    – Misha
    Commented Oct 10, 2013 at 5:33

1 Answer 1


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.

  • $\begingroup$ Entschuldigt. Mine Deutsch ist nicht sehr gut! $\endgroup$ Commented Oct 10, 2013 at 3:36
  • 1
    $\begingroup$ Nichts ueber dein Deutsch zu sagen, aber each triangle which maps onto (0,1,\infty) in the upper half-plane thus becomes an ideal hyperbolic triangle. Actually, a hyperbolic triangle is made like follows: you take three circles in the upper half-plane {(x,y)|y>0} such that all of them are orthogonal to {y=0}. Then their respective segments form the sides of a triangle. This is a hyperbolic triangle. However, each hyperbolic triangle is uniquely determined by its internal angles or side lengths, so you may be looking at a more general object which you name a circular triangle. $\endgroup$ Commented Oct 10, 2013 at 4:16
  • $\begingroup$ It does not have to be in the upper halfplane. When I say fractional-linear transformation, I mean SL(2,C). In fact there are three types of circular triangles, depending on the sum of angles: those equivalent by SL(2,C) to hyperbolic, flat or spherical (geodesic) triangles. $\endgroup$ Commented Oct 11, 2013 at 3:07

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