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Algebraic geometry on imaginary triangles. (English) Zbl 1411.14034

An imaginary triangle is defined as a pair consisting of a triple \((a,b,c)\) of complex numbers, the side lengths, and an equivalence class \([\alpha, \beta, \gamma]\) of corresponding angles that is only determined modulo \(2\pi\) and up to multiplication with \(-1\) and satisfies the constraint \(\alpha + \beta + \gamma \equiv \pi \mod 2\pi\). Moreover, it is required that the laws of sines and cosines hold.
If no side length vanishes, there is a unique angle class \([\alpha, \beta, \gamma]\). If a side length vanishes, suitable angles exist only if the remaining two sides are equal up to sign. In this case, even infinitely many angles do exist. Hence, every point of the complex projective plane \(\mathbb{C}P^2\) minus the lines \(a = 0\), \(b = 0\), \(c = 0\) plus the “gate points” \(E_{\pm} = [0,1,\pm 1]\), \(F_{\pm} = [1,0,\pm 1]\), \(G_{\pm} = [1,\pm 1, 0]\) corresponds to at least one complex triangle up to scaling and precisely the gate points give rise to an infinity of complex triangles.
The author studies triangles for which \(\alpha : \beta = p : q\) where \(p\) and \(q\) are two relatively prime positive integers. This generalizes the notion of isosceles triangles (\(p = q = 1\)). These triangles correspond to a rational curve \(\mathcal{C}_{p,q}\) in \(\mathbb{C}P^2\). A rational parametric equation is given by \([U_{p-1}, U_{q-1}, U_{p+q+1}]\) with \(U_n\) denoting the \(n\)-th Chebyshev polynomial of the second kind. While a rational parametric equation provides rational solution triangles (Pythagorean triples), an algebraic equation for \(\mathcal{C}_{p,q}\) can be viewed as Pythagorean theorem. It can be computed for any \(p\), \(q\) by an iteration process involving a quadratic Cremona transformation of \(\mathbb{C}P^2\) that sends triangles with angles \(\alpha\), \(\beta\) to triangles with angles \(\alpha\), \(\beta - \alpha\), coordinate swaps and removal of extraneous factors that can be computed a priori.

MSC:

14H50 Plane and space curves
14Q05 Computational aspects of algebraic curves
14E07 Birational automorphisms, Cremona group and generalizations
51M04 Elementary problems in Euclidean geometries

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References:

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