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Surfaces containing two circles through each point. (English) Zbl 1416.51003

Summary: We find all analytic surfaces in 3-dimensional Euclidean space such that through each point of the surface one can draw two transversal circular arcs fully contained in the surface (and analytically depending on the point). The search for such surfaces traces back to the works of Darboux from XIXth century. We prove that such a surface is an image of a subset of one of the following sets under some composition of inversions:
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the set \(\{p+q:p\in \alpha ,q\in \beta \}\), where \(\alpha ,\beta \) are two circles in \(\mathbb{R}^3\);
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the set \(\{2\frac{p \times q^{}}{|p+q|^2}:p\in \alpha ,q\in \beta ,p+q\ne 0\}\), where \(\alpha ,\beta \) are circles in the unit sphere \({S}^2\);
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the set \(\{(x,y,z): Q(x,y,z,x^2+y^2+z^2)=0\}\), where \(Q\in \mathbb{R}[x,y,z,t]\) has degree 2 or 1.
The proof uses a new factorization technique for quaternionic polynomials.

MSC:

51B10 Möbius geometries
13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
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