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Cylinders through five points: Complex and real enumerative geometry. (English) Zbl 1195.68106
Botana, Francisco (ed.) et al., Automated deduction in geometry. 6th international workshop, ADG 2006, Pontevedra, Spain, August 31–September 2, 2006. Revised papers. Berlin: Springer (ISBN 978-3-540-77355-9/pbk). Lecture Notes in Computer Science 4869. Lecture Notes in Artificial Intelligence, 80-97 (2007).
Summary: It is known that five points in \(\mathbb R^{3}\) generically determine a finite number of cylinders containing those points. We discuss ways in which it can be shown that the generic (complex) number of solutions, with multiplicity, is six, of which an even number will be real-valued and hence correspond to actual cylinders in \(\mathbb R^{3}\). We partially classify the case of no real solutions in terms of the geometry of the five given points. We also investigate the special case where the five given points are coplanar, as it differs from the generic case for both complex and real-valued solution cardinalities.
For the entire collection see [Zbl 1132.68006].

68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
51M04 Elementary problems in Euclidean geometries
51N20 Euclidean analytic geometry
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
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