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In geometry, a pencil is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a plane, or the set of circles that pass through two given points in a plane.
A pencil is a particular kind of linear system of divisors on , namely a one-parameter family, parametrised by the projective line.This means that in the case of a complex algebraic variety, a Lefschetz pencil is something like a fibration over the Riemann sphere; but with two qualifications about singularity.
In any affine space (including a Euclidean space) the set of lines parallel to a given line (sharing the same direction) is also called a pencil, and the vertex of each pencil of parallel lines is a distinct point at infinity; including these points results in a projective space in which every pair of lines has an intersection.
If one pencil is elliptic, its perpendicular pencil is hyperbolic, and vice versa; in this case the two pencils form a set of Apollonian circles. The pencil of circles perpendicular to a parabolic pencil is also parabolic; it consists of the circles that have the same common tangent point but with a perpendicular tangent line at that point. [4]
The set of all points on a line, called a projective range, has as its dual a pencil of lines, the set of all lines on a point, in two dimensions, or a pencil of hyperplanes in higher dimensions. A line segment on a projective line has as its dual the shape swept out by these lines or hyperplanes, a double wedge .
Pencils are a common theme in many geometry publications throughout history, though the term is less commonly used today. This article does not explicitly refer to pencils, though some of the constructions found herein, and in projective geometry more broadly, do in fact implicitly use the notion of a pencil, often by different terminology or ...
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Each curve in the pencil passes through the nine points of the complex projective plane whose homogeneous coordinates are some permutation of 0, –1, and a cube root of unity. There are three roots of unity, and six permutations per root, giving 18 choices for the homogeneous coordinates of each point, but they are equivalent in pairs giving ...
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