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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.
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 ...
A perspectivity: ′ ′ ′ ′, In projective geometry the points of a line are called a projective range, and the set of lines in a plane on a point is called a pencil.. Given two lines and in a projective plane and a point P of that plane on neither line, the bijective mapping between the points of the range of and the range of determined by the lines of the pencil on P is called a ...
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.
A beam compass and a regular compass Using a compass A compass with an extension accessory for larger circles A bow compass capable of drawing the smallest possible circles. A compass, also commonly known as a pair of compasses, is a technical drawing instrument that can be used for inscribing circles or arcs.
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In geometry, inversive geometry is the study of inversion, ... which is a surface of revolution contains a pencil of circles which is mapped onto a pencil of circles ...