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There is a map from a curve to its dual, sending each point to the point dual to its tangent line. If C is algebraic then so is its dual and the degree of the dual is known as the class of the original curve. The equation of the dual of C, given in line coordinates, is known as the tangential equation of C.
A collection of puzzles involving numbers, logic, and probability 1962 Nov: Some puzzles based on checkerboards: 1962 Dec: Some simple tricks and manipulations from the ancient lore of string play: 1963 Jan: The author pays his annual visit to Dr. Matrix, the numerologist: 1963 Feb: Curves of constant width, one of which makes it possible to ...
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These sets can be used to define a plane dual structure. Interchange the role of "points" and "lines" in C = (P, L, I) to obtain the dual structure. C ∗ = (L, P, I ∗), where I ∗ is the converse relation of I. C ∗ is also a projective plane, called the dual plane of C. If C and C ∗ are isomorphic, then C is called self-dual.
A curve in this context is defined by a non-degenerate algebraic equation in the complex projective plane. Lines in this plane correspond to points in the dual projective plane and the lines tangent to a given algebraic curve C correspond to points in an algebraic curve C * called the dual curve.
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In a cyclic polygon, longer sides correspond to larger exterior angles in the dual (a tangential polygon), and shorter sides to smaller angles. [citation needed] Further, congruent sides in the original polygon yields congruent angles in the dual, and conversely. For example, the dual of a highly acute isosceles triangle is an obtuse isosceles ...
The vector spaces () and () are dual to each other by the intersection pairing, and the nef cone is (by definition) the dual cone of the cone of curves. [ 6 ] A significant problem in algebraic geometry is to analyze which line bundles are ample , since that amounts to describing the different ways a variety can be embedded into projective space.