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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.
1. If a line meets a cubic curve in 3 points, the residual intersections of the tangents of these points with the cubic all lie on a line, called the satellite line of the original line. See Salmon (1879, p. 127). 2. A certain plane curve of degree (n–1)(n–2) constructed from a plane curve of degree n and a generic
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 ...
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.
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.
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Toggle Mathematics (Geometry) subsection. 1.1 Algebraic curves. 1.1.1 Rational curves. ... Space-filling curve (Peano curve) See also List of fractals by Hausdorff ...