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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 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.
In mathematics, the Cayley–Bacharach theorem is a statement about cubic curves (plane curves of degree three) in the projective plane P 2. The original form states: Assume that two cubics C 1 and C 2 in the projective plane meet in nine (different) points, as they do in general over an algebraically closed field. Then every cubic that passes ...
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.
In Euclidean and projective geometry, five points determine a conic (a degree-2 plane curve), just as two (distinct) points determine a line (a degree-1 plane curve).There are additional subtleties for conics that do not exist for lines, and thus the statement and its proof for conics are both more technical than for lines.
To an abelian variety A over a field k, one associates a dual abelian variety (over the same field), which is the solution to the following moduli problem. A family of degree 0 line bundles parametrised by a k -variety T is defined to be a line bundle L on A × T {\displaystyle A\times T} such that
The iTunes description for Crickler 2 states that this take on the crossword puzzle genre is an "adaptive" experience, that automatically adjusts itself to your own skill level and knowledge.
Thus, the dual of a non-degenerate point conic is a subset of points of a dual line and not an oval curve (in the dual plane). So, only in the case that Char ≠ 2 {\displaystyle \operatorname {Char} \neq 2} is the dual of a non-degenerate point conic a non-degenerate line conic.