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Curves, dual to each other; see below for properties. In projective geometry, a dual curve of a given plane curve C is a curve in the dual projective plane consisting of the set of lines tangent to C. There is a map from a curve to its dual, sending each point to the point
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A.
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 dual of an isogonal (vertex-transitive) polygon is an isotoxal (edge-transitive) polygon. For example, the (isogonal) rectangle and (isotoxal) rhombus are duals. In a cyclic polygon, longer sides correspond to larger exterior angles in the dual (a tangential polygon), and shorter sides to smaller angles.
Download QR code; Print/export Download as PDF; Printable version; In other projects ... This is a gallery of curves used in mathematics, by Wikipedia page ...
The cube and regular octahedron are dual graphs of each other. According to Steinitz's theorem, every polyhedral graph (the graph formed by the vertices and edges of a three-dimensional convex polyhedron) must be planar and 3-vertex-connected, and every 3-vertex-connected planar graph comes from a convex polyhedron in this way.
The construction can be easily generalized to nodal curves with multiple nodes. This is used in the construction of the Hodge bundle on the compactified moduli space of curves: it allows us to extend the relative canonical sheaf over the boundary which parametrizes nodal curves. The Hodge bundle is then defined as the direct image of a relative ...