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If the degree of the curve is d then the degree of the polar is d − 1 and so the number of tangents that can be drawn through the given point is at most d(d − 1). The dual of a line (a curve of degree 1) is an exception to this and is taken to be a point in the dual space (namely the original line).
dual 1. The dual of a projective space is the set of hyperplanes, considered as another projective space. 2. The dual curve of a plane curve is the set of its tangent lines, considered as a curve in the dual projective plane. 3. A dual number is a number of the form a+εb where ε has square 0. Semple & Roth (1949, p.268)
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, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these ...
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.
Let X be a Riemann surface.Then the intersection number of two closed curves on X has a simple definition in terms of an integral. For every closed curve c on X (i.e., smooth function :), we can associate a differential form of compact support, the Poincaré dual of c, with the property that integrals along c can be calculated by integrals over X:
If the curve is non-singular the geometric genus and the arithmetic genus are equal, but if the curve is singular, with only ordinary singularities, the geometric genus is smaller. More precisely, an ordinary singularity of multiplicity r {\displaystyle r} decreases the genus by 1 2 r ( r − 1 ) {\displaystyle {\frac {1}{2}}r(r-1)} .
An isoptic is the set of points for which two tangents of a given curve meet at a fixed angle (see below). An isoptic of two plane curves is the set of points for which two tangents meet at a fixed angle. Thales' theorem on a chord PQ can be considered as the orthoptic of two circles which are degenerated to the two points P and Q.