Ads
related to: dual curve in geometry examples worksheet 1 answers
Search results
Results From The WOW.Com Content Network
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).
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.
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)
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.
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 ...
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 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.
The negativity of the canonical line bundle makes projective spaces prime examples of Fano varieties, equivalently, their anticanonical line bundle is ample (in fact very ample). Their index ( cf. Fano varieties ) is given by I n d ( P n ) = n + 1 {\displaystyle \mathrm {Ind} (\mathbb {P} ^{n})=n+1} , and, by a theorem of Kobayashi-Ochiai ...