On a Theorem of Intersecting Conics
نویسنده
چکیده
Given two conics over an infinite field that intersect at the origin, a line through the origin will, in general intersect both conic sections once more each, at points C and D. As the line varies we find that the midpoint of C and D traces out a curve, which is typically a quartic. Intuitively, this locus is the “average” of the two conics from the perspective of an observer at the origin. We give necessary and sufficient conditions for this locus to be a point, line, line minus a point, or a conic itself.
منابع مشابه
Real Enumerative Geometry and Effective Algebraic Equivalence
Determining the common zeroes of a set of polynomials is further complicated over nonalgebraically closed fields such as the real numbers. A special case is whether a problem of enumerative geometry can have all its solutions be real. We call such a problem fully real. Little is known about enumerative geometry from this perspective. A standard proof of Bézout’s Theorem shows the problem of int...
متن کاملPlane Conics in Algebraic Geometry
We first examine points and lines within projective spaces. Then we classify affine conics based on the classification of projective conics. Based on the parametrization of conics, we also prove two easy cases of Bézout’s Theorem. In the end we turn to the discussion of the space of conics and the notion of pencils of conics.
متن کاملMarden theorem and Poncelet-Darboux curves
The Marden theorem of geometry of polynomials and the great Poncelet theorem from projective geometry of conics by their classical beauty occupy very special places. Our main aim is to present a strong and unexpected relationship between the two theorems. We establish a dynamical equivalence between the full Marden theorem and the Poncelet-Darboux theorem. By introducing a class of isofocal def...
متن کاملEnumerative Geometry for Real Varieties
Of the geometric figures in a given family satisfying real conditions, some figures are real while the rest occur in complex conjugate pairs, and the distribution of the two types depends subtly upon the configuration of the conditions. Despite this difficulty, applications ([7],[28],[32]) may demand real solutions. Fulton [11] asked how many solutions of an enumerative problem can be real, and...
متن کاملProjective bundles
A projective bundle in PG(2, q) is a collection of q + q + 1 conics that mutually intersect in a single point and hence form another projective plane of order q. The purpose of this paper is to investigate the possibility of partitioning the q5 − q2 conics of PG(2, q) into q2(q − 1) disjoint projective bundles. As a by-product we obtain a multiplier theorem for perfect difference sets that gene...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2011