pencil of conics造句

• A pencil of conics can represented algebraically in the following way.
• All these degenerate conics may occur in pencils of conics.
• In a projective plane defined over an algebraically closed field any two conics meet in four points ( counted with multiplicity ) and so, determine the pencil of conics based on these four points.
• Furthermore, the four base points determine three line pairs ( degenerate conics through the base points, each line of the pair containing exactly two base points ) and so each pencil of conics will contain at most three degenerate conics.
• A striking application of such a family is in which gives a geometric solution to a quartic equation by considering the pencil of conics through the four roots of the quartic, and identifying the three degenerate conics with the three roots of the resolvent cubic.
• Such families arise naturally  given four points in general linear position ( no three on a line ), there is a pencil of conics through them ( five points determine a conic, four points leave one parameter free ), of which three are degenerate, each consisting of a pair of lines, corresponding to the \ textstyle { \ binom { 4 } { 2, 2 } = 3 } ways of choosing 2 pairs of points from 4 points ( counting via the multinomial coefficient ).