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A conic is the curve obtained as the intersection of a plane, called the cutting plane, with the surface of a double cone (a cone with two nappes).It is usually assumed that the cone is a right circular cone for the purpose of easy description, but this is not required; any double cone with some circular cross-section will suffice.
In mathematics, the matrix representation of conic sections permits the tools of linear algebra to be used in the study of conic sections. It provides easy ways to calculate a conic section's axis , vertices , tangents and the pole and polar relationship between points and lines of the plane determined by the conic.
The Steiner conic or more precisely Steiner's generation of a conic, named after the Swiss mathematician Jakob Steiner, is an alternative method to define a non-degenerate projective conic section in a projective plane over a field. The usual definition of a conic uses a quadratic form (see Quadric (projective geometry)). Another alternative ...
In the real projective plane a von Staudt conic is a conic section in the usual sense. In more general projective planes this is not always the case. Karl Georg Christian von Staudt introduced this definition in Geometrie der Lage (1847) as part of his attempt to remove all metrical concepts from projective geometry.
From the fact, that the affine image of a conic section is a conic section of the same type (ellipse, parabola,...), one gets: Any plane section of an elliptic cone is a conic section. Obviously, any right circular cone contains circles. This is also true, but less obvious, in the general case (see circular section).
As an example, count the conic sections tangent to five given lines in the projective plane. [4] The conics constitute a projective space of dimension 5, taking their six coefficients as homogeneous coordinates, and five points determine a conic, if the points are in general linear position, as passing through a given point imposes a linear ...
In Euclidean geometry, a circumconic is a conic section that passes through the three vertices of a triangle, [1] and an inconic is a conic section inscribed in the sides, possibly extended, of a triangle. [2] Suppose A, B, C are distinct non-collinear points, and let ABC denote the triangle whose vertices are A, B, C.
In algebraic geometry, the conic sections in the projective plane form a linear system of dimension five, as one sees by counting the constants in the degree two equations. The condition to pass through a given point P imposes a single linear condition, so that conics C through P form a linear system of dimension 4.