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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.
It is usual, when dealing with dual and common conic sections, to call the common conic section a point conic and the dual conic a line conic. In the case that the underlying field has = all the tangents of a point conic intersect in a point, called the knot (or nucleus) of the conic. Thus, the dual of a non-degenerate point conic is a subset ...
The name "parabola" is due to Apollonius, who discovered many properties of conic sections. It means "application", referring to "application of areas" concept, that has a connection with this curve, as Apollonius had proved. [1] The focus–directrix property of the parabola and other conic sections was mentioned in the works of Pappus.
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).
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.
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 ...
More generally, when the directrix is an ellipse, or any conic section, and the apex is an arbitrary point not on the plane of , one obtains an elliptic cone [4] (also called a conical quadric or quadratic cone), [5] which is a special case of a quadric surface.
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.