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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.
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 enumerative geometry, Steiner's conic problem is the problem of finding the number of smooth conics tangent to five given conics in the plane in general position. If the problem is considered in the complex projective plane CP 2 , the correct solution is 3264. [ 1 ]
Another classic problem in enumerative geometry, of similar vintage to conics, is the Problem of Apollonius: a circle that is tangent to three circles in general determines eight circles, as each of these is a quadratic condition and 2 3 = 8. As a question in real geometry, a full analysis involves many special cases, and the actual number of ...
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.
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.
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