Search results
Results From The WOW.Com Content Network
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.
requiring a conic to pass through a point imposes a linear condition on the coordinates: for a fixed (,), the equation + + + + + = is a linear equation in (,,,,,); by dimension counting , five constraints (that the curve passes through five points) are necessary to specify a conic, as each constraint cuts the dimension of possibilities by 1 ...
The equation for a conic section with apex at the origin and tangent to the y axis is + (+) = alternately = + (+) where R is the radius of curvature at x = 0. This formulation is used in geometric optics to specify oblate elliptical ( K > 0 ), spherical ( K = 0 ), prolate elliptical ( 0 > K > −1 ), parabolic ( K = −1 ), and hyperbolic ( K ...
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.
In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section – though it may be degenerate, and all conic sections arise in this way. The equation will be of the form A x 2 + B x y + C y 2 + D x + E y + F = 0 with A , B , C not all zero. {\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0{\text{ with ...
Equations. If one describes the ellipse in the x-y-plane in the common way by the equation + =, then the corresponding focal hyperbola in the x-z-plane has equation
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.