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  2. Conic section - Wikipedia

    en.wikipedia.org/wiki/Conic_section

    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.

  3. Eccentricity (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Eccentricity_(mathematics)

    In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape. One can think of the eccentricity as a measure of how much a conic section deviates from being circular. In particular: The eccentricity of a circle is 0. The eccentricity of an ellipse which is not a circle is between 0 and 1.

  4. Matrix representation of conic sections - Wikipedia

    en.wikipedia.org/wiki/Matrix_representation_of...

    The center of a conic, if it exists, is a point that bisects all the chords of the conic that pass through it. This property can be used to calculate the coordinates of the center, which can be shown to be the point where the gradient of the quadratic function Q vanishes—that is, [8] = [,] = [,].

  5. Conic constant - Wikipedia

    en.wikipedia.org/wiki/Conic_constant

    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 ...

  6. Cone - Wikipedia

    en.wikipedia.org/wiki/Cone

    It is an affine image of the right-circular unit cone with equation + = . 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.

  7. Rotation of axes in two dimensions - Wikipedia

    en.wikipedia.org/wiki/Rotation_of_axes_in_two...

    A non-degenerate conic section given by equation can be identified by evaluating . The conic section is: [ 13 ] an ellipse or a circle, if B 2 − 4 A C < 0 {\displaystyle B^{2}-4AC<0} ;

  8. Analytic geometry - Wikipedia

    en.wikipedia.org/wiki/Analytic_geometry

    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 ...

  9. Confocal conic sections - Wikipedia

    en.wikipedia.org/wiki/Confocal_conic_sections

    A pencil of confocal ellipses and hyperbolas is specified by choice of linear eccentricity c (the x-coordinate of one focus) and can be parametrized by the semi-major axis a (the x-coordinate of the intersection of a specific conic in the pencil and the x-axis). When 0 < a < c the conic is a hyperbola; when c < a the conic is an ellipse.