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The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse. A circle is a special case of an ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a ...
The discriminant B 2 – 4AC of the conic section's quadratic equation (or equivalently the determinant AC – B 2 /4 of the 2 × 2 matrix) and the quantity A + C (the trace of the 2 × 2 matrix) are invariant under arbitrary rotations and translations of the coordinate axes, [14] [15] [16] as is the determinant of the 3 × 3 matrix above.
The vertices of a central conic can be determined by calculating the intersections of the conic and its axes — in other words, by solving the system consisting of the quadratic conic equation and the linear equation for alternately one or the other of the axes. Two or no vertices are obtained for each axis, since, in the case of the hyperbola ...
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 ...
A hyperbola and its conjugate may be constructed as conic sections obtained from an intersecting plane that meets tangent double cones sharing the same apex. Each cone has an axis, and the plane section is parallel to the plane formed by the axes. Using analytic geometry, the hyperbolas satisfy the symmetric equations
A,C: vertices of the ellipse and foci of the hyperbola E,F: foci of the ellipse and vertices of the hyperbola Focal conics: two parabolas A: vertex of the red parabola and focus of the blue parabola F: focus of the red parabola and vertex of the blue parabola. In geometry, focal conics are a pair of curves consisting of [1] [2] either
A family of conic sections of varying eccentricity share a focus point and directrix line, including an ellipse (red, e = 1/2), a parabola (green, e = 1), and a hyperbola (blue, e = 2). The conic of eccentricity 0 in this figure is an infinitesimal circle centered at the focus, and the conic of eccentricity ∞ is an infinitesimally separated ...
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 ...