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A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes considered a
In homogeneous coordinates, each conic section with the equation of a circle has the form + + = It can be proven that a conic section is a circle exactly when it contains (when extended to the complex projective plane) the points I(1: i: 0) and J(1: −i: 0).
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 ...
The standard form of the equation of a central conic section is obtained when the conic section is translated and rotated so that its center lies at the center of the coordinate system and its axes coincide with the coordinate axes. This is equivalent to saying that the coordinate system's center is moved and the coordinate axes are rotated to ...
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 ...
A conic is defined as the locus of points for each of which the distance to the focus divided by the distance to the directrix is a fixed positive constant, called the eccentricity e. If 0 < e < 1 the conic is an ellipse, if e = 1 the conic is a parabola, and if e > 1 the conic is a hyperbola.
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). The intersection of an elliptic cone with a concentric sphere is a spherical conic.
Geometric objects with a well-defined axis include circles (any line through the center), spheres, cylinders, [2] conic sections, and surfaces of revolution. Concentric objects are often part of the broad category of whorled patterns , which also includes spirals (a curve which emanates from a point, moving farther away as it revolves around ...