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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 called as a
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 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 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).
This can be simplified in various ways, to conform to more specific cases, such as the equation = for a circle with a center at the pole and radius a. [ 15 ] When r 0 = a or the origin lies on the circle, the equation becomes r = 2 a cos ( φ − γ ) . {\displaystyle r=2a\cos(\varphi -\gamma ).}
It is also possible to describe all the conic sections as loci of points that are equidistant from a single focus and a single, circular directrix. For the ellipse, both the focus and the center of the directrix circle have finite coordinates and the radius of the directrix circle is greater than the distance between the center of this circle ...
The eccentricity of the conic section obtained by reciprocation is the ratio of the distances between the two circles' centers to the radius r of reciprocation circle C. If B and C represent the points at the centers of the corresponding circles, then = ¯.
The lateral surface area of a right circular cone is = where is the radius of the circle at the bottom of the cone and is the slant height of the cone. [4] The surface area of the bottom circle of a cone is the same as for any circle, . Thus, the total surface area of a right circular cone can be expressed as each of the following: