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A circle of finite radius has an infinitely distant directrix, while a pair of lines of finite separation have an infinitely distant focus. In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape.
The ellipse thus generated has its second focus at the center of the directrix circle, and the ellipse lies entirely within the circle. For the parabola, the center of the directrix moves to the point at infinity (see Projective geometry). The directrix "circle" becomes a curve with zero curvature, indistinguishable from a straight line.
The vertices are (±a, 0) and the foci (±c, 0). Define b by the equations c 2 = a 2 − b 2 for an ellipse and c 2 = a 2 + b 2 for a hyperbola. For a circle, c = 0 so a 2 = b 2, with radius r = a = b. For the parabola, the standard form has the focus on the x-axis at the point (a, 0) and the directrix the line with equation x = −a. In ...
describes a right circular conoid with the unit circle of the x-y-plane as directrix and a directrix plane, which is parallel to the y--z-plane. Its axis is the line (,,) . Special features: The intersection with a horizontal plane is an ellipse.
In mathematics, a directrix is a curve associated with a process generating a geometric object, such as: Directrix (conic section) Directrix (generatrix)
In geometry, a surface S in 3-dimensional Euclidean space is ruled (also called a scroll) if through every point of S, there is a straight line that lies on S. Examples include the plane , the lateral surface of a cylinder or cone , a conical surface with elliptical directrix , the right conoid , the helicoid , and the tangent developable of a ...
The vertex A is equidistant from the focus F and from the directrix. Since C is on the directrix, the y coordinates of F and C are equal in absolute value and opposite in sign. B is the midpoint of FC. Its x coordinate is half that of D, that is, x/2.
The pole of a line is the inversion of its closest point to the circle C, whereas the polar of a point is the converse, namely, a line whose closest point to C is the inversion of the point. 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 ...