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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.
A circle of finite radius has an infinitely distant directrix, while a pair of lines of finite separation have an infinitely distant focus. Alternatively, one can define a conic section purely in terms of plane geometry: it is the locus of all points P whose distance to a fixed point F (called the focus ) is a constant multiple e (called the ...
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 this case one could have used the apex as the directrix, i.e. = (,,) and = (, ,) as the line directions. For any cone one can choose the apex as the directrix. This shows that the directrix of a ruled surface may degenerate to a point.
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 directrix is often taken as a plane curve, in a plane not containing the apex, but this is not a requirement. [1] In general, a conical surface consists of two congruent unbounded halves joined by the apex. Each half is called a nappe, and is the union of all the rays that start at the apex and pass through a point of some fixed space curve ...
In mathematics, a directrix is a curve associated with a process generating a geometric object, such as: Directrix (conic section) Directrix (generatrix)
A cone can be generated by moving a line (the generatrix) fixed at the future apex of the cone along a closed curve (the directrix); if that directrix is a circle perpendicular to the line connecting its center to the apex, the motion is rotation around a fixed axis and the resulting shape is a circular cone.