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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. 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 ...
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.
More generally, for any collection of points P i, weights w i, and constant C, one can define a circle as the locus of points X such that (,) =.. The director circle of an ellipse is a special case of this more general construction with two points P 1 and P 2 at the foci of the ellipse, weights w 1 = w 2 = 1, and C equal to the square of the major axis of the ellipse.
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 ...
The directrix of a conic section can be found using Dandelin's construction. Each Dandelin sphere intersects the cone at a circle; let both of these circles define their own planes. The intersections of these two parallel planes with the conic section's plane will be two parallel lines; these lines are the directrices of the conic section.
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.
canal surface: directrix is a helix, with its generating spheres pipe surface: directrix is a helix, with generating spheres pipe surface: directrix is a helix. In geometry and topology, a channel or canal surface is a surface formed as the envelope of a family of spheres whose centers lie on a space curve, its directrix.
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. [3]