So in the hyperbola, A is the focus and CD the directrix when the ratio HA:HK is constant for all points of the curve; and in the parabola, A is the focus and CD the directrix when the ratio BA:BC is constant.
A point so related to a conic section and certain straight line called the directrix that the ratio of the distance between any point of the curve and the focus to the distance of the same point from the directrix is constant.
The distance from each point of the ellipse to one of the foci, and to the directrix adjacent to that focus, are to each other as the eccentricity is to the major axis.
Construction of a paraboloid of raccordement to a ruled surface defined by two directrices and a directrix cone.
The osculating plane of the directrix at the point of contact touches the surface along this edge.
Illustration] The surface formed by revolving the catenary about itsdirectrix is named the alysseide.
Thus AB is the directrix of the parabola VED, of which F is the focus.
Illustration: Directrix of a Parabola] DIREC'TRIX, a fixed line that is required for the description of a curve.
The directrix of a parabola is a line perpendicular to the axis produced, and at a distance from the vertex equal to the distance of the vertex from the focus.
Defn: A point so related to a conic section and certain straight line called the directrix that the ratio of the distace between any point of the curve and the focus to the distance of the same point from the directrix is constant.
Conic Sections) Defn: The line drawn through a focus of a conic section parallel to the directrixand terminated both ways by the curve.
Every catenary lying between them has its directrix higher, and every catenary lying beyond them has its directrix lower than that of the two catenaries.
Since the tension is measured by the height above the directrix these two catenaries have the same directrix.
Hence a catenoid whose directrix coincides with the axis of revolution has at every point its principal radii of curvature equal and opposite, so that the mean curvature of the surface is zero.
Now let us consider the surfaces of revolution formed by this system of catenaries revolving about the directrix of the two catenaries of equal tension.
The radius of curvature of a catenary is equal and opposite to the portion of the normal intercepted by the directrix of the catenary.
Let P be any point on the directrix through which a line is drawn meeting the conic in the points A and B (Fig.
We have thus the property often taken as the definition of a conic: The ratio of the distances from a point on the conic to the focus and the directrix is constant.
For instance, in the case of the parabola, the distance of any particle from the directrix is equal to its distance from the focus.
As necessary as the directrix is to the curve, so are the corresponding laws to the State.
In an ellipse there is less likelihood of his straying away from the course which the directrix points out, on account of the two-fold guidance which he receives from the two foci.
If a parabola roll on another parabola, their vertices coinciding, the focus of the first traces out the directrix of the second.
Similarly, in an hyperbola a vertex is nearer to the directrix than to the focus.
In a parabola the vertex lies halfway betweendirectrix and focus.
If we next draw through A and B lines parallel to TF, then the points A1, B1 where these cut the directrix will be harmonic conjugates with regard to P and the point where FT cuts the directrix.
It follows in an ellipse the ratio between the distance of a point from the focus to that from the directrix is less than unity, in the parabola it equals unity, and in the hyperbola it is greater than unity.
The centre of a circle is its focus, and its directrix has gone to infinity, having no special direction.
Hence each focus lies within a conic; and a directrix does not cut the conic.
Hence the theorem: The ratio of the distances of any point on a conic from a focus and the corresponding directrix is constant.
