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A consequence is that the equation (in ,) of the parabola determined by 3 points = (,), =,,, with different x coordinates is (if two x coordinates are equal, there is no parabola with directrix parallel to the x axis, which passes through the points) =.
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 ...
Conic sections of varying eccentricity sharing 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 pair of lines.
In an ellipse, the semi-major axis is the geometric mean of the distance from the center to either focus and the distance from the center to either directrix. The semi-minor axis of an ellipse runs from the center of the ellipse (a point halfway between and on the line running between the foci) to the edge of the ellipse. The semi-minor axis is ...
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 ...
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It has been proved that the Kiepert hyperbola is the hyperbola passing through the vertices, the centroid and the orthocenter of the reference triangle and the Kiepert parabola is the parabola inscribed in the reference triangle having the Euler line as directrix and the triangle center X 110 as focus. [1]