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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.
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 ...
The area formula is intuitive: start with a circle of radius (so its area is ) and stretch it by a factor / to make an ellipse. This scales the area by the same factor: π b 2 ( a / b ) = π a b . {\displaystyle \pi b^{2}(a/b)=\pi ab.} [ 18 ] However, using the same approach for the circumference would be fallacious – compare the integrals
The eccentricity of a circle is defined to be zero and its focus is the center of the circle, but its directrix can only be taken as the line at infinity in the projective plane. [ 2 ] The eccentricity of an ellipse can be seen as a measure of how far the ellipse deviates from being circular.
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.
Examples: The orthoptic of a parabola is its directrix (proof: see below),; The orthoptic of an ellipse + = is the director circle + = + (see below),; The orthoptic of a hyperbola =, > is the director circle + = (in case of a ≤ b there are no orthogonal tangents, see below),
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 ...
English: A diagram showing some basic properties of an ellipse, including The constant of proportionality is the eccentricity, e. D, called the directrix. The ratios e=f/a=a/d=PF/PD are always constant. *The eccentricity is always between 0 and 1. At zero, the ellispe becomes a circle, at 1 the ellipse becomes a parabola.