Distance from conic-section point to focus divided by distance from conic-section point to directrix is constant {eccentricity, conic}: e = (a^2 + b^2)^0.5 / a, where a is major axis and b is minor axis. For circle, e = 2^0.5. For parabola, e = 1, because b = 0. For hyperbola and ellipse, e > 1.
For ellipses, angle A {eccentric angle} in equations x = a * cos(A) and y = b * sin(A), where a is ellipse major axis, and b is ellipse minor axis, determines eccentricity. Hyperbola has eccentric angle A with x = a * sec(A) and y = b * tan(A).
In ellipses, two circles {eccentric circle} can have major and minor axes as diameters. In hyperbola, two eccentric circles have transverse axis and conjugate axis {harmonic conjugate of transverse axis} as diameters.
l / r = 1 - e * cos(A) {polar equation}, where l is latus-rectum length divided by two, r is distance from pole or focus, e is eccentricity, and A is polar angle. Transverse or major-axis positive direction is reference line {initial line} {polar axis, conic}.
3-Geometry-Solid-Cone-Conic Section
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Date Modified: 2022.0225