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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 ...
As above, for e = 0, the graph is a circle, for 0 < e < 1 the graph is an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola. The polar form of the equation of a conic is often used in dynamics ; for instance, determining the orbits of objects revolving about the Sun. [ 20 ]
This is the equation of an ellipse (<) or a parabola (=) or a hyperbola (>). All of these non-degenerate conics have, in common, the origin as a vertex (see diagram). All of these non-degenerate conics have, in common, the origin as a vertex (see diagram).
The ellipse, parabola, and hyperbola are viewed as conics in projective geometry, and each conic determines a relation of pole and polar between points and lines. Using these concepts, "two diameters are conjugate when each is the polar of the figurative point of the other."
The semi-minor axis (minor semiaxis) of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section. For the special case of a circle, the lengths of the semi-axes are both equal to the radius of the circle.
Then for the ellipse case of AC > (B/2) 2, the ellipse is real if the sign of K equals the sign of (A + C) (that is, the sign of each of A and C), imaginary if they have opposite signs, and a degenerate point ellipse if K = 0. In the hyperbola case of AC < (B/2) 2, the hyperbola is degenerate if and only if K = 0.
The diameter of the unit hyperbola represents a frame of reference in motion with rapidity a where tanh a = y/x and (x,y) is the endpoint of the diameter on the unit hyperbola. The conjugate diameter represents the spatial hyperplane of simultaneity corresponding to rapidity a. In this context the unit hyperbola is a calibration hyperbola [3 ...
For elliptical orbits, a simple proof shows that gives the projection angle of a perfect circle to an ellipse of eccentricity e. For example, to view the eccentricity of the planet Mercury (e = 0.2056), one must simply calculate the inverse sine to find the projection angle of 11.86 degrees. Then, tilting any circular object by that angle ...