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The length of the chord through one focus, perpendicular to the major axis, is called the latus rectum. One half of it is the semi-latus rectum. A calculation shows: [4] = = (). The semi-latus rectum is equal to the radius of curvature at the vertices (see section curvature).
The latus rectum is defined similarly for the other two conics – the ellipse and the hyperbola. The latus rectum is the line drawn through a focus of a conic section parallel to the directrix and terminated both ways by the curve. For any case, is the radius of the osculating circle at the vertex. For a parabola, the semi-latus rectum, , is ...
The length of the chord through one of the foci, perpendicular to the major axis of the hyperbola, is called the latus rectum. One half of it is the semi-latus rectum. A calculation shows =. The semi-latus rectum may also be viewed as the radius of curvature at the vertices.
where e is the eccentricity and l is the semi-latus rectum. 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]
The length of the semi-major axis a of an ellipse is related to the semi-minor axis's length b through the eccentricity e and the semi-latus rectum, as follows: b = a 1 − e 2 , ℓ = a ( 1 − e 2 ) , a ℓ = b 2 . {\displaystyle {\begin{aligned}b&=a{\sqrt {1-e^{2}}},\\\ell &=a(1-e^{2}),\\a\ell &=b^{2}.\end{aligned}}}
The universal parabolic constant is the red length divided by the green length. The universal parabolic constant is a mathematical constant. It is defined as the ratio, for any parabola, of the arc length of the parabolic segment formed by the latus rectum to the focal parameter. The focal parameter is twice the focal length. The ratio is ...
Also shown are: semi-major axis a, semi-minor axis b and semi-latus rectum p; center of ellipse and its two foci marked by large dots. For θ = 0°, r = r min and for θ = 180°, r = r max. Mathematically, an ellipse can be represented by the formula:
When calculating the length of a short north-south line at the equator, the circle that best approximates that line has a radius of (which equals the meridian's semi-latus rectum), or 6335.439 km, while the spheroid at the poles is best approximated by a sphere of radius , or 6399.594 km, a 1% difference. So long as a spherical Earth is assumed ...