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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 endpoints (,) of the minor axis lie at the height of the asymptotes over/under the hyperbola's vertices. Either half of the minor axis is called the semi-minor axis, of length b . Denoting the semi-major axis length (distance from the center to a vertex) as a , the semi-minor and semi-major axes' lengths appear in the equation of the ...
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 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.
A conic section with one focus on the pole and the other somewhere on the 0° ray (so that the conic's major axis lies along the polar axis) is given by: = where e is the eccentricity and is the semi-latus rectum (the perpendicular distance at a focus from the major axis to the curve).
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).
As approaches from above, the limit of the pencil of confocal hyperbolas degenerates to the relative complement of that line segment with respect to the x-axis; that is, to the two rays with endpoints at the foci pointed outward along the x-axis (an infinitely flat hyperbola). These two limit curves have the two foci in common.