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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.
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.
The principal axis is the line joining the foci of an ellipse or hyperbola, and its midpoint is the curve's center. A parabola has no center. The linear eccentricity (c) is the distance between the center and a focus. The latus rectum is the chord parallel to the directrix and passing through a focus; its half-length is the semi-latus rectum (ℓ).
Considering the pencils of confocal ellipses and hyperbolas (see lead diagram) one gets from the geometrical properties of the normal and tangent at a point (the normal of an ellipse and the tangent of a hyperbola bisect the angle between the lines to the foci). Any ellipse of the pencil intersects any hyperbola orthogonally (see diagram).
The eccentricity is directly related to the angle between the asymptotes. With eccentricity just over 1 the hyperbola is a sharp "v" shape. At = the asymptotes are at right angles. With > the asymptotes are more than 120° apart, and the periapsis distance is greater than the semi major axis. As eccentricity increases further the motion ...
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).
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 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 ...