conjugate axis

Semi-minor axis

[sem-ee-mahy-ner, sem-ahy-, sem-ee-, sem-ahy-]

In geometry, the semi-minor axis (also semiminor axis) is a line segment associated with most conic sections (that is, with ellipses and hyperbolas). One end of the segment is the center of the conic section, and it is at right angles with the semi-major axis. It is one of the axes of symmetry for the curve: in an ellipse, the shorter one; in a hyperbola, the one that does not intersect the hyperbola.

Ellipse

The semi-minor axis of an ellipse is one half of the minor axis, running from the center, halfway between and perpendicular to the line running between the foci, and to the edge of the ellipse. The minor axis is the longest line that runs perpendicular to the major axis.

It is related to the semi-major axis a through the eccentricity e and the semi-latus rectum l, as follows:

b = a sqrt{1-e^2},!

al=b^2,!.

A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping l fixed. Thus a and b tend to infinity, a faster than b.

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Hyperbola

The length of the semi-minor axis of a hyperbola is the distance from a top, along the tangent line, to each asymptote; if this is in the y-direction it is b in this equation of the hyperbola:

frac{left(x-h right)^2}{a^2} - frac{left(y-k right)^2}{b^2} = 1.

It is related to the semi-major axis through the eccentricity, as follows:

b = a sqrt{e^2-1}.

Note that in a hyperbola b can be larger than a.

The conjugate axis of a hyperbola runs in the same direction as the Semi-major axis.

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