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In geometry and linear algebra, a principal axis is a certain line in a Euclidean space associated with a ellipsoid or hyperboloid, generalizing the major and minor axes of an ellipse or hyperbola. The principal axis theorem states that the principal axes are perpendicular, and gives a constructive procedure for finding them. Mathematically ...
which, as follows from basic trigonometric identities, are equivalent expressions (i.e. the formula for S oblate can be used to calculate the surface area of a prolate ellipsoid and vice versa). In both cases e may again be identified as the eccentricity of the ellipse formed by the cross section through the symmetry axis. (See ellipse).
An ellipse (red) obtained as the intersection of a cone with an inclined plane. Ellipse: notations Ellipses: examples with increasing eccentricity. In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant.
By the principal axis theorem, the two eigenvectors of the matrix of the quadratic form of a central conic section (ellipse or hyperbola) are perpendicular (orthogonal to each other) and each is parallel to (in the same direction as) either the major or minor axis of the conic.
For example, to study the equations of ellipses and hyperbolas, the foci are usually located on one of the axes and are situated symmetrically with respect to the origin. If the curve (hyperbola, parabola , ellipse, etc.) is not situated conveniently with respect to the axes, the coordinate system should be changed to place the curve at a ...
The axes of symmetry or principal axes are the transverse axis (containing the segment of length 2a with endpoints at the vertices) and the conjugate axis (containing the segment of length 2b perpendicular to the transverse axis and with midpoint at the hyperbola's center). [6]
If the ellipse is rotated about its major axis, the result is a prolate spheroid, elongated like a rugby ball. The American football is similar but has a pointier end than a spheroid could. If the ellipse is rotated about its minor axis, the result is an oblate spheroid, flattened like a lentil or a plain M&M.
In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis ( major semiaxis ) is the longest semidiameter or one half of the major axis, and thus runs from the centre, through a focus , and ...