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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.
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.
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.
As described in the tennis racket theorem, rotation of an object around its first or third principal axis is stable, while rotation around its second principal axis (or intermediate axis) is not. The motion is simplified in the case of an axisymmetric body, in which the moment of inertia is the same about two of the principal axes.
The semi-minor axis of an ellipse runs from the center of the ellipse (a point halfway between and on the line running between the foci) to the edge of the ellipse. The semi-minor axis is half of the minor axis. The minor axis is the longest line segment perpendicular to the major axis that connects two points on the ellipse's edge. The semi ...
In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci F 1 {\displaystyle F_{1}} and F 2 {\displaystyle F_{2}} are generally taken to be fixed at − a {\displaystyle -a} and + a {\displaystyle +a} , respectively, on the x ...
By the principal axis theorem, the plane admits a Cartesian coordinate system with its origin at the midpoint between foci and its axes aligned with the axes of the confocal ellipses and hyperbolas. If c {\displaystyle c} is the linear eccentricity (half the distance between F 1 {\displaystyle F_{1}} and F 2 {\displaystyle F_{2}} ), then in ...
Principal axis may refer to: Principal axis (crystallography) Principal axis (mechanics) Principal axis theorem; See also. Aircraft principal axes;