Minor AxisEdit
Minor axis is a geometric term most often encountered in the study of ellipses and other conic sections. It denotes the shorter principal diameter of an ellipse, or more generally the axis that lies perpendicular to the major axis and passes through the shape’s center. In a coordinate system aligned with the ellipse’s axes, the minor axis runs through the center and connects the two points where the ellipse meets its narrowest direction. This axis is a key element in describing an ellipse’s shape and orientation, and it plays a central role in applications ranging from celestial mechanics to optical design.
In the simplest, standard form, an ellipse with semi-major axis a and semi-minor axis b is described by the equation x^2/a^2 + y^2/b^2 = 1 in a frame where the major axis aligns with the x-direction. The major axis length is 2a and the minor axis length is 2b. When a = b, the ellipse is a circle and the minor axis can be any diameter; when a > b, the ellipse is elongated in the major axis direction, and the minor axis is the direction of the smallest radius. The endpoints of the minor axis are located at (0, ±b) in the axis-aligned coordinate system. For a rotated ellipse, the same principle applies: the minor axis remains perpendicular to the major axis and passes through the center.
Definition and geometric interpretation
- The minor axis is the line segment through the center of the ellipse in the direction of the smallest radius. It is perpendicular to the major axis, which is the line segment through the center in the direction of the largest radius.
- The endpoints of the minor axis lie on the ellipse at equal distances from the center, each at a distance b, the semi-minor axis, from the center.
- The lengths of the axes are related to the ellipse’s eccentricity e by b = a sqrt(1 − e^2); equivalently, e = sqrt(1 − (b^2/a^2)). These relationships connect the visual impression of “how stretched” the ellipse is with the algebra of its defining parameters.
Equations and coordinate representations
- Aligned ellipse: in a coordinate frame where the major axis is the x-axis, the standard form is x^2/a^2 + y^2/b^2 = 1 with a ≥ b; the major axis length is 2a and the minor axis length is 2b. The endpoints of the minor axis are at (0, ±b).
- Rotated ellipse: if the ellipse is rotated by an angle, the equation acquires cross terms, but the geometric notions of major and minor axes persist. The directions of these axes are eigenvectors of the quadratic form describing the ellipse, and their lengths are still 2a and 2b.
- In many contexts, the semi-major axis a and semi-minor axis b are the fundamental descriptors of the ellipse’s size in orthogonal directions, while the eccentricity e gauges how far the ellipse deviates from a circle.
Occurrences in natural phenomena and design
- Elliptical orbits in celestial mechanics: planets and satellites often trace elliptical paths around a focus, with the major axis drawn through the two foci. The majority of the orbit’s size information is encoded in the major axis, but the minor axis still constrains the cross-sectional shape of the orbit’s projection. The relationship b^2 = a^2 (1 − e^2) ties the minor axis to the orbit’s eccentricity and the size parameter a. See orbital mechanics and eccentricity for related discussions.
- Optics and engineering: elliptical shapes appear in lens and mirror design, where the major and minor axes define the cross-sectional geometry that influences focusing properties and stability. Elliptical mirrors and caustics leverage the geometry of axes to achieve desired imaging or beam-shaping outcomes. See optics for broader context.
- Data representation and modeling: ellipses arise in statistics and multivariate analysis as representations of confidence regions and covariance structures. In such cases, the major and minor axes reflect principal directions of variation, often tied to the eigenvalues and eigenvectors of the covariance matrix. See statistics and covariance matrix for related topics.
Historical notes and terminology
The terms major axis and minor axis originate in classical geometry to distinguish the longest and shortest principal diameters of an ellipse. The nomenclature is stable across many languages and mathematical traditions, even as practical usage has broadened to fields such as physics and engineering. In informal discussion, the ellipse’s axes are sometimes described in terms of symmetry directions or principal radii, but the underlying concepts remain tied to the same pair of perpendicular directions through the center.