ellipticity

Ellipticity is a dimensionless quantity that measures how much a two‑dimensional shape or field deviates from circular symmetry. In geometry, the most common reference is an ellipse with semi-major axis a and semi-minor axis b. The traditional eccentricity e_e = sqrt(1 − (b^2 / a^2)) describes how elongated the ellipse is, ranging from 0 for a circle (a = b) to 1 for a highly elongated shape (b → 0). A related quantity is the flattening f = 1 − b/a, often used in astronomy and planetary science.

In image analysis and astronomy, ellipticity can also be defined from second moments of a brightness distribution. The ellipticity vector has components (e1, e2) that encode elongation and orientation: e1 = (Ixx − Iyy)/(Ixx + Iyy) and e2 = 2 Ixy/(Ixx + Iyy). The magnitude e = sqrt(e1^2 + e2^2) gives a scale of ellipticity independent of direction.

Applications vary by field. In astronomy, ellipticity describes the apparent shapes of galaxies and is a key observable in weak gravitational lensing, where distortions reveal information about intervening matter. In optics, ellipticity characterizes the polarization state of light: linear polarization corresponds to zero ellipticity, circular polarization to maximum ellipticity, and elliptical polarization to intermediate values, with the polarization ellipse describing the axes and orientation. In mathematics and physics, ellipticity also denotes a class of partial differential equations that are well-posed and non-degenerate.

Examples: a circle has ellipticity zero; a very elongated ellipse has ellipticity near one. See also eccentricity, flattening, and polarization ellipse.