elliptisch

Elliptisch is a German adjective that denotes something relating to or shaped like an ellipse. The term comes from Greek elleipsis, meaning "a leaving out," via Latin ellipticus, and entered mathematical and general usage to describe forms, properties, or methods associated with ellipses.

In geometry and astronomy, elliptisch describes objects and motions tied to ellipses. An ellipse is the set of points for which the sum of distances to two foci is constant. Many planetary orbits are approximately elliptical with the Sun at a focus. The term also appears in discussions of eccentricity, axes, and other ellipse-related features.

In mathematics, elliptisch relates to several concepts named after the ellipse. Elliptic integrals arise from arc lengths of ellipses and generally cannot be expressed by elementary functions. Elliptic functions are doubly periodic complex functions, with applications in various areas of analysis and number theory. Elliptic curves are smooth cubic curves used in algebraic geometry and number theory; they play a central role in modern cryptography, notably in elliptic-curve cryptography. The broad use of elliptisch reflects these connections to the properties and structures that emerge from or generalize the idea of an ellipse.

In linguistics and rhetoric, elliptisch describes sentences or phrases that omit one or more words because they are understood from context. Elliptical phrasing can create conciseness or stylistic effects.

In analysis and partial differential equations, elliptisch refers to elliptic operators, whose principal symbol is positive definite. Such operators are associated with well-posed boundary-value problems and smoothness results for solutions.