EllipseEdit
An ellipse is a smooth, closed plane curve that arises in geometry as the set of all points for which the sum of the distances to two fixed points, called the foci, is constant. This simple distance rule yields a symmetric shape that is wider in one direction than a circle and has a distinctive, easily described structure. Ellipses appear naturally in a range of physical and practical contexts, from the orbits of celestial bodies to the shaping of optical and architectural components. For the broader family of curves formed by slicing a double cone with a plane, the ellipse sits alongside the parabola and hyperbola as one of the classical conic sections conic sections.
In its most familiar form, an ellipse can be centered at the origin and oriented with its major axis along the x-axis. Its standard Cartesian equation is x^2/a^2 + y^2/b^2 = 1, where a > b > 0. The two principal axes meet at the center and define a major axis of length 2a and a minor axis of length 2b. The distance from the center to each focus is c, with c^2 = a^2 - b^2, so the foci lie at (±c, 0) in the standard position. The eccentricity e = c/a satisfies 0 ≤ e < 1, capturing how stretched the ellipse is: e = 0 gives a circle, while e approaching 1 gives a highly elongated shape. The basic focal property, the major/minor axes, and the eccentricity together determine much of an ellipse’s geometry focus, major axis, minor axis, eccentricity.
Geometry and properties
- Focus–sum property: every point on the ellipse has the same sum of distances to the two foci. This defining feature is one of the most frequently cited geometric characterizations of the ellipse and underpins many practical constructions and proofs focus.
- Axes and center: the ellipse is symmetric with respect to its major and minor axes, and its center is the midpoint of the line segment joining the foci.
- Relationship to the circle: the circle is the special case e = 0, which occurs when a = b. In that case the two foci coincide at the center.
- Tangents and normals: tangents to the ellipse have a well-behaved reflective property, and the normal directions pass through the corresponding auxiliary points; these features have applications in optics and engineering.
- Rotations and general placement: in the plane, a general ellipse need not be aligned with the coordinate axes; rotating and translating the standard form yields the most general equation of an ellipse in the plane.
The ellipse also admits several equivalent descriptions that are useful in analysis and applications. One common parametrization is x = a cos t, y = b sin t for t in [0, 2π], which traces the curve once as t goes through a full cycle. In polar coordinates with one focus at the origin, the ellipse can be written as r(θ) = a(1 − e^2)/(1 − e cos θ), emphasizing how the radius changes with angle around that focus polar coordinates.
Constructions and representations
- String-and-pins method: a classical construction uses two pins placed at the foci and a taut string of length 2a; tracing the string with a pencil keeps the sum of distances to the pins constant as the pencil moves, producing a perfect ellipse.
- Algebraic forms: the standard form x^2/a^2 + y^2/b^2 = 1 is the simplest representation, while a rotated ellipse in general position has an equation containing an xy term and cross-terms that reflect its orientation.
- Geometric synthesis: ellipses can be generated as the locus of points where the projection of a circle onto a line is foreshortened in a fixed way, a perspective useful in computer graphics and design.
History and mathematical context
The ellipse has a deep place in the history of geometry. It is one of the conic sections studied by Apollonius of Perga in antiquity, who introduced the terminology and distinguished the ellipse from the parabola and hyperbola as sections of a cone formed by slicing with a plane. Later scholars extended these ideas into analytic geometry, culminating in the explicit Cartesian and polar forms used today. The modern prominence of ellipses in celestial mechanics originates with Johannes Kepler, whose laws describe planetary orbits as ellipses (with the sun at one focus) in the two-body problem. Isaac Newton’s law of gravitation then explained why gravity yields such conic-orbit shapes under suitable conditions, connecting geometry to physics Kepler's laws.
In more applied settings, the ellipse’s optical properties—specifically, that light emanating from one focus reflects to the other—made it a natural choice for focusing devices, including certain telescopes and acoustic reflectors. The ellipse’s mathematical tractability, paired with its practical relevance, has ensured its continued use in engineering, design, and numerical methods.
Applications
- Astronomy and orbital dynamics: many planetary and satellite orbits are well modeled by ellipses, especially in regimes where perturbations are small and the two-body approximation is strong; the orbital period and shape are tied to the ellipse’s parameters orbital mechanics.
- Optics and acoustics: elliptical mirrors and resonators exploit the focus property to concentrate or redirect waves with high efficiency.
- Engineering and design: ellipses appear in mechanical linkages, gear teeth profiles, and architectural forms where a smooth, closed, convex curve is desired.
- Computer graphics and vision: ellipses are fundamental primitives in rendering, collision detection, and shape analysis, with numerous algorithms for fitting ellipses to data and transforming them under scaling, rotation, or translation.