Crossword clues for hyperbola
The Collaborative International Dictionary
Hyperbola \Hy*per"bo*la\, n. [Gr. ?, prop., an overshooting, excess, i. e., of the angle which the cutting plane makes with the base. See Hyperbole.] (Geom.) A curve formed by a section of a cone, when the cutting plane makes a greater angle with the base than the side of the cone makes. It is a plane curve such that the difference of the distances from any point of it to two fixed points, called foci, is equal to a given distance. See Focus. If the cutting plane be produced so as to cut the opposite cone, another curve will be formed, which is also an hyperbola. Both curves are regarded as branches of the same hyperbola. See Illust. of Conic section, and Focus.
Douglas Harper's Etymology Dictionary
1660s, from Latinized form of Greek hyperbole "extravagance," literally "a throwing beyond" (see hyperbole). Perhaps so called because the inclination of the plane to the base of the cone exceeds that of the side of the cone.
Wiktionary
n. (context geometry English) A conic section formed by the intersection of a cone with a plane that intersects the base of the cone and is not tangent to the cone.
WordNet
n. an open curve formed by a plane that cuts the base of a right circular cone
Wikipedia
In mathematics, a hyperbola (plural hyperbolas or hyperbolae) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse. A circle is a special case of an ellipse). If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.
Hyperbolas arise in many ways: as the curve representing the function f(x) = 1/x in the Cartesian plane, as the path followed by the shadow of the tip of a sundial, as the shape of an open orbit (as distinct from a closed elliptical orbit), such as the orbit of a spacecraft during a gravity assisted swing-by of a planet or more generally any spacecraft exceeding the escape velocity of the nearest planet, as the path of a single-apparition comet (one travelling too fast ever to return to the solar system), as the scattering trajectory of a subatomic particle (acted on by repulsive instead of attractive forces but the principle is the same), and so on.
Each branch of the hyperbola has two arms which become straighter (lower curvature) further out from the center of the hyperbola. Diagonally opposite arms, one from each branch, tend in the limit to a common line, called the asymptote of those two arms. So there are two asymptotes, whose intersection is at the center of symmetry of the hyperbola, which can be thought of as the mirror point about which each branch reflects to form the other branch. In the case of the curve f(x) = 1/x the asymptotes are the two coordinate axes.
Hyperbolas share many of the ellipses' analytical properties such as eccentricity, focus, and directrix. Typically the correspondence can be made with nothing more than a change of sign in some term. Many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids (saddle surfaces), hyperboloids ("wastebaskets"), hyperbolic geometry ( Lobachevsky's celebrated non-Euclidean geometry), hyperbolic functions (sinh, cosh, tanh, etc.), and gyrovector spaces (a geometry proposed for use in both relativity and quantum mechanics which is not Euclidean).
Usage examples of "hyperbola".
The abandoned Bussard ramjet crossed over the sun and curved inward, following a shallow hyperbola which would take it through the plane of the planets.
The ships followed a flat hyperbola across the planetless skies and transited to another new sky, this time a properly unfamiliar one.
They were startled by a sudden engine burn of two seconds, the ship responding to gravitometer feedback to the program governing their trajectory, a hyperbola in and out of the shock bubble.
It will not be captured, but the hyperbola of its orbit is narrow and it will come within an astronomical unit.
Apollonius of Perga, the mathematician who demonstrated the forms of the conic sections* - ellipse, parabola and hyperbola - the curves, as we now know, followed in their orbits by the planets, the comets and the stars.
On one page of the notebook I drew to the best of my ability the three conic sections with their axes and centers: an ellipse, a parabola, and an hyperbola.
The rectangle changed every five minutes, and a new point appeared on the hyperbola, showing the new position of the Snarkhunter instrument package.
Armies, nations, prime ministers, presidents and geniuses fall around it, not in steady planetlike orbits but in the crazy careening ellipses and hyperbolae of comets and stray asteroids.
It is not so much a dying city as an ungovernable one, seven million souls moving in seven million orbits under spectacular centrifugal pressures that threaten at any moment to make hyperbolas of us all.
The Orion drives had kicked the asteroids into fairly flat hyperbolas involving far less transit time than the years simple Hohmann transfer orbits would have taken, and those same drives continued to accelerate them steadily.
The teetering spires and hyperbolas loom so tall their lower stories disappear below in a haze of ramparts and sparkling viaducts and spans that meld with distance to a golden ether.
The planet rotates, it's inclined on its axis and precesses, it's also describing a large ellipse around the Sun, and the Sun is moving on the inward leg of a hyperbola with the galactic core at one focus, and the Galaxy—"