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Hyperbola. A hyperbola is an open curve with two branches, the intersection of a plane with both halves of a double cone. The plane does not have to be parallel to the axis of the cone; the hyperbola will be symmetrical in any case. Hyperbola (red): features. In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its ...
In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape. One can think of the eccentricity as a measure of how much a conic section deviates from being circular. In particular: The eccentricity of a circle is 0. The eccentricity of an ellipse which is not a circle is between 0 and 1.
In geometry, two conic sections are called confocal if they have the same foci. Because ellipses and hyperbolas have two foci, there are confocal ellipses, confocal hyperbolas and confocal mixtures of ellipses and hyperbolas. In the mixture of confocal ellipses and hyperbolas, any ellipse intersects any hyperbola orthogonally (at right angles).
A radial hyperbolic trajectory is a non-periodic trajectory on a straight line where the relative speed of the two objects always exceeds the escape velocity. There are two cases: the bodies move away from each other or towards each other. This is a hyperbolic orbit with semi-minor axis = 0 and eccentricity = 1.
Focus (geometry) Geometric point from which certain types of curves are constructed. Point F is a focus point for the red ellipse, green parabola and blue hyperbola. In geometry, focuses or foci (/ ˈfoʊkaɪ /; sg.: focus) are special points with reference to which any of a variety of curves is constructed. For example, one or two foci can be ...
Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including ...
In celestial mechanics, Lambert's problem is concerned with the determination of an orbit from two position vectors and the time of flight, posed in the 18th century by Johann Heinrich Lambert and formally solved with mathematical proof by Joseph-Louis Lagrange. It has important applications in the areas of rendezvous, targeting, guidance, and ...
F: focus of the red parabola and vertex of the blue parabola. In geometry, focal conics are a pair of curves consisting of [1] [2] either. an ellipse and a hyperbola, where the hyperbola is contained in a plane, which is orthogonal to the plane containing the ellipse. The vertices of the hyperbola are the foci of the ellipse and its foci are ...