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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 ...
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 ...
The linear eccentricity of an ellipse or hyperbola, denoted c (or sometimes f or e), is the distance between its center and either of its two foci. The eccentricity can be defined as the ratio of the linear eccentricity to the semimajor axis a : that is, e = c a {\displaystyle e={\frac {c}{a}}} (lacking a center, the linear eccentricity for ...
In addition to the eccentricity (e), foci, and directrix, various geometric features and lengths are associated with a conic section. The principal axis is the line joining the foci of an ellipse or hyperbola, and its midpoint is the curve's center. A parabola has no center. The linear eccentricity (c) is the distance between the center and a ...
A ray through the unit hyperbola x 2 − y 2 = 1 at the point (cosh a, sinh a), where a is twice the area between the ray, the hyperbola, and the x-axis. For points on the hyperbola below the x-axis, the area is considered negative (see animated version with comparison with the trigonometric (circular) functions).
The semi-major axis (major semiaxis) is the longest semidiameter or one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. The semi-minor axis (minor semiaxis) of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section.
The hyperbolic angle parametrizes the unit hyperbola, which has hyperbolic functions as coordinates. In mathematics, hyperbolic angle is an invariant measure as it is preserved under hyperbolic rotation. The hyperbola xy = 1 is rectangular with semi-major axis , analogous to the circular angle equaling the area of a circular sector in a circle ...
The inversion of hyperbola yields a lemniscate. The lemniscate is symmetric to the line connecting its foci F 1 and F 2 and as well to the perpendicular bisector of the line segment F 1 F 2. The lemniscate is symmetric to the midpoint of the line segment F 1 F 2. The area enclosed by the lemniscate is a 2 = 2c 2.