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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.
[2] Given discrete random variables X {\displaystyle X} with image X {\displaystyle {\mathcal {X}}} and Y {\displaystyle Y} with image Y {\displaystyle {\mathcal {Y}}} , the conditional entropy of Y {\displaystyle Y} given X {\displaystyle X} is defined as the weighted sum of H ( Y | X = x ) {\displaystyle \mathrm {H} (Y|X=x)} for each possible ...
In the Cartesian plane, these pairs lie on a hyperbola, and when the double sum is fully expanded, there is a bijection between the terms of the sum and the lattice points in the first quadrant on the hyperbolas of the form xy = k, where k runs over the integers 1 ≤ k ≤ n: for each such point (x,y), the sum contains a term g(x)h(y), and ...
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).
This pair of hyperbolas share the asymptotes y = x and y = −x. When the conjugate of the unit hyperbola is in use, the alternative radial length is =. The unit hyperbola is a special case of the rectangular hyperbola, with a particular orientation, location, and scale.
Indeed, if x and y are positive integers satisfying this equation, then x/y is an approximation of √ 2. The numbers x and y appearing in these approximations, called side and diameter numbers, were known to the Pythagoreans, and Proclus observed that in the opposite direction these numbers obeyed one of these two equations. [5]
xy = 1 with y = 0 as asymptote. When reflected in the x-axis, a line y = mx becomes y = −mx. In this case the lines are hyperbolic orthogonal if their slopes are additive inverses. x 2 − y 2 = 1 with y = x as asymptote. For lines y = mx with −1 < m < 1, when x = 1/m, then y = 1. The point (1/m, 1) on the line is reflected across y = x to ...
The curve represents xy = 1. A hyperbolic angle has magnitude equal to the area of the corresponding hyperbolic sector, which is in standard position if a = 1. In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of xy = 1 in Quadrant I of the Cartesian plane.