Hyperbolas are the two-branched mirror conic
A hyperbola is a smooth plane curve made of two mirror-image branches, like a pair of endless bows. It appears when a plane slices both halves of a double cone without touching the apex, and it traces paths from a sundial's shadow tip to the orbit of a comet escaping the Sun.
There are several equivalent definitions. It is the set of points whose distances to two fixed foci differ by a constant amount. It is also a curve where, at every point, the lines to the two foci reflect into each other across the tangent. And it is the solution set of certain two-variable quadratic equations, such as the reciprocal relation y = A/x. Besides sundial shadows, it describes the open orbit of any body moving faster than the escape velocity of the nearest gravitating mass and the scattering path of a subatomic particle.
Moving outward from the centre, each branch's two arms flatten. Diagonally opposite arms, one from each branch, approach a common straight line, the asymptote, so every hyperbola has two asymptotes crossing at its centre of symmetry. For y = A/x with A positive the curve is a rectangular hyperbola lying in the first and third quadrants, with the coordinate axes as asymptotes and vertices at (√A, √A) and (−√A, −√A); flipping the sign of A moves it into the second and fourth quadrants. As with the ellipse, the distance to a focus divided by the distance to the matching directrix equals the eccentricity, c/a, and many ellipse formulas carry over with a single sign change.
Menaechmus found hyperbolas while working on the problem of doubling the cube, calling them sections of obtuse cones. Apollonius of Perga is believed to have coined the name in his Conics, from the Greek for excessive, the same root as hyperbole. The terms ellipse, deficient, and parabola, applied, share an older Pythagorean origin in comparing rectangles of fixed area with a line segment.
The curve lends its name to saddle-shaped hyperbolic paraboloids, hyperboloids, the hyperbolic functions sinh and cosh, and Lobachevsky's hyperbolic geometry.
Source: Hyperbola
