hyperbola Geom.
(haɪˈpɜːbələ)
[a. mod.L. hyperbola, ad. Gr. ὑπερβολή the name of the curve, lit. excess (cf. hyperbole), f. ὑπερβάλλειν to exceed (ὑπέρ over + βάλλειν to throw). In F. hyperbole.
The hyperbola was so named either because the inclination of its plane to the base of the cone exceeds that of the side of the cone (see ellipse), or because the side of the rectangle on the abscissa equal to the square of the ordinate is longer than the latus rectum.]
One of the conic sections; a plane curve consisting of two separate, equal and similar, infinite branches, formed by the intersection of a plane with both branches of a double cone (i.e. two similar cones on opposite sides of the same vertex). It may also be defined as a curve in which the focal distance of any point bears to its distance from the directrix a constant ratio greater than unity. It has two foci, one for each branch, and two asymptotes, which intersect in the centre of the curve, midway between the vertices of its two branches. (Often applied to one branch of the curve.)
| 1668 Phil. Trans. III. 643 The Area of one Hyperbola being computed, the Area of all others may be thence argued. 1692 Bentley Boyle Lect. viii. 267 They would not have moved in Hyperbola's, or in Ellipses very eccentric. 1706 W. Jones Syn. Palmar. Matheseos 256 The Sections of the opposite Cones will be equal Hyperbolas. 1728 Pemberton Newton's Philos. 232 With a velocity still greater the body will move in an hyperbola. 1828 Hutton Course Math. II. 102 The section is an hyperbola, when the cutting plane makes a greater angle with the base than the side of the cone makes. 1885 G. L. Goodale Phys. Bot. (1892) 381 note, If the outline of the growing plant is a hyperbola, the periclinals will be confocal hyperbolas, with the same axis but different parameter. |
b. Extended (after Newton) to algebraic curves of higher degrees denoted by equations analogous to that of the common hyperbola.
| 1727–41 Chambers Cycl. s.v., Infinite Hyperbola's, or Hyperbola's of the higher kinds, are those defined by the equation ay{supm} + n = bz{supm}(a + x)n. Ibid., As the hyperbola of the first kind or order has two asymptotes, that of the second kind or order has three, that of the third, four, etc. 1753 ― Cycl. Supp. s.v., Hyperbolas of all degrees may be expressed by the equation x{supm} yn = a{supm} + n. 1852 [see hyperbolic 2]. |