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In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the ...
A chain hanging from points forms a catenary. The silk on a spider's web forming multiple elastic catenaries.. In physics and geometry, a catenary (US: / ˈ k æ t ən ɛr i / KAT-ən-err-ee, UK: / k ə ˈ t iː n ər i / kə-TEE-nər-ee) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field.
M-shape, the shape that resembles the capital letter M (interchangeable with the W-shape) N-shape, the shape that resembles the capital letter N (interchangeable with the Z-shape) O-shape, the shape that resembles the capital letter O. O-ring; P-shape, the shape that resembles the capital letter P. P-trap, a P-shaped pipe under a sink or basin
Hemihelix, a quasi-helical shape characterized by multiple tendril perversions; Tendril perversion (a transition between back-to-back helices) Seiffert's spiral; Slinky spiral; Space cardioid; Twisted cubic; Viviani's curve
The point (,) is the vertex of the parabola. Pencil of confocal parabolas From the definition of a parabola , for any point P {\displaystyle P} not on the x -axis, there is a unique parabola with focus at the origin opening to the right and a unique parabola with focus at the origin opening to the left, intersecting orthogonally at the point P ...
The values of u, v, w such that g(u, v, w) = 0 are the homogeneous coordinates of the points of the completion of the curve in the projective plane and the points of the initial curve are those such that w is not zero. An example is the Fermat curve u n + v n = w n, which has an affine form x n + y n = 1. A similar process of homogenization may ...
In this position, the hyperbolic paraboloid opens downward along the x-axis and upward along the y-axis (that is, the parabola in the plane x = 0 opens upward and the parabola in the plane y = 0 opens downward). Any paraboloid (elliptic or hyperbolic) is a translation surface, as it can be generated by a moving parabola directed by a second ...
In this simple approximation, the trajectory takes the shape of a parabola. Generally when determining trajectories, it may be necessary to account for nonuniform gravitational forces and air resistance (drag and aerodynamics). This is the focus of the discipline of ballistics.