Parabola

Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity.

[2] Designs were proposed in the early to mid-17th century by many mathematicians, including René Descartes, Marin Mersenne,[3] and James Gregory.

The latus rectum is the line drawn through a focus of a conic section parallel to the directrix and terminated both ways by the curve.

(the left side of the equation uses the Hesse normal form of a line to calculate the distance

We will call its radius r. Another perpendicular to the axis, circular cross-section of the cone is farther from the apex A than the one just described.

For any given cone and parabola, r and θ are constants, but x and y are variables that depend on the arbitrary height at which the horizontal cross-section BECD is made.

It is proved in a preceding section that if a parabola has its vertex at the origin, and if it opens in the positive y direction, then its equation is y = ⁠x2/4f⁠, where f is its focal length.

This discussion started from the definition of a parabola as a conic section, but it has now led to a description as a graph of a quadratic function.

In order to prove the directrix property of a parabola (see § Definition as a locus of points above), one uses a Dandelin sphere

This conclusion about reflected light applies to all points on the parabola, as is shown on the left side of the diagram.

The following properties of a parabola deal only with terms connect, intersect, parallel, which are invariants of similarities.

This can be done with calculus, or by using a line that is parallel to the axis of symmetry of the parabola and passes through the midpoint of the chord.

If these quantities are signed, the length of the arc between any two points on the parabola is always shown by the difference between their values of s. The calculation can be simplified by using the properties of logarithms:

If a body traces the path of the parabola due to an inverse square force directed towards S, the area SVB increases at a constant rate as point B moves forward.

The construction can be extended simply to include the case where neither radius coincides with the axis SV as follows.

By Book 1, Proposition 16, Corollary 6 of Newton's Principia, the speed of a body moving along a parabola with a force directed towards the focus is inversely proportional to the square root of the radius.

The above construction was devised by Isaac Newton and can be found in Book 1 of Philosophiæ Naturalis Principia Mathematica as Proposition 30.

The focal length can be determined by a suitable parameter transformation (which does not change the geometric shape of the parabola).

Negative fractional powers correspond to the implicit equation xp yq = k and are traditionally referred to as higher hyperbolas.

The parabolic trajectory of projectiles was discovered experimentally in the early 17th century by Galileo, who performed experiments with balls rolling on inclined planes.

Another hypothetical situation in which parabolas might arise, according to the theories of physics described in the 17th and 18th centuries by Sir Isaac Newton, is in two-body orbits, for example, the path of a small planetoid or other object under the influence of the gravitation of the Sun.

Long-period comets travel close to the Sun's escape velocity while they are moving through the inner Solar system, so their paths are nearly parabolic.

Approximations of parabolas are also found in the shape of the main cables on a simple suspension bridge.

[20][21] Under the influence of a uniform load (such as a horizontal suspended deck), the otherwise catenary-shaped cable is deformed toward a parabola (see Catenary § Suspension bridge curve).

Unlike an inelastic chain, a freely hanging spring of zero unstressed length takes the shape of a parabola.

The best-known instance is the parabolic reflector, which is a mirror or similar reflective device that concentrates light or other forms of electromagnetic radiation to a common focal point, or conversely, collimates light from a point source at the focus into a parallel beam.

The principle of the parabolic reflector may have been discovered in the 3rd century BC by the geometer Archimedes, who, according to a dubious legend,[22] constructed parabolic mirrors to defend Syracuse against the Roman fleet, by concentrating the sun's rays to set fire to the decks of the Roman ships.

Today, paraboloid reflectors can be commonly observed throughout much of the world in microwave and satellite-dish receiving and transmitting antennas.

In this case, the centrifugal force causes the liquid to climb the walls of the container, forming a parabolic surface.

Aircraft used to create a weightless state for purposes of experimentation, such as NASA's "Vomit Comet", follow a vertically parabolic trajectory for brief periods in order to trace the course of an object in free fall, which produces the same effect as zero gravity for most purposes.