Parabola

The focus is ${\displaystyle \left(0,{\frac {1}{4a}}\right)}$,

the focal length ${\displaystyle {\frac {1}{4a}}}$, the semi-latus rectum is ${\displaystyle p={\frac {1}{2a}}}$,

the vertex is ${\displaystyle (0,0)}$,

the directrix has the equation ${\displaystyle y=-{\frac {1}{4a}}}$,

the at point ${\displaystyle (x_{0},ax_{0}^{2})}$ has the equation ${\displaystyle y=2ax_{0}x-ax_{0}^{2}}$.

tangent

For ${\displaystyle e=0}$ the conic is a circle (osculating circle of the pencil),

for ${\displaystyle 0<e<1}$ an ellipse,

for ${\displaystyle e=1}$ the parabola with equation ${\displaystyle y^{2}=2px,}$

for ${\displaystyle e>1}$ a hyperbola (see picture).

Consider the pencil at the vertex ${\displaystyle S(0,0)}$ and the set of lines ${\displaystyle \Pi _{y}}$ that are parallel to the y axis.

1. Let ${\displaystyle P=(x_{0},y_{0})}$ be a point on the parabola, and ${\displaystyle A=(0,y_{0})}$, ${\displaystyle B=(x_{0},0)}$.
2. The line segment ${\displaystyle {\overline {BP}}}$ is divided into n equally spaced segments, and this division is projected (in the direction ${\displaystyle BA}$) onto the line segment ${\displaystyle {\overline {AP}}}$ (see figure). This projection gives rise to a projective mapping ${\displaystyle \pi }$ from pencil ${\displaystyle S}$ onto the pencil ${\displaystyle \Pi _{y}}$.
3. The intersection of the line ${\displaystyle SB_{i}}$ and the i-th parallel to the y axis is a point on the parabola.

For a point (pole) on the parabola, the polar is the tangent at this point (see picture: ${\displaystyle P_{1},\ p_{1}}$).

For a pole ${\displaystyle P}$ outside the parabola the intersection points of its polar with the parabola are the touching points of the two tangents passing ${\displaystyle P}$ (see picture: ${\displaystyle P_{2},\ p_{2}}$).

For a point within the parabola the polar has no point with the parabola in common (see picture: ${\displaystyle P_{3},\ p_{3}}$ and ${\displaystyle P_{4},\ p_{4}}$).

The intersection point of two polar lines (for example, ${\displaystyle p_{3},p_{4}}$) is the pole of the connecting line of their poles (in example: ${\displaystyle P_{3},P_{4}}$).

Focus and directrix of the parabola are a pole–polar pair.

Image is inverted. AB is x axis. C is origin. O is center. A is (x, y). OA = OC = R. PA = x. CP = y. OP = (R − y). Other points and lines are irrelevant for this purpose.

The radius of curvature at the vertex is twice the focal length. The measurements shown on the above diagram are in units of the latus rectum, which is four times the focal length.

${\displaystyle {\vec {f}}_{0}}$ is a point of the parabola,

${\displaystyle {\vec {f}}_{1}}$ is a tangent vector at point ${\displaystyle {\vec {f}}_{0}}$,

${\displaystyle {\vec {f}}_{2}}$ is parallel to the axis of the parabola (axis of symmetry through the vertex).

elliptical ,

cone

parabolic ,

cylinder

elliptical ,

paraboloid

hyperbolic paraboloid,

of one sheet,

hyperboloid

hyperboloid of two sheets.

A bouncing ball captured with a stroboscopic flash at 25 images per second. The ball becomes significantly non-spherical after each bounce, especially after the first. That, along with spin and air resistance, causes the curve swept out to deviate slightly from the expected perfect parabola.

Parabolic trajectories of water in a fountain.

The path (in red) of Comet Kohoutek as it passed through the inner Solar system, showing its nearly parabolic shape. The blue orbit is the Earth's.

The supporting cables of suspension bridges follow a curve that is intermediate between a parabola and a catenary.

The Rainbow Bridge across the Niagara River, connecting Canada (left) to the United States (right). The parabolic arch is in compression and carries the weight of the road.

Parabolic arches used in architecture

Parabolic shape formed by a liquid surface under rotation. Two liquids of different densities completely fill a narrow space between two sheets of transparent plastic. The gap between the sheets is closed at the bottom, sides and top. The whole assembly is rotating around a vertical axis passing through the centre. (See Rotating furnace)

Solar cooker with parabolic reflector

Parabolic antenna

Parabolic microphone with optically transparent plastic reflector used at an American college football game.

Array of parabolic troughs to collect solar energy

Edison's searchlight, mounted on a cart. The light had a parabolic reflector.

Physicist Stephen Hawking in an aircraft flying a parabolic trajectory to simulate zero gravity

Degenerate conic

Dome § Paraboloid dome

Parabolic partial differential equation

Quadratic equation

Quadratic function

Universal parabolic constant

Confocal conic sections § Confocal parabolas

Lockwood, E. H. (1961). A Book of Curves. Cambridge University Press.

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Parabola"

"Parabola". MathWorld.

Weisstein, Eric W.

Interactive parabola-drag focus, see axis of symmetry, directrix, standard and vertex forms

at cut-the-knot

Archimedes Triangle and Squaring of Parabola

at cut-the-knot

Two Tangents to Parabola

at cut-the-knot

Parabola As Envelope of Straight Lines

at cut-the-knot

Parabolic Mirror

at cut-the-knot

Three Parabola Tangents

at cut-the-knot

Focal Properties of Parabola

at cut-the-knot

Parabola As Envelope II

at Dynamic Geometry Sketches, interactive dynamic geometry sketch.

The similarity of parabola

Frans van Schooten: Mathematische Oeffeningen, 1659