Equations of parabola

Cartesian
In the following equations $h$ and $k$ are the coordinates of the vertex, $(h,k)$ , of the parabola and $p$ is the distance from the vertex to the focus and the vertex to the directrix.

Vertical axis of symmetry

$(x - h)^2 = 4p(y - k) \,$

$y =\frac{(x-h)^2}{4p}+k\,$

$y = ax^2 + bx + c \,$

where

$a = \frac{1}{4p}; \ \ b = \frac{-h}{2p}; \ \ c = \frac{h^2}{4p} + k; \ \$

$h = \frac{-b}{2a}; \ \ k = \frac{4ac - b^2}{4a}$ .

Parametric form:

$x(t) = 2pt + h; \ \ y(t) = pt^2 + k \,$

Horizontal axis of symmetry

$(y - k)^2 = 4p(x - h) \,$

$x =\frac{(y - k)^2}{4p} + h;\ \,$

$x = ay^2 + by + c \,$

where

$a = \frac{1}{4p}; \ \ b = \frac{-k}{2p}; \ \ c = \frac{k^2}{4p} + h; \ \$

$h = \frac{4ac - b^2}{4a}; \ \ k = \frac{-b}{2a}$ .

Parametric form:

$x(t) = pt^2 + h; \ \ y(t) = 2pt + k \,$

General parabola

The general form for a parabola is

$(\alpha x+\beta y)^2 + \gamma x + \delta y + \epsilon = 0 \,$

This result is derived from the general conic equation given below:

$Ax^2 +Bxy + Cy^2 + Dx + Ey + F = 0 \,$

and the fact that, for a parabola,

$B^2=4AC \,$ .

The equation for a general parabola with a focus point F(u, v), and a directrix in the form

$ax+by+c=0 \,$

$\frac{\left(ax+by+c\right)^2}{{a}^{2}+{b}^{2}}=\left(x-u\right)^2+\left(y-v\right)^2 \,$

Latus rectum, semilatus rectum, and polar coordinates

In polar coordinates, a parabola with the focus at the origin and the directrix parallel to the y-axis, is given by the equation

$r (1 + \cos \theta) = l \,$

where l is the semilatus rectum: the distance from the focus to the parabola itself, measured along a line perpendicular to the axis of symmetry. Note that this equals the perpendicular distance from the focus to the directrix, and is twice the focal length, which is the distance from the focus to the vertex of the parabola.

The latus rectum is the chord that passes through the focus and is perpendicular to the axis of symmetry. It has a length of 2l.

Gauss-mapped form

A Gauss-mapped form: $(\tan^2\phi,2\tan\phi)$ has normal $(\cos\phi,\sin\phi)$ .

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Saturday, 6 October 2012