Parabola

History

See the History section of Conic Sections page .

Description

Parabola is a member of conic sections, along with hyperbola and ellipse. Parabola can be thought of as a limiting case of ellipse or hyperbola. Note that parabola is not a family of curves. The impression that some parabola are more curved is because we are looking at different scale of the curve. Similarly, part of a large circle appears to be a line may induce us to conclude that there are different shapes of circles.

Like ellipse and hyperbola, there are many ways to define parabola. A common definition defines it as the locus of points P such that the distance from a line (called the directrix) to P is equal to the distance from P to a fixed point F (called the focus). As a conic section, the eccentricity of Parabola is 1.

The axis of a parabola is a line perpendicular to its directrix and passing its focus. Vertex of the parabola is the intersection of the parabola and its axis.

Formula

Parametric: {t, 1/4 t^2}, -∞ < t < ∞
Cartesian: y == 1/4 x^2.

For the given formulas, vertexes is at {0,0}, focus is at {0,1}.

Properties

Point and Tangent Construction

Let F be a given point and d be a give line. Let B := Point[d]. Let t := LineBisector[B,F]. Let b := Perpendicular[B,d]. Let P := Intersect[b,t] Since length[segment[B,P]]==length[segment[P,F]], P is a point on parabola. Further, t is the tangent at P.

parabola tangent const — Parabola Point Tangent Construction

Invariant under certain Dilation

Parabola have the property that when scaled (streching/shrinking) along a direction parallel or perpendicular to its axis, the curve remain unchanged. (For example, line also have this property, but circle do not. A streched line is still a line, but a streched circle is no longer a circle) When a parabola is streched along the directrix “a” units and along the axis by “b” units, the resulting curve is the original parabola scaled in both direction by “a^2/b”.

Given a parametrization of a parabola {xf[t], yf[t]} with vertex at Origin and focus along the y-axis, its focus is {0, xf[t]^2/(4 yf[t]) }.

Optical Property

A radiant point at the focus will reflect off the parabola into parallel lines. The figure shows three parabolas, two of which share a common focus.

parabolaReflect — Parabola with a Moving Light Source

car headlight — A car's headlight (Honda Civic 2000).

Tangents of Probala

Any set of tangents on the parabola will always cut a arbitrary tangent into the same proportion. That is, suppose you pick three tangents call them a, b, c. Now pick a arbitrary tangent x. Tagents a, b, c will cut x into segments with certain proportions. Now pick any other tangent x1, it will be cut into the same proportions. Thus, the envelope of lines with a positive constant sum of intercepts is a segment of parabola.