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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
parabola
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(Definition)
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A parabola is the locus of points  in the Euclidean plane which are equidistant from a given line  , called the directrix, and a given point  not on the directrix, called the focus.
To obtain a simple equation for the parabola, assume that the directrix is parallel to the  -axis, the focus is on the  -axis, and the directrix and focus are the same distance from the origin. By reflecting the plane if necessary, this means that there is a positive number  such that the equation of the directrix  is  and the position of the focus  is  . Then the condition that a point  is equidistant from  and  can be interpreted as the equation
Since
 , the above equation simplifies to
Below is the graph of a parabola for  :
From the equation
we can immediately observe some important properties of the parabola. First, since  is an even function, the parabola is symmetric with respect to the  -axis; this can also be deduced directly from the geometric definition of the parabola. Second, notice that the coefficient of  in the equation of the parabola is inversely proportional to  , the distance between the focus and the directrix. So this distance controls how rapidly the function
 grows. As  tends to zero, the parabola becomes flatter and flatter, tending to the straight line  in the degenerate case  . On the other hand, as  increases, the curvature of the parabola at 0 increases. When 
tends to infinity, the parabola begins to resemble a hairpin more and more until it suddenly becomes a single point, the origin, in the degenerate case
 .
The parabola is a conic section with eccentricity 1. All parabolas are similar, which follows directly from the definition of parabola.
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"parabola" is owned by mps. [ full author list (3) ]
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(view preamble)
Cross-references: similar, eccentricity, conic section, infinity, function, coefficient, even function, properties, graph, positive, plane, origin, distance, parallel, equation, line, Euclidean plane, points, locus
There are 32 references to this entry.
This is version 5 of parabola, born on 2007-05-22, modified 2007-10-27.
Object id is 9444, canonical name is Parabola2.
Accessed 2334 times total.
Classification:
| AMS MSC: | 51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry) | | | 53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space) |
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Pending Errata and Addenda
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{Graph of $\displaystyle y=\frac{1}{4}x^2$} \end{pspicture}](http://images.planetmath.org:8080/cache/objects/9444/l2h/img18.png "\begin{pspicture}(-4.5,-2.5)(4.5,4.5) \psaxes{<->}(0,0)(-4.5,-1.5)(4.5,4.5) \rpu... ...){.} \rput[b](0,-2.2){Graph of $\displaystyle y=\frac{1}{4}x^2$} \end{pspicture}")
