Parabola: Difference between revisions

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Synthetically, a '''parabola''' is the locus of points in a plane that are equidistant from a given line (the ''directrix'') and a given point (the ''focus'').  Alternatively, a parabola is a [[conic section]] obtained as the intersection of a right circular cone with a plane parallel to a generator of the cone.
A '''parabola''' is the planar [[curve]] formed by the points that lie as far from a given line (the ''directrix'') as from a given point (the ''focus'').  Alternatively, a parabola is the curve you get when intersecting a right circular [[cone]] with a plane parallel to a generator of the cone (a line on the cone which goes through its apex); thus, a parabola is a [[conic section]].


Let <math>d</math> be a line and <math>F</math> a point.  In the special case when <math>F</math> is a point of <math>D</math>, the "parabola" with directrix <math>D</math> and focus <math>F</math> is the line through <math>F</math> that is perpendicular to <math>D</math>. In the language of conic sections, this corresponds to the case when the plane contains a generator of the cone.
If the focus lies on the directrix, then the "parabola" is in fact a line. In the language of conic sections, this corresponds to the case when the plane contains a generator of the cone.  


To avoid this degenerate case, assume that <math>F</math> does not lie in <math>D</math>, let <math>\Pi</math> be the unique plane containing <math>F</math> and <math>D</math> and let <math>P</math> be the parabola with focus <math>F</math> and directrix <math>d</math>. The line <math>s</math> through <math>F</math> and perpendicular to <math>D</math> is called the ''axis'' of the the parabola <math>P</math> and is the unique line of symmetry of <math>P</math>. The unique point <math>V</math> of <math>s</math> that is equidistant from <math>F</math> and <math>D</math> lies on <math>P</math> and is known as the ''vertex'' of the parabola, and the distance <math>FV</math> (half the distance from <math>F</math> to <math>D</math>) is called the ''focal distance'' of the parabola.
To avoid this degenerate case, we assume that the focus lies not on the directrix. The line through the focus and [[perpendicular]] to the directrix is called the ''axis'' of the parabola. It is the unique line of symmetry of the parabola. The parabola has one point that lies on the axis. This point is called the ''vertex'' of the parabola. The distance between the focus and the vertex is called the ''focal distance'' of the parabola. It is the same as the distance between the vertex and the directrix, and half the distance from the focus to the directrix.


Now let <math>P'</math> be any other parabola in <math>\Pi</math> be with the same focal distance as <math>P</math>.  Let <math>F'</math> be its focus and <math>D'</math> its directrix.  Then there is a unique, orientation-preserving [[rigid motion]] of <math>\Pi</math> taking <math>F</math> to <math>F'</math> and <math>D</math> to <math>D'</math> and therefore, the parabola <math>P</math> to the parabola <math>P'</math>.  In other words, any two parabolas with the same focal distance are congruent.
The shape of a parabola is determined by the focal distance. All parabola with the same focal distance are [[congruence (geometry)|congruent]], meaning that given any parabola can be moved to any other parabola with the same focal distance by a [[rigid motion]].

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A parabola is the planar curve formed by the points that lie as far from a given line (the directrix) as from a given point (the focus). Alternatively, a parabola is the curve you get when intersecting a right circular cone with a plane parallel to a generator of the cone (a line on the cone which goes through its apex); thus, a parabola is a conic section.

If the focus lies on the directrix, then the "parabola" is in fact a line. In the language of conic sections, this corresponds to the case when the plane contains a generator of the cone.

To avoid this degenerate case, we assume that the focus lies not on the directrix. The line through the focus and perpendicular to the directrix is called the axis of the parabola. It is the unique line of symmetry of the parabola. The parabola has one point that lies on the axis. This point is called the vertex of the parabola. The distance between the focus and the vertex is called the focal distance of the parabola. It is the same as the distance between the vertex and the directrix, and half the distance from the focus to the directrix.

The shape of a parabola is determined by the focal distance. All parabola with the same focal distance are congruent, meaning that given any parabola can be moved to any other parabola with the same focal distance by a rigid motion.

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