Parabola: Difference between revisions
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Let <math>d</math> be a line and <math>F</math> a point. In the degenerate 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. | Let <math>d</math> be a line and <math>F</math> a point. In the degenerate 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. | ||
To avoid the 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. | To avoid the 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> is called the ''focal distance'' of the parabola. | ||
Now let <math>F'</math> be a point in <math>\Pi</math> and <math>d'</math> a line in <math>\Pi</math> such that the distance from <math>F'</math> to <math>d'</math> equals the distance from <math>F</math> to <math>d</math>. 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 with focus <math>F'</math> and directrix <math>d'</math>. | Now let <math>F'</math> be a point in <math>\Pi</math> and <math>d'</math> a line in <math>\Pi</math> such that the distance from <math>F'</math> to <math>d'</math> equals the distance from <math>F</math> to <math>d</math>. 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 with focus <math>F'</math> and directrix <math>d'</math>. In other words, any two parabolas with the same focal distance are congruent. | ||
Revision as of 19:37, 9 December 2007
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.
Let be a line and a point. In the degenerate when is a point of , the "parabola" with directrix and focus is the line through that is perpendicular to . In the language of conic sections, this corresponds to the case when the plane contains a generator of the cone.
To avoid the degenerate case, assume that does not lie in , let be the unique plane containing and and let be the parabola with focus and directrix . The line through and perpendicular to is called the axis of the the parabola and is the unique line of symmetry of . The unique point of that is equidistant from and lies on and is known as the vertex of the parabola, and the distance is called the focal distance of the parabola.
Now let be a point in and a line in such that the distance from to equals the distance from to . Then there is a unique, orientation-preserving rigid motion of taking to and to and therefore, the parabola to the parabola with focus and directrix . In other words, any two parabolas with the same focal distance are congruent.
