Parabola: Difference between revisions
imported>Holger Kley (New page: 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 [[...) |
imported>Holger Kley No edit summary |
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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< | 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. | ||
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>. | ||
Revision as of 20:32, 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.
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 .
