Conic Sections as Loci of Points: What is it?
A Mathematical Droodle
What if applet does not run? |
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Copyright © 1996-2018 Alexander BogomolnyConic Sections as Loci of Points
There are three proper conic sections: ellipse, parabola and hyperbola. Each can be defined in several ways. On this page we illustrate their geometric definition: the curves are defined as loci of points.
For ellipse and hyperbola, their are two special points - their foci - in terms of which the definitions are set. Denote the foci F_{1} and F_{2}.
Ellipse is a set (locus) of points M the sum of whose distances to F_{1} and F_{2} is constant:
for some constant c. |
Hyperbola is a set (locus) of points M for which the absolute value of the difference of the distances to F_{1} and F_{2} is constant:
for some constant c. |
The geometric definition appears to place parabola aside of ellipse and hyperbola; it's definition use a point and a line, not two points. However, other definitions of the conics are more symmetrical. Asymmetry of the geometric definition is due more to the limitations of our descriptive means than the intrinsic properties of the curves.
Parabola is a set (locus) of points M equally distant from a given point F (focus) and a given line d (directrix):
where the distance from M to d is the length of the perpendicular from M to d. |
The applet below illustrates the foregoing definitions. Since the circle is the locus of points at the same distance from a center point, the intersection of two circles is located at the prescribed distances from their centers.
What if applet does not run? |
Imagine a family of circles around two points F_{1} and F_{2} whose radii differ by a fixed unit of measurement. For a given constant c (Shift in the applet), we consider circles F_{1}(R_{1}) with radius R_{1} from the first family and F_{2}(R_{2}) with radius R_{2} from the second, such that
R_{1} + R_{2} = 2c. |
According to the definition, the points of intersection of such pairs - one circle from each of the two families - lie on an ellipse. Changing parameters c generates a family of confocal ellipses, i.e. ellipses with the same foci.
For hyperbola, the applet displays the points of intersection of the circles that satisfy
R_{1} - R_{2} = 2c, |
where c, the Shift, is allowed to be negative. The hyperbola, in fact, consists of two branches that are combined into a single expression:
|R_{1} - R_{2}| = 2|c|. |
For parabola, the applet displays the intersections of a family of circles with a family of straight lines.
References
- V. Gutenmacher, N. Vasilyev, Lines and Curves: A Practical Geometry Handbook , Birkhauser; 1 edition (July 23, 2004)
Conics
- Conic Sections
- Conic Sections as Loci of Points
- Construction of Conics from Pascal's Theorem
- Cut the Cone
- Dynamic construction of ellipse and other curves
- Joachimsthal's Notations
- MacLaurin's Construction of Conics
- Newton's Construction of Conics
- Parallel Chords in Conics
- Theorem of Three Tangents to a Conic
- Three Parabolas with Common Directrix
- Butterflies in a Pencil of Conics
- Ellipse
- Parabola
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Copyright © 1996-2018 Alexander Bogomolny
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