Conic Sections as Loci of Points: What is it?
A Mathematical Droodle

 

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


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Explanation

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Copyright © 1996-2017 Alexander Bogomolny

Conic 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 F1 and F2.

 

Ellipse is a set (locus) of points M the sum of whose distances to F1 and F2 is constant:

  |F1M| + |F2M| = 2c,

for some constant c.

 

Hyperbola is a set (locus) of points M for which the absolute value of the difference of the distances to F1 and F2 is constant:

  ||F1M| - |F2M|| = 2c,

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):

  |FM| = dist(M, d),

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.

 

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

Imagine a family of circles around two points F1 and F2 whose radii differ by a fixed unit of measurement. For a given constant c (Shift in the applet), we consider circles F1(R1) with radius R1 from the first family and F2(R2) with radius R2 from the second, such that

  R1 + R2 = 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

  R1 - R2 = 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:

  |R1 - R2| = 2|c|.

For parabola, the applet displays the intersections of a family of circles with a family of straight lines.

References

  1. V. Gutenmacher, N. Vasilyev, Lines and Curves: A Practical Geometry Handbook , Birkhauser; 1 edition (July 23, 2004)

Conics

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