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.

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.

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.

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

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

where c, the Shift, is allowed to be negative. The hyperbola, in fact, consists of two branches that are combined into a single expression:

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

References