A key feature of a real circle is that it can be constructed metrically as the locus of points at a constant positive distance from a fixed point F. The initial object of this chapter is to present a metric construction that produces parabolas, real ellipses, and hyperbolas. The construction involves a fixed point F, and a fixed line D not passing through F. We will show that the standard parabolas, real ellipses, and hyperbolas of Chapter 4 can all be constructed in this way. Indeed, combining this with the classification of conics in Chapter 15, we will see that any parabola, real ellipse, or hyperbola has this property. The importance of the construction lies in the fact that constructible conics have interesting metric properties, of physical significance. That leads us to geometry which might otherwise remain unnoticed.

Focal Constructions

For the purposes of this text a focal construction (or just construction) is a choice of a point F (the focus), a line D not through F (the directrix), and a positive constant e (the eccentricity). We consider a variable point P subject to the constraint that its distance from F is proportional to its distance from D, where the constant of proportionality is e.