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Plane Loci

Apollonius was also interested in the problem of plane loci. Plane loci are curves that are defined by their distances from other objects in the plane. For example, a circle is a plane locus, because every point on a circle is the same distance from the point at its center. Conic sections are also plane loci. A parabola, for instance, can be defined as the set of points that are an equal distance from a point called the focus and a line called the directrix:

Defining curves by their distances from other points and lines is a lot like graphing those curves. Unfortunately, Apollonius' book on plane loci has been lost. We know that he wrote about them, because other people wrote commentaries on his work, or included references in their own books. But we still don't know exactly what Apollonius said.

The Greek scholar who made the most advances in the problem of plane loci is Pappus, who lived in Alexandria, Egypt. He was born around 290 AD, hundreds of years after Apollonius had lived. Pappus discussed plane loci defined by the ratios and products of the distance from a point on the locus to between three and six lines. Mathematicians were still working on the "Pappus problem" of finding plane loci more than a thousand years later!


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