The Conic as a Locus of Lines

We defined a conic as the set of points of intersection of corresponding lines in two projective pencils. This prompts an inquiry into the dual figure: the set of lines joining corresponding points on two projective point ranges. Before we can do this, we need to define some terms. A line is tangent to a curve (set of points) if exactly one point lies on both the line and the curve. A set of points, or curve, is the envelope of a set of lines if each point lies on exactly one line and each line lies on exactly one point. Then each line in the set will be tangent to the curve.

The envelopes that we will want to study have a familiar interpretation in string art. We set up hooks on two straight objects, and place strings between corresponding hooks, noting the curve that the strings trace out. Durer gave the following diagram:

The curve on the right, traced out by lines connecting corresponding points on the horizontal and vertical lines, is a parabola. (For more on Durer, click here.) We can describe it easily using coordinates; it is the locus of lines connecting points (-a,0) to (-13,a).

Connecting points (a,0) to (0,1/a) in a similar way gives a hyperbola:

Connecting points (a,0) to (1/a,1) gives an ellipse:

So now it seems that the envelope of the lines connecting corresponding points on two projective point ranges is a conic. Now we define such an envelope as a line conic, and need to prove that any line conic is a point conic and vice versa. We can prove this now using coordinates and calculus; we will present synthetic proofs in following sections.