the focal-distance definitions

There is a hands-on way to draw an ellipse with two pins and a loop of string: pin the loop at two points, pull it taut with a pencil, and slide the pencil around — the curve it draws is an ellipse. This works because of the focal-distance definition, which builds the ellipse and the hyperbola directly from the two foci, with no cone and no directrix in sight.

For the ellipse: pick two foci F_1 and F_2; the ellipse is the set of all points P whose two focal distances add to a fixed constant, |PF_1| + |PF_2| = 2a. The pinned string has fixed length, so the two distances from the pencil to the pins always sum to the same amount — exactly this rule. For the hyperbola, replace the sum with a difference: it is the set of all points P with | |PF_1| - |PF_2| | = 2a, a fixed constant. (One branch is closer to F_1, the other closer to F_2; the absolute value collects both.) In both cases 2a turns out to be the distance between the two vertices, the length of the major or transverse axis.

These definitions are wonderfully concrete and are why these curves appear in nature and engineering: planets obey the constant-sum law (Kepler), and the constant-difference law underlies hyperbolic navigation, where the time-difference of two signals fixes you on a hyperbola. The focal-distance definition, the focus-directrix property, and the plane-section definition all describe exactly the same curves — the Dandelin spheres prove they agree.