If we erase a plane curve after drawing several of its tangent lines, the eye can still make out the shape of the original curve as the envelope of all those lines. Our brains have, in effect, found an approximate solution to a problem in differential geometry: in the general case, determining the curve known as the envelope, which is tangent to a one-parameter family of curves, parametrized by λ, with equation f (x, y, λ) = 0. Analytically, this means solving the system:
in which the second equation characterizes the point of tangency. For a family of lines, the moving point on the envelope is then the intersection of two lines. Conics have been extensively studied since Apollonius, and their tangent lines are easy to construct, making them suitable for construction by folding.
Conics encircled
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Conics—plane curves defined analytically by a quadratic equation—are, geometrically, the sections of a surface of revolution generated by a line. If this line intersects the axis of revolution, the result is a cone whose sections, depending on the orientation of the plane, are ellipses, parabolas or hyperbolas. If the line lies off the axis and is parallel to it, the result is a cylinder, whose sections can only be ellipses (including circles). Otherwise, the result is a diabolo-shaped one-sheeted hyperboloid of revolution. The degenerate cases produce a line, two lines or a point (see Tangente SUP 75, 2014).
A conic generally has two foci. For a circle, they coincide; for a parabola, one of them strays off to infinity. Let 2c denote the distance between the foci and 2a the distance between the points where the conic meets the focal axis. The characteristic parameter of a conic is its eccentricity e = c / a. For an ellipse, 0 < e < 1. For a hyperbola, e > 1. For a circle, e = 0, while for the improbable parabola, e = 1. We shall now examine a general construction of conics that also determines the tangent line at any point.
