Family reunion

Some geometric figures are a delight to behold: a host of skilfully arranged lines, intertwined with a graceful curve that seems barely to touch each one. What mathematical theory lies behind them? The theory of envelopes of lines.

Written by

BERTRAND HAUCHECORNE

PublishedMarch 25, 2019UpdatedMay 16, 20261 min read

Between 1620 and 1670, the most eminent mathematicians, including René Descartes, Pierre de Fermat and John Wallis, sought a method for determining the equation of the tangent to a curve. Isaac Newton and Gottfried Wilhelm Leibniz's introduction of differential calculus in the late 17^th century solved the problem brilliantly. The inverse problem arose immediately: given a family of lines, can we find a curve to which they are all tangent?

From geometry to analysis ---------------------------

Let us take a simple geometric example. Consider perpendicular coordinate axes (Ox) and (Oy)

and slide a line segment [AB] of length a, with A on the (Ox) axis and B on the (Oy) axis:

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