Characteristic of Envelope of Family of Curves

Theorem

Let $\FF$ be a one-parameter family of curves defined by a parameter $m$.

Let elements $A$ and $B$ of $\FF$ have parameters which differ by a small amount $\delta m$.

If $\delta m$ is sufficiently small, $A$ and $B$ will intersect.

The locus of the points of intersection of elements of $\FF$ as $\delta m \to 0$ becomes the envelope of $\FF$.

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): envelope: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): envelope: 1.