Equation of Envelope of Family of Curves

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Theorem

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

Let $E$ be the equation defining $\FF$.

Let $E'$ be the equation obtained by equating to zero the partial derivative of $E$ with respect to $m$.


The equation defining the envelope of $\FF$ can then be found by eliminating $m$ between $E$ and $E'$.


Proof


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Examples

Example: $y = 2 m x + m^2$

Consider the family of curves $\FF$ embedded in the Cartesian plane defined by the equation $E$:

$E: \quad y = 2 m x + m^2$

where $m$ is the parameter of $\FF$.


The envelope of $\FF$ is the parabola whose equation is:

$y = -x^2$


Sources

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