Envelopes of Families of Curves




Suppose we have a family of curves MATH, where $\QTR{cal}{C}_{s}$ has parametrization
MATH
Their envelope $\QTR{cal}{E}$, if it exists, is a (single) curve which is tangent to each $\QTR{cal}{C}_{s}$ where it intersects it. Suppose $\QTR{cal}{E}$ does exist, and let $t=h(s)$ pick out the intersection of $\QTR{cal}{C}_{s} $ with $\QTR{cal}{E}$. In other words, MATH is the point where $\QTR{cal}{C}_{s}$ meets $\QTR{cal}{E}$, and where they have a common tangent line. Thus, the envelope $\QTR{cal}{E}$ is parametrized by MATH.

The tangent vector to $\QTR{cal}{E}$ is given by MATH. We can compute this by noting that the two components of $E(s)$ are the compositions
MATH

Applying the (multivariable) chain rule, we get
MATH
(All partials are calculated at $t=h(s)$.)

On the other hand, the tangent vector to $\QTR{cal}{C}_{s}$ at this point is simply MATH, so we have
MATH

Equating the components of these vectors, we get two equations which we can use to eliminate $h^{\prime}(s)$, obtaining the equation MATH. Since $\lambda \neq 0$, we get the (partial) differential equation


MATH

In practice, this equation enables us to find $t$ as a function of $s\ $; i.e. $t=h(s)$.