EDB — 23Y

[23Y]

• Suppose the curves in the plane are described by the equation in implicit form $F(x,y,a)=0$; that is, fixed the parameter $a$, the curve is the locus

  $$\{(x,y):F(x,y,a)=0\};$$

  Then the envelope is obtained by expliciting the variable $a$ from the equation $\frac{\partial }{\partial a}F(x,y,a)=0$ and substituting it into the $F(x,y,a)=0$.

• For simplicity, consider curves that are functions of the abscissa. Let $y=f(x,a)=f_a(x)$ be a family of functions, with $x\in I,a\in J$ (open intervals), then $y=g(x)$ is the envelope of $f_a$ if the graph of $g$ is covered by the union of the graphs of $f_a$ and the curve $g$ is tangent to every $f_a$ where it touches it. More precisely, for every $x\in I$ there is $a\in J$ for which $g(x)=f(x,a)$, and also, for every choice of $a$ that satisfies $g(x)=f(x,a)$, we have $g^{\prime }(x)=f^{\prime }(x,a)$.