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

  \[ \{ (x,y) : F(x,y,a)=0\} \quad ; \]

  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 \(x∈ I,a∈ J\) (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 \(x∈ I\) there is \(a∈ J\) for which \(g(x)=f(x,a)\), and also, for every choice of \(a\) that satisfies \(g(x)=f(x,a)\), we have \(g'(x)=f'(x,a)\).