We define a new coordinate system $R = x + y,$ $S = x - y,$ in which $R$ and $S$ are tilted at 45 degrees relative to $x$ and $y.$ This gives us \begin{eqnarray*} x &=& \frac{1}{2}(R+S) \\ y &=& \frac{1}{2}(R-S). \end{eqnarray*} Substituting this into our expression for the envelope curve $$y(x)=(1-\sqrt{x})^2$$ gives \begin{eqnarray*}\frac{1}{2}(R-S) & = & \left(1-\sqrt{\frac{1}{2}(R+S)}\right)^2 \\ \frac{1}{2}(R-S) & = & 1+\frac{1}{2}(R+S) - 2\sqrt{\frac{1}{2}(R+S)} \\ 0 &=& 1+S -\sqrt{2(R+S)}\\ 2R+2S &=& 1+S^2+2S \\ 2R &=& 1+S^2\\ $R &=& \frac{1}{2}+\frac{S^2}{2}. \end{eqnarray*} This is the equation of a parabola.
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Changing the variables
Share this pageWe define a new coordinate system $R = x + y,$ $S = x - y,$ in which $R$ and $S$ are tilted at 45 degrees relative to $x$ and $y.$ This gives us \begin{eqnarray*} x &=& \frac{1}{2}(R+S) \\ y &=& \frac{1}{2}(R-S). \end{eqnarray*} Substituting this into our expression for the envelope curve $$y(x)=(1-\sqrt{x})^2$$ gives \begin{eqnarray*}\frac{1}{2}(R-S) & = & \left(1-\sqrt{\frac{1}{2}(R+S)}\right)^2 \\ \frac{1}{2}(R-S) & = & 1+\frac{1}{2}(R+S) - 2\sqrt{\frac{1}{2}(R+S)} \\ 0 &=& 1+S -\sqrt{2(R+S)}\\ 2R+2S &=& 1+S^2+2S \\ 2R &=& 1+S^2\\ $R &=& \frac{1}{2}+\frac{S^2}{2}. \end{eqnarray*} This is the equation of a parabola.