Changing the variables


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

  $\displaystyle  x  $ $\displaystyle = $ $\displaystyle  \frac{1}{2}(R+S)  $    
  $\displaystyle y  $ $\displaystyle = $ $\displaystyle  \frac{1}{2}(R-S).  $    

Substituting this into our expression for the envelope curve

  \[ y(x)=(1-\sqrt {x})^2 \]    

gives

  $\displaystyle \frac{1}{2}(R-S)  $ $\displaystyle  =  $ $\displaystyle  \left(1-\sqrt {\frac{1}{2}(R+S)}\right)^2  $    
  $\displaystyle \frac{1}{2}(R-S)  $ $\displaystyle  =  $ $\displaystyle  1+\frac{1}{2}(R+S) - 2\sqrt {\frac{1}{2}(R+S)}  $    
  $\displaystyle 0  $ $\displaystyle = $ $\displaystyle  1+S -\sqrt {2(R+S)} $    
  $\displaystyle 2R+2S  $ $\displaystyle = $ $\displaystyle  1+S^2+2S  $    
  $\displaystyle 2R  $ $\displaystyle = $ $\displaystyle  1+S^2 $    
  $\displaystyle $R $ $ $\displaystyle = $ $\displaystyle  \frac{1}{2}+\frac{S^2}{2}.  $    

This is the equation of a parabola.

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