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 x=12(R+S)y=12(R−S). Substituting this into our expression for the envelope curve y(x)=(1−x)2 gives 12(R−S)=(1−12(R+S))212(R−S)=1+12(R+S)−212(R+S)0=1+S−2(R+S)2R+2S=1+S2+2S2R=1+S2$R=12+S22. This is the equation of a parabola. Back to main article