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Given a family of curves $$F=(x-t)^2+y^2-\frac{1}{2}t^2,$$ I am trying to compute the envelope of this family.

The envelope is described by the equations $$F=0, \\ \dfrac{\partial F}{\partial t} = -2(x-t)-t = 0.$$

To eliminate $t$, I computed a Gröbner basis for $I=\left\langle F,\dfrac{\partial F}{\partial t}\right\rangle.$ Namely, $$ \{ g_1=x^2-y^2,g_2=t-2x \}.$$ So $I\cap \mathbb R[x,y]= \langle g_1 \rangle$, and the envelope lies on the curve $x^2-y^2=0.$

Now, I want to draw a picture to illustrate my answer by Mathematica, (I also cannot imagine what the picture of that family looks like), but I don't know how to write the code. Could you give me a hand?

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In gnuplot, which is freely available, the commands

set parametric
plot [-4:4] [-4:4] for [a=-100:100] 0.4*a/sqrt(2)*cos(t)+0.4*a,0.4*a/sqrt(2)*sin(t) lt 1 notitle

will generate a figure of your family of curves like this: enter image description here

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Not too hard to do in Mathematica:

ContourPlot[Evaluate[Table[(x - t)^2 + y^2 - t^2/2 == 0,
            {t, -5, 5, 1/10}]], {x, -8, 8}, {y, -8, 8}]

circle envelope 1

or, if you exploit the fact that the generator curves are in fact circles,

Graphics[Table[Circle[{t, 0}, Abs[t]/Sqrt[2]], {t, -5, 5, 1/10}], 
 Frame -> True]

circle envelope 2

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