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?


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


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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.