This is an exercise on page 150 of Cox/Little/O'Shea's Ideal, varieties and algorithms: an introduction to computational algebraic geometry and commutative algebra, 3rd ed.
I get lost in this problem...
For part (a), I find the equation of the envelope to be $g_1=-27 x^3 + 16 x^5 - 72 x^3 y + 16 x^3 y^2 - 64 x y^3=0,$ by computing a Groebner basis for $
The whole Groebner basis is $\{g_1=-27 x^3 + 16 x^5 - 72 x^3 y + 16 x^3 y^2 - 64 x y^3, g_2=27 x^2 - 16 x^4 + 72 t x y + 24 x^2 y + 32 t x y^2 - 16 x^2 y^2, g_3=9 t x^2 - 6 x^3 + 4 t x^2 y - 8 x y^2, -9 x^2 + 8 t x^3 - 24 t x y - 4 x^2 y, g_4=3 t x - 2 x^2 + 2 t^2 y - 2 y^2, g_5=3 t^2 x - 2 t x^2 + x y, g_6=2 t^3 - x - 2 t y\}.$
I don't know how to show that "the envelope is the union of two varieties" algebraically. I plotted the picture of of $(x - t)^2 + (y - t^2)^2 =t^2$, which looks like
Note that the image consists of two separate parts, so one can "imagine" that "the envelope is the union of two varieties." But how can we prove it formally?
For part (b), first I notice that $g_6=0$ is quadratic in $t$, so that a given partial solution $(x,y)$ extends in at most two ways to a complete solution.
Then I tried to find the singular points of $V(g_1)$, which are given by $g_1=\frac{\partial g_1}{\partial x}=\frac{\partial g_2}{\partial x}=0.$ In order to find the solutions of these equations, I computed a Groebner basis for them, which is $\{h_1=729 y^6 + 3888 y^7 + 4320 y^8 + 1792 y^9 + 256 y^{10}, h_2=729 x y^3 + 3888 x y^4 + 4320 x y^5 + 1792 x y^6 + 256 x y^7, h_3=531441 x^2 + 419904 y^3 - 1119744 y^4 + 3234816 y^5 - 694270080 y^6 - 931055616 y^7 - 414746624 y^8 - 61521920 y^9\}$
Note that $h_1$ includes only $y$, so from $h_1=0$, we get $y=-\frac{9}{4},-\frac{1}{4},0.$ Since the leading coefficient of $h_3$ in x is a constant, these partial solutions of $y$ extends. Then I am lost. I don't know the outline of how to solve this problem.
Above is all of my work. Looking forward to your advice and help. Thanks in advance.