# How to fix resizing between Graphics and Plot when using Inset

I have the following code:

g[u_,p_] = Graphics{
Circle[{2 Sqrt[u], u}, u],
Circle[{-2 Sqrt[u], u}, u],
Point[{2 Sqrt[u], u}],
Point[{-2 Sqrt[u], u}],
Inset[p,{0,1},{0,0},{15,10}]
}


and I want to draw the parabola that goes through the centers of both of these circles, for every $$u \neq 0$$ with its value in $$0$$ equal to $$1$$. We can see that, if we let $$y$$ to be the needed curve, that $$y(2\sqrt{u}) = u, \forall u \neq 0$$ and $$y(0) = 1$$. By making $$u:= u^2$$, we see that $$y(2u) = u^2$$, so by making again $$u := u/2$$ we see that $$y(u) = u^2/4,$$ and we also need to subtract one because we map $$(0,1)$$ to $$(0,0)$$.

However, if I try to plot this by saying

f[u_] = g[u, Plot[u^2/4 - 1,{u,-100,100}]


I don't really get what I want. I suspect this is because of the resizing done by Graphics and Plot. I have tried using PlotRange and PlotRangeClipping, but nothing works.

Is there any way I can do this or do I need another method to draw this curve? As a matter of fact, can it be done without Inset?

A couple of changes:

• do not use Inset to combine the two plots; use Show instead, which will plot both on the same coordinate system;
• I think that there may have been a small error in your calculation; it seems to work if I plot $$u^2/4$$ instead of $$u^2/4-1$$;
• finally, a practical consideration: you will want to reduce the range over which you plot your parabola, or your circles become too small (here I used $$-5,5$$).

With those changes:

ClearAll[g]
g[u_] := Graphics@{
Circle[{2 Sqrt[u], u}, u],
Circle[{-2 Sqrt[u], u}, u],
Point[{2 Sqrt[u], u}],
Point[{-2 Sqrt[u], u}]
};

Show[
Plot[u^2/4, {u, -5, 5}],
g[0.7]
]


... and just for fun:

ClearAll[gcombo]
gcombo[u_] := Show[
Plot[x^2/4, {x, -5, 5}],
Graphics@
Through[{Point, Map[Circle[#, u] &]}[{# Sqrt[u], u} & /@ {2, -2}]]
]

Animate[
gcombo[u],
{u, 0, 2},
AnimationDirection -> ForwardBackward,
AnimationRunning -> False
]


• Thank you so much!
– hit
Commented Mar 29, 2019 at 19:12