Suppose we have a wire made of an ideal spring steel. If we bend it and then release any external force it will straighten itself into a perfect line.

Suppose also that the wire is attached to some set of points in the plane. The wire can rotate in these points slide through the points but can not escape out of these points.

How to simulate this?

I made a code when we have only two fixed points – (large points in image). The small points are ends of the wire. The length of wire is constant all the time.

```
Manipulate(
Plot({x^2, 1/a x^2 + 1 - 1/a}, {x, -5, 5},
Epilog -> {Point({{b /.
FindRoot(
b Sqrt(1 + (4 b^2)/a^2) + 1/2 a ArcSinh((2 b)/a) -
1/2 (4 Sqrt(17) + ArcSinh(4)), {b, 1}),
1/a b^2 + 1 - 1/a /.
FindRoot(
b Sqrt(1 + (4 b^2)/a^2) + 1/2 a ArcSinh((2 b)/a) -
1/2 (4 Sqrt(17) + ArcSinh(4)), {b, 1})}, {-b /.
FindRoot(
b Sqrt(1 + (4 b^2)/a^2) + 1/2 a ArcSinh((2 b)/a) -
1/2 (4 Sqrt(17) + ArcSinh(4)), {b, 1}),
1/a b^2 + 1 - 1/a /.
FindRoot(
b Sqrt(1 + (4 b^2)/a^2) + 1/2 a ArcSinh((2 b)/a) -
1/2 (4 Sqrt(17) + ArcSinh(4)), {b, 1})}}),
Point({{-2, 4}, {2, 4}}), PointSize -> Large,
Point({{-1, 1}, {1, 1}})}, AspectRatio -> Automatic,
PlotRange -> {-1, 5}), {a, 1, 100})
```

This is only a pseudo simulation and it has nothing to do with reality, I only wanted to demonstrate how it might look.

Instead of two fixed points we can have 3, 4 or more points. I have no idea how to simulate it.