Source
r/mathriddles•
sobe86

Find the shortest curve

Hard

X-posting this one: https://www.reddit.com/r/math/s/i3Tg9I8Ldk (spoilers), I'll reword the original.

 1.⁠ ⁠Find a curve of minimal length that intersects any infinite straight line that intersects the unit circle in at least one point. Said another way, if an infinite straight line intersects the unit circle, it must also intersect this curve.

 2.⁠ ⁠Same conditions, but you may use multiple curves. (I think this is probably the more interesting of the two)

For example the unit circle itself works, and is (surely) the shortest closed curve, but a square circumscribing the unit circle, minus one side, also works and is more efficient (6 vs 2 pi).

This is an open question, no proven lower bound has been given that is close to the best current solutions, which as of writing are

  1. !2 + pi ~ 5.14!<

  2. !2 + sqrt(2) + pi / 2 ~ 4.99 !<

respectively

Posted Dec 27, 2023, 12:07 AM UTC · edited

610
plutrichor•

New upper bound for (2): >!4.819!<, obtained by >!optimizing the construction that gave the 2 + sqrt(2) + pi/2 bound.!< >!See https://www.desmos.com/calculator/3hanyxusfv for details.!<

3
Deathranger999•

I might be dumb, but for both questions isn’t the unit circle itself, of length pi = 3.14, such a curve?

1
QuagMath•

The curve made of a line segment from (1,1) to (0,1), the left half of the unit circle, and then the line segment from (0,-1) to (1,-1) is shorter and I believe satisfies this property, which >! appears to be equal to OP’s upper bound !<

2
Deathranger999•

Yep, good call. I just forgot how circles work.

1
pichutarius•

C = 2pi(1) = 6.28

1
Deathranger999•

Wow I’m a fucking idiot. Thanks.

1
ThrowAwayPureVPNDM•

The curve needs to be connected?

1
sobe86OP•

For 1. yes, for 2. you may use multiple curves, not necessarily overlapping

1
ThrowAwayPureVPNDM•

What do you mean by curve? Piecewise C1 or any regularity? Can they be degenerate, like points?

1
sobe86OP•

I didn't make up the question, but I'd recommend piecewise C1 if you're going to try and justify it having a finite length

1
Pit_shost•

Based on everyone else's ideas, there is a shape that brings (2) down to 2 x sqrt(3) = 3.4641...

It's past midnight, will post updates later.

1