24-Ruby IV

Question

The problem of cockroach races

Forum|Forum|8 years ago
September 20, 2017
6 replies
17498 views

The cockroach needs to run from start (from different positions of the start line) to finish-line in minimum time.
On the field of running there is a section in the form of a circle (see Fig.), where the running speed
differs by V1> V2 or V1 <V2.
Which route (trace) from the start line to the finish line should the cockroach use from different positions?

Better to get numerical, symbolical and plot solution (with animation!).

Other

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LucMeekes

23-Emerald IV

Not formulated clear enough:

- Is finish anywhere on the bottom line?

- Does the full circle need to be run in all cases, or is running (part of) the lower half of the circle enough?

- Does the full horizontal track (at the top) need to be run?

Luc

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LucMeekes

23-Emerald IV

Ah, now I see it.

The run is from top to bottom, and the question is if it's wise to run across the circle, or to avoid it.

Luc

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ValeryOchkov

Author

24-Ruby IV

Two cases

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RichardJ

19-Tanzanite

The case of V2<V1 doesn't make much sense. In that case it's clear that the cockroach must avoid the circle. Then there are an infinite number of equivalent solutions where the cockroach starts at some point on the start line and runs perpendicular to that to the finish line.

If V2>V1 the solution is obvious. Start above the center of the circle, run perendicular to the start line to the finish line. That's the shortest possible path, with as much of it as possible in the circle.

Edit:. You added a new post while I was typing, and now I see that the start point is not a variable, but fixed. That makes the solutions less trvial

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LucMeekes

23-Emerald IV

Valery's question is to solve it for any starting position (horizontal) along the start line.

I bet it's related to an optical problem.

Intriguing, but I have to crack a coconut first...

Luc

D

DJF

16-Pearl

While I don't have the time to solve it, I think my approach would be a monte carlo.

Given - starting position, v1, v2 ,s, r

Approach - Walk in a straight line at a random angle (0.0001 to 179.9999 deg - 90 deg being straight to the finish). Then you need logic on if you hit the circle or not. If you hit the circle, begin walking at a 2nd random angle (0 to 180 deg again) until you hit the circle edge again. Then straight down to the finish. Calculate the time for each attempt.

I'm assuming the question involves straight lines only. But if the penalty or bonus of V2 was sufficient then following the edge of the curve would win. (If V2 was ~0 then you'd want to walk around it) That makes the real answer more difficult to calculate.

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ValeryOchkov

Author

24-Ruby IV

I would like to have someone:

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Werner_E

25-Diamond I

The result sure depends on the ratio v1/v2.

Furthermore you'll have to explain, as dferry pointed out, if you are just after straight lines or if you will also allow a curved path along the circumstance of the circle.

D

DJF

16-Pearl

While it's not my problem, I'd suggest the following constraints and clarifications

a.) travel in straight lines only

b.) If the circle is entered, 2 direction changes total are allowed (when you enter circle and when you leave)

c.) If the circle is not entered, 1 direction change is allowed (allowing you to avoid the circle)

d.) the roach is infinitely thin (allowing it to pass tangent to circle)

Just as I finished typing I see your later constraints. But I figured I'd send this anyway.

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ValeryOchkov

Author

24-Ruby IV

All is allowed for the minimal time of run from start tp finish!

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Werner_E

25-Diamond I

@ValeryOchkov wrote:

All is allowed for the minimal time of run from start tp finish!

Really? In your answer to Fred you confirmed that start and finish point must be vertically aligned. If thats true, none of dferrys three paths would be allowed!??

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LucMeekes

23-Emerald IV

It took a little longer. Unlike (possibly) Valery, I don't have cockroaches in the house. I do have silverfish (Lepisma saccharina) in my house, but they are much harder to train....

Anyway, here's the input:

And the output is shown below, for s=1, r=1 v1=1 and v2=2 and b positions ranging from -2 to 2:

Luc

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ValeryOchkov

Author

24-Ruby IV

Luc! I showed your animation to my cockroaches. They liked it. But one (in the middle - the egghead - as you and me) asked to start the animation from the point (case) when they run faster, bypassing the circle - from x =-3 to 3 for example. And for thr case when v.circle<v.field!