## 400 Reputation

13 years, 178 days

## Tests of suggestions by vv and Rouben Ro...

@Rouben Rostamian  vv:- your suggestion alone permits the boat to cross the same river (same velocity profile, same destination across from the departure) with the boat's speed reduced from 1.0 to 0.82. The time of crossing increases from 1.15 to 1.52 seconds.

Rouben Rostamian:- your suggestion, while also using vv's suggestion, permits the boat to cross from [0,0] to [1,-0.3] provided its speed is increased from 1.0 to 1.1.

Both of your suggestions allow me to test the limits of the Maple solution. I greatly enjoy such exploration! I thank you both.

## Observations...

@Rouben Rostamian  Thank you for reading and commenting on my worksheet.

After examining your solution to the least time crossing, here are a few observations:

I animated the boat (as a plot arrow) as it crossed the river following the least time path. It always headed upstream (against the current) with remarkably little variation in its bearing (the value of alpha).

I tried reducing the boat's speed, but any value below 0.935 produced this error message from dsolve;

Error, (in dsolve/numeric/bvp) initial Newton iteration is not converging

This same error occurred if I moved the destination upstream from the starting point, however a destination downstream from the starting point produced a correct least time path.

## I am astounded...

@Rouben Rostamian  Thank you for your wonderful analysis and solution to my question. I will take lots of time to analyze it, including substituting my worksheet's river velocity function for yours.

@Carl Love Please see my latest reply to Rouben Rostamian. I hope you too can now access Rivercrossing.mw

## The worksheet Rivercrossing should now b...

@Rouben Rostamian  Following Carl Love's advice I renamed my worksheet as one word and it appears to have uploaded successfully. Please let me know if you can now access it.

@Rouben Rostamian  Sorry, I have tried the upload of the worksheet several times and it fails each time, so I deleted the non-working link. Is there another way I can make the worksheet available to you?

In addition I have altered the question's wording to try to clarify that the worksheet animates one river crossing path, namely the boat always heads towards its destination. I would also like to animate a least time path for crossing from the starting point to the destination, which I assume requires a functional definition of the boat's constantly changing heading.

## Thank you....

@phil2 Your fix solves this problem.

## Thank you Thomas...

@Thomas Richard I copied the Sievert routine from a book on bubbles by John Oprea. He was using Maple 5 or 6.

## Thank you, vv...

@vv Your fix works very well.

## Only when quit is selected in the DEBUG ...

@phil2 This occurs on all worksheets when "quit" is selected in the Maple Debugger window. Below is a very simple example of this output. When "continue' is selected and the procedure completes normal execution only a blank text entry area is added to the worksheet.

 >
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 1, 1, 2, 3 A:    2   plot3d([Rad*cos(theta)*sin(phi), Rad*sin(theta)*sin(phi), Rad*cos(phi)],theta = 0 .. Pi,phi = 0 .. 2*Pi,scaling = constrained) ============================================

## Thank you!...

@Kitonum It seems your method is more general in that the conditions within the map clause can be any that the programmer desires e.g. the cap could have an elliptical shape.

## Thank you, Rouben...

@Rouben Rostamian  Thanks, your procedure will help me in my current project. I believe my azimuth and polar are equivalent to your longitude and colatitude.

## Much thanks...

@vv Thanks to you my Maple education continues.

## How does procedure Apollo exit?...

@vv Your programming here is amazing!

Unfamiliar with recursion, I can't see how Apollo ends. In particular how can the final display command execute apparently without the preceding gen(1,2,3,4) command executing for the second time?

## Thank you Carl...

@Carl Love Calling a Mobius strip single-edged reveals my almost total lack of knowledge of topology. The Catalan surface seemed different because it has two cusp-like edges between curved portions of the surface.

Thanks to you, I looked up the definition of homeomorphic in Wikipedia and am now just slightly less naive with topology!

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