Author Topic: Another Step Forward  (Read 70209 times)

DerekSmith

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Another Step Forward
« on: June 06, 2008, 11:32:58 PM »
When I first came to the IGKT Website, I expected to find a complete database of all known rational knots and a means of filtering through them to find the knot which had piqued my interest.  I was disappointed that I did not find such a resource, but perhaps I should not have been dissappointed with the IGKT, because I could not find such a resource anywhere, perhaps because to date such a repository and locational tool probably did not exist.

As I thought about how you might search such a repository (when one eventually does exist) I was struck with the fact that knots are so three dimensionally complex, we would probably be stuck with having to find a person with the experience and mental caliber of  Budworth or Lehmanian proportions, send them the knot and beg their time to find its 'name' for you.  After a while it struck me that knots could be roughly categorised by making some measure of their complexity - how many times the cords crossed to make the knot.  My 'Overs Index' was born and quickly dashed with a "Read Budworth's 'Knots in Crime' - he did this years ago"  And sure enough, so he had - 120 knots categorised out across 24 groups based on the number of 'crossings' the knot exhibited when laid out flat and as simply as possible.

I started to build on Geoffrey's 120 knots but quickly came to realise that the 80 : 20 rule (or worse) was going to get to work here as most of the knots fell ever more often into certain popular 'pots'.  The Crossings count was a useful segregator but was no where near sensitive enough to be anything other than a crude segregation.

Further thought led me to realise that probably the next important distinction between knots was how 'complete' their crossings were.  The most 'complete' or as I came to think of it 'saturated' a knot could not possibly be more 'tangled'  - every over crossing was followed by an under crossing - they were fully saturated, but a pile of cord which might have the same number of crossings would effectively have zero saturation because as you pulled on the ends, nothing held on to anything else and the whole thing just unravelled.  This way two knots belonging to the 'Six' crossing pot say, the Reef and the Granny, could now be split into two sub groups, the Reef was 6:10 while the Granny was 6:12.  But it was still fairly poor.  Using this method the 20 knots in Budworth's V1 class could only be resolved into five sub classes and did only marginally better with the 19 knots in his VIII class which could be resolved into eight subclasses but we still had over half the knots in just two of those sub classes.

Allied to the limited ability of the Overs Index to be able to locate any particular knot was the almost universal call of 'It's just too complicated - the man in the street can't do it'.  So, although the Overs Index was potentially better than nothing, in reality it was a useless as nothing because virtually no one could count the crossings and the saturations....   and so it languished until FCB4 came into being.

If you have read the post, then you will be familar with Frank Brown's cypher method for exchanging a knot diagram and the little drawing program that has been written to allow knot diagrams to be easily drawn and recreated from their cypher files.  While writing the program, it struck me that it would be relatively straightforward to add in a function to count the crossings and work out the saturation - at least this would remove one of the hurdles to using the OI - just draw out your knot and the OI gets worked out for you -- handy.

As I worked on how to calculate the OI, I started by creating a 'picture' of the path of the cord by writing down a '1' if the cord went over and writing down a '0' when it went under.  From this pattern, the 'Crossings' was simply the number of 1's and the saturation was twice this number minus the number of consecutive events.  Taking the following diagram of the overhand knot, starting with the WP symbol, first the cord goes over (so a '1'), then under (so a '0')etc. its full sequence is 101010 giving an OI of 3:6



You have probably seen it already, but it took me a couple of days to realise that the sequence was just a binary number that reflects the passage of the cord through the knot - it IS the knot.  As nobody can remember binary numbers, I converted it to decimal and when I saw what 101010 was in decimal I new - sure as my name is not Zaphod Beedlebrock - that I was onto something significant.

THE BINARY SIGNATURE OF THE SIMPLEST KNOT - THE OVERHAND KNOT - IS     42    AKA THE ULTIMATE ANSWER TO THE ULTIMATE QUESTION !!!!

Naturally I immediately started looking at other knot diagrams and working out their binary signatures.  In the following diagram, the cord and the bent cord have no crossings and have a signature of zero.  Make a loop and it has a signature of 2.  Two piled loops has a signature of 56, but with an OI of 3:2 it is so unsaturated that it just falls to pieces a non knot.  But change one little crossing shown in green in the red diagram and the unsaturated pile turns to a fully saturated OH 3:6 with a signature of 42.



I checked the 20 knots in Budworth's Group VI  Unfortunately eight of them were all fully saturated so they scored 101010101010 or 2730 in decimal, however the remainder were all different variations except for a small group - the Reef, Thief, Strap and Phoebe counting in at 2898.

By the time I had got up to the 19 knots in group VIII there were 17 discrete groups  --  almost complete seperation.

How easy is it to get this index number - well, if you have downloaded the FCB4 utility, then if you can draw out the diagram, you can either work it out by hand, or email me the cypher file and I will send the result back to you.  Alternatively, in a couple of weeks I should have managed to write the code to do the calculations automatically, and then it will be as simple as drawing the diagram.

Here is one I did earlier - the Sliding Grip Hitch  - OI 12:19  binary index  10,048,725



Imagine one day, when the database is built, draw the diagram and the utility could link you to the wiki database and take you directly to the page that features the knot(s) with that binary signature.

"One day my son, meat will come in little boxes"  --  and  "one day you will just have to draw a knot to find out all about it"

Derek

squarerigger

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Re: Another Step Forward
« Reply #1 on: June 07, 2008, 12:36:09 AM »
Thanks Derek,

If not yet a conclusion, then at least a step along the path - great!  I loved the reference to Zaphod B - would that life were really that easy!  Incredible! ;D

SR

Dan_Lehman

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Re: Another Step Forward
« Reply #2 on: June 08, 2008, 05:47:17 AM »
I don't know about Beeblebrox (what was it Derek said?),
but let me take Marvin's position on this latest message about
the good ol' "OI Index":  Oh, no, not another one.   ::)

Lemme ask:  will topologically equal knots have the same OI Index?
Well, no, they can't or you'd have 0 for the TIB (tiable in-the bight) knots,
and that's no fun.

But this necessary flattening--taking to 2 dimensions--of knots still
strikes me as an arbitrariness likely to lose the entity's uniqueness.
E.g., I can manipulate (with some increasing reliability, now, at long
last after getting a clue or two) a "Fig.9" knot (the near-Stevedore)
into THREE forms, each able to survive in rope/cord to my heavy
loading (and, I'd think, to rupture--but I don't go there).  So, they are
certainly distinct and durable in this distinctness (useful?, you might
ask, wellllll ... ).  But I must wonder at how this OI-ing would treat them.
(#525 is one, and the near-Fig.8/Stevedore is another which should be
obvious to form; the 3rd form will take an artful sketch transmitted by me).

But, hey, so far we've not enlisted Godel numbers!

 :)

DerekSmith

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Re: Another Step Forward
« Reply #3 on: June 08, 2008, 12:10:54 PM »

Lemme ask:  will topologically equal knots have the same OI Index?
Well, no, they can't or you'd have 0 for the TIB (tiable in-the bight) knots,
and that's no fun.


Hi Dan,

My nickname is 'The Dot Counter' because of my fascination with the details of a problem - I think we may have found yours - 'Marvin' - because of your encyclopaedic knowledge of knots (and much much more).

So Marvin - what happens when we tie the simple OH knot in the bight?  The answer is quite inciteful.

First, the crossings goes up by the square of the number of cords, so 3 crossings become 12 crossings, but the saturation remains the same because it is the same knot, however it looses one degree because the bight causes the cord to make two consecutive unders (overs) as the cord leaves, then re-enters the knot so OI 12:11



As for the Binary index.  Well, a three crossing knot is described by a six digit binary number, but a 12 crossing knot requires a 24 digit binary number to record its overs and unders which makes it a big number for such a little (simple) knot.. The BI for the OH in the bight is 110011001100001100110011 (trace it out for yourself)  --  in decimal notation that's 13419315.  It's a big number, (and for me unmemorable, particularly in contrast with the OH index of 42 which I shall probably remember long into senility).  But what does that matter, it's only an index and some of the URL's which we use to navigate the internet are hugely cumbersome, however, they work, especially when they remain largely within the internet (i.e. transparent(UK) to us).

So to the Marvinesque quip 'Oh No, not ANOTHER overs index' I would have to respond, No, not another OI, but a more detailed extraction of discrete information from the existing OI.  And that of course means that it is subject to all the criticisms quite rightly laid against the original versions of the OI and its precursor the Crossings Index.

In 'Knots and Crime', Budworth acknowledged "It is possible to be misled over the precise number of crossing points a knot has.  Try to arrive at the lowest possible number - extra and unnecessary ones can be created by accidental rearrangements of the knot, or simply by miscounting - but be prepared to search knot groups one or two crossing points more or less than the knot under consideration.".  You have also added to this the observation that one given diagram can be dressed into a number of final knot forms, all of which would be represented by the same diagram and therefore the same index.

This system is still flawed and far from perfect, but with today's tools we can to some large extent mitigate against those flaws.  For example - knowing that a knot can be 'laid out' in three popular configurations, the BI's of all three can be pointed to the same knot information page - much the way that a website owner will point  a number of similar url's at their website to help folks who have mistyped the url or search phrase.

To do this though, the project will need the skills of someone like yourself to identify the probable variants needed for inclusion.

I believe the world of knotting needs a Library and simple indexing system and that the Wiki in conjunction with the OI and the BI offer an imperfect route to create this.  I also believe that these tools are all we are likely to have access to until technology brings forth a 3D camera able to look inside a knot to reveal its dressed structure alongside 3D graphical search technology.  I look forward to the day that technology arrives, but until then I hope we can organise our world and learn a lot using the simplistic tools of two and a half dimensional diagrams, the OI and the BI to help us make the call -

'I'll name that knot in one'.

However, even with these tools, it is still a job requiring a team effort to pull it off, and for that, as Geoffrey indicated, the Guild members are perfect candidates.

We have 'The dot counter' to help create the tools.
We might have 'Marvin' to map out the obstacles.
We are still going to need a 'Co-ordinator', a 'Librarian' and numerous 'Draftsmen'.

Anyone tempted?

Derek

DaveRoot

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Re: Another Step Forward
« Reply #4 on: June 27, 2008, 06:01:41 PM »
 
When I first came to the IGKT Website, I expected to find a complete database of all known rational knots and a means of filtering through them to find the knot which had piqued my interest.

Here's another idea for filtering through knots, which might be intuitive for most people.

Imagine a "Knot Finder" website with several high-level categories, such as:

   Fixed Loops (with a representative image, e.g. Bowline and/or Spanish Bowline)
   Hitches (with a representative image, e.g. Clove Hitch and/or Two Half Hitches)
   Bends (with a representative image)
   (etc.)
 
The user can click on the "Fixed Loops" category (for example) to drill down to the subcategories, such as:

   End-Line (with a representative image)
   Mid-Line (with a representative image)

Then the user clicks the "End-Line" category (for example) to drill down to the subcategories, such as:

   Single Loops (with a representative image)
   Double Loops (with a representative image)
   Multi Loops (with a representative image)

Say the user clicks the "Single Loops" category.  Perhaps there might be subcategories such as:

   Bowlines (with a representative image)
   Other (with a representative image)

If the user clicks the "Other" category (for example), this will bring up a page with a list of non-Bowline single-loop end-line knots, with a picture of each knot (similar to http://www.layhands.com/Knots/Knots_KnotsIndex.htm#SingleLoops).  Clicking the Figure-Eight picture (for example) will bring up a page with information about the knot, such as front views and back views and exploded views of the knot, explanations of how to tie it (and its variations), various names for the knot, the pro's and con's of the knot for various applications, the Overs Index information, and so on.

One of the benefits of this approach is that people without a stitch of knot-knowledge can compare the pictures in the Knot Finder with the knot that they're holding in their hand, and be able to find out about the knot.  Another benefit is its simplicity, making it easy to develop.

Instead of a numeric value for indexing the knot (or in addition to a numeric value), an indexing value can be derived from the knot's "path," such as: Loop-End-Single-Other-FigureEight, for example.

That's the bare-bones germ of an idea, in case it seems to have potential.

Dave

squarerigger

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Re: Another Step Forward
« Reply #5 on: June 28, 2008, 02:27:53 AM »
Brilliant!  And ideally suited to a forum like this one.  Now, who can get it started and is there enough space....

SR

DerekSmith

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Re: Another Step Forward
« Reply #6 on: June 28, 2008, 10:25:00 AM »
Hi Dave,

So good to hear from you again after so long.  I hope you and your family are well.

I agree totally with your suggestion, because the problem will not, I believe, be solved by one technique alone.  To achieve this goal, we will almost certainly have to utilise some sort of amalgam of techniques.

Tell me though, do you think your proposed approach might be amenable to a decision tree style of approach?

Is the knot tied to something?         If it is branch to the hitches and continue with questions and diagrams to further classify.
Is the knot tied with two cords?      If it is branch to the bends etc.
Is it tied in the middle of a cord?      etc.
Does it have a loop or loops?           etc.

If you can think up the questions and the structure of the decision tree, then we could easily build a prototype into the wiki pages.

Fancy giving it a try?

Derek

DerekSmith

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Re: Another Step Forward
« Reply #7 on: June 28, 2008, 04:38:34 PM »
Dan's concerns about topology and the fact that the deformation of a knot needed to 'lay it out' in readiness for diagramming making it into a different knot.

Taking those concerns and then taking another step forward.

Budworth with the Crossings Index, and then latterly Root and Smith with the Overs Index, all extolled the importance of reducing a knot to its simplest (i.e. least crossings) form in order to establish some uniformity in counting the crossings.  Others, including Dan, warned that this process destroyed the structural elements of a knot deemed necessary to be created during the knot dressing.

Naturally, Dan is right, and I must now conclude that the 'Rationalisation' of the knot was WRONG.  And, as it turns out, it was also totally unnecessary.

I was recently proof reading the PACI 'Knot Study Guide'  When I was reminded yet again of the importance of the correct dressing structure for the Fig. 8 loop knot (I don't know how many times I need reminding of something before it finally gels into an 'Oh YES' moment - but this was one of them.)

Up until now, the Fig 8 loopknot has been depicted like this ;



With an Overs Index of 16:15 --  BUT  - it is NOT the Fig 8 loopknot used in climbing applications.

So, I loosened slightly the correctly dressed Fig 8 Loop Knot, sufficiently to see the path of the cords but keeping exactly the dressed structure and drew it out again.



Now it is a much more complex knot -- OI 20:15, yet almost intuitively, we can see that it is a more truethful representation of the real knot.  If we tied this structure, all we would have to do would be to draw it up to make the final knot, keeping all the cords in the same relative positions to create the correctly dressed Fig 8 loopknot.

Now when you click through to the Library page it goes straight to details of the correctly dressed Fig 8 loop knot.

You might have noticed the word 'Find' in the above image, just after the long Binary Signature string.  The FCB program now allows you to draw the diagram, then click 'Find' and if a page with the same Binary Signature exists in the Knot Library Wiki, it is loaded immediately for you in order to be able to compare its details with the knot you have tried to find.

The program is not quite ready for release because it only works with single cord knots at the moment and will have to be able to cope with two cord knots before it is ready for use.

But essentially, here we have it  -- See it, draw it, click it, go straight to its library page.



"One day, meat will come in little boxes"

Derek

DaveRoot

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Re: Another Step Forward
« Reply #8 on: June 30, 2008, 06:01:33 PM »
So good to hear from you again after so long.  I hope you and your family are well.

Thanks, we're doing well!


Tell me though, do you think your proposed approach might be amenable to a decision tree style of approach?

Is the knot tied to something?         If it is branch to the hitches and continue with questions and diagrams to further classify.
Is the knot tied with two cords?      If it is branch to the bends etc.
Is it tied in the middle of a cord?      etc.
Does it have a loop or loops?           etc.

If you can think up the questions and the structure of the decision tree, then we could easily build a prototype into the wiki pages.

When I visit the support pages of various businesses, some of them ask questions in order to "help" me narrow down my search (e.g. http://support.skype.com/?_a=knowledgebase).  However, I find that it takes some thought to read through the questions and determine which one fits my needs.

In contrast, if we have just a few top-level categories (Loops, Hitches, Bends, etc.) with some well-chosen pictures next to each category (e.g. showing a Clove Hitch and a Two Half Hitches, which are different ways of tying to an object), then the Knot Finder page is likely to be cleaner and easier to navigate.  That would be my preference, but others might prefer the questions approach b/c some people are more visual or more auditory, etc.

Maybe someone can put together a few prototypes with different navigation schemes so that we can test out the look and feel of different approaches.

Dave


« Last Edit: June 30, 2008, 06:02:48 PM by DaveRoot »

DerekSmith

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Re: Another Step Forward
« Reply #9 on: July 15, 2008, 01:51:50 PM »
Well, it has been a slog, but now version 4.2 is available for download from here.

https://knotcyphers.pbwiki.com/The+FCB+Cypher

It calculates the OI and the binary Signatures for both one cord and two cord knots.

There is just one requirement and that is the cord depicted in cord #1 MUST have a working end in order to trigger the OI and the Binary Sig functions.  This means that knots TIB must be assigned a temporary WE in order to have the utility calculate the signatures.

So please give it a try.  Download it, it does not need to be installed it does not register itself into your registry and it does not load any dll's which could interfere with other programs on your PC.  Just download it and double click the icon to run it.

You can use it to diagram out knots and having created a diagram, just click on the 'Find' button to go straight to The Knot Library to see if a page exists yet for the diagram you have created.  If it has not been entered into the library yet, then please save the diagram and post the .cyp file on this forum so that other members can look at the knot and so that it can be recorded into the Library.

Your comments on the utility would be most appreciated.

Derek

DaveRoot

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Re: Another Step Forward
« Reply #10 on: July 15, 2008, 08:27:57 PM »
Derek,

Nice utility for drawing knots!  Making a knot diagram was quick and simple....the hardest part was laying out my rope so that the knot only consisted of orthogonal lines!  ;D

I have been playing with the Grapple Hitch, so I used it as my test case with FCB4.  I used different colors in different sections of the knot in order to help visualize the knot, and I made two different diagrams to visualize the knot in different ways (I decided that I wanted to keep the Standing Part straight, which is why I made a second diagram).  Here's the file contents:

-----------------------------
FCB Version 4.x


0:0
 daj lae maj dbf mbf dcf mcf ddh mdh dek fej ffh mfh dgi egg fgl mgh dio mio
 cai kai cbr kbr ccf kcf cdh kdh cef eeh kef lei meg nel cgf kgf lgk ngj chk dhg ehl khk lhe mhg nhl
 bdi edj jdi lde ndj bef jef bfk cfg dfe efg gfa jfk kfg lfg nfa
10
0
-----------------------------


Some thoughts for future releases:

1. It would be helpful if the active tile in the pallet is highlighted.

2. An Undo feature would be handy.

3. At first I didn't see how to delete a tile in the drawing area.  Maybe a blank pallet tile (or a right-click menu with a Delete option) would be more intuitive.

4. It would be very helpful to be able to highlight one or more tiles in the drawing area (e.g. CTRL+click) and move them left/right/up/down in order to make more room!

5. When we can highlight multiple tiles in the drawing area, it would be nice to have the ability to Copy/Paste those tiles in another place, especially if we can then rotate them or mirror-image them.

6. Instead of placing the cursor in the "Draw cord #2" text box to change colors, it would be more intuitive to have something like radio buttons ("Blue", "Red", "Black," etc.) for changing colors.

7. For me, the dotted-line tiles make it easy to visualize that the rope is behind a spar.  Similarly, I suspect that dotted-line tiles would make it easy to visualize that the rope is behind another section of rope.  The current "crossing" tiles cause me to do some extra mental processing since they look different than the "spar crossing" tiles.

8. Sometimes it might be useful if multiple vertical spars (or multiple horizontal spars) can be placed in the drawing area.

9. File/New should remove all of the spars in the drawing area.

10. When we close the application, it would nice to be prompted if we have made some modifications which we haven't yet saved.

11. It would be useful to be able to write comments in the drawing area (and be able to draw an arrow from a comment to the particular section of the knot).

12. To help make the knots easier to visualize in FCB, what if you use actual photographs of rope for your pallet tiles instead of line drawings?!  ;D


Again, nice work!

Dave
 

DaveRoot

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Re: Another Step Forward
« Reply #11 on: July 15, 2008, 09:14:18 PM »
I wonder if any of these "3D rotating cube" applets can be tweaked to make 3D rotating knots?   8)

http://www.google.com/search?q=3d+rotating+cube&hl=en&sa=2

Dave

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Re: Another Step Forward
« Reply #12 on: July 16, 2008, 05:25:03 AM »
Derek:

Your program appears to work well under Linux (using Wine).

Regards,
Snowman

DerekSmith

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Re: Another Step Forward
« Reply #13 on: July 16, 2008, 03:03:44 PM »
Hi Dave,

Thank you for such a thorough review of the utility and for such an excellent list of 'ToDo's'.  A couple - the highlight, copy paste and drag/drop ideas were already on the ToDo list, I just don't know how to do them yet (very much a case of learning as I go).

No.9 - remove old spars when New is initiated is a bug - well spotted.  I have corrected the fault straight away.

The other ideas are all excellent and will go into the ToDo list for action.  Do you have any preference for order? i.e. if you could have one but none of the others which would it be sort of approach.

Re No. 12 - photo's instead of drawings.  'A long time ago' (seems like it at least) when the utility was very much at the stage of trying to make a cheap (i.e. FREE) alternative to Frank Browns use of a top end CAD program to make diagrams, I started by using literally small .gif images of segments of cords.  Although it was simple enough, it consumed large amounts of memory and meant that there was no way of progressing to multi cord diagrams without the need for an army of image tiles, so I dropped the image route and moved to representing the cords by lines.  There are advantage to be had by using the lines approach that have not been developed yet but which are very much on the books for future development.

The first of these is that by using line drawings for the cord we can make the thickness of the cord change, so we can 'bulk up' the cord to better 'see' the knot.  Further in the future, there is the opportunity of making the cord displace itself as it is made thicker, so the knot effectively 'dresses' itself, then we can reduce down the cord thickness again to allow us to see into the structure of the dressed knot.  This of course will be when we are dealing with 3D diagrams and are no longer limited to the restrictive four faces of the square that we are constrained within by today's version of FCB.  I have on the drawing board for version five the outline for using the cube instead of the square, the cube will  give us direct access to 3D, albeit still very restricted to the simplistic 'any direction so long as it is at right angles' type of approach.

Also, it was the move to a line diagram that led me to the perception of how to establish the essence of the 'path' of the cord through the knot and from that the ability to create the algorithms for the Binary Signature and the Overs Index.  Again, by retaining the cord as a line concept, we are also allowing ourselves access in the future to 3D generation of the knot image with full rendering of the cord surface texture and curvature.  this is way beyond my abilities today, but hopefully we will be able to attract someone with 3D skill to the project if it ever gains sufficient credibility.

The other nice potential of staying with a 'line based model' is that we can potentially drop the tile pallet all together and simply click through from face to face and have the utility 'join the dots' for us.  Again, this is on the drawing board for version 5.

Strangely enough, I am beginning to find that the restriction of drawing out knots in this two dimensional manner is actually able to bring insight to various structures.  For example, in your two diagrams for the Grapple Hitch, in one the SP is kept straight, and in the other it is displaced out to the side.  These diagrams each represent two forms of the grapple.  One which is little more than a noose and the other which indeed has the ability to hold under load.

I have put the grapple page into the Library now, so if you click the Find button you should go to its page (which needs detail adding to it).

By the way, did you find that if you right click anywhere on the drawing grid, the tile and the cord are copied and can then be used to paint new tiles and this includes the blank tile which can be used for erase?  Also, that the Overs Index and Bin Sig only compute when a working end has been assigned to cord #1?

Thanks for the feedback, without it, this project would be poorer.

Derek

DaveRoot

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Re: Another Step Forward
« Reply #14 on: July 16, 2008, 06:56:48 PM »
By the way, did you find that if you right click anywhere on the drawing grid, the tile and the cord are copied and can then be used to paint new tiles and this includes the blank tile which can be used for erase?

I didn't notice the right-click functionality.  It's handy, so my "blank tile" idea is not urgent (or not needed).  But for first-time users it's not the most intuitive feature.  What if you add a status bar at the bottom of the screen?  When the mouse is in the drawing area then you could place a message in the status bar such as "Right-click to copy a tile."  When the mouse is in the pallet area then you could place a message in the status bar such as "Left-click to select a tile."

By the way, does FCB need a screen resolution greater than 1024x768?  Looks like part of the FCB window is cut off (e.g. the "Fin..." button at the bottom right is cut off).

   
Also, that the Overs Index and Bin Sig only compute when a working end has been assigned to cord #1?

I saw this in the Help file, but in my case I wanted to use different colors for different sections of a single cord.  Since I created two diagrams, I'm assuming that the Overs Index and Bin Sig were not accurate, is that correct?


Updates on my suggestions:

1. It would be helpful if the active tile in the pallet is highlighted.

   Nice to have, but not urgent.  Makes FCB a bit more intuitive for first-time users.

2. An Undo feature would be handy.

   Nice to have, but not urgent.

3. At first I didn't see how to delete a tile in the drawing area.  Maybe a blank pallet tile (or a right-click menu with a Delete option) would be more intuitive.

   This is addressed in my comments above.
   
4. It would be very helpful to be able to highlight one or more tiles in the drawing area (e.g. CTRL+click) and move them left/right/up/down in order to make more room!

   This would be my #2 choice.  If you find a way to highlight multiple tiles then perhaps a set of up/down/left/right buttons can be used for moving those tiles.  The right-click functionality is useful when trying to move tiles, but it's easy to introduce errors into the diagram this way.

5. When we can highlight multiple tiles in the drawing area, it would be nice to have the ability to Copy/Paste those tiles in another place, especially if we can then rotate them or mirror-image them.

   This would be my #3 choice.

6. Instead of placing the cursor in the "Draw cord #2" text box to change colors, it would be more intuitive to have something like radio buttons ("Blue", "Red", "Black," etc.) for changing colors.

   Nice to have, but not urgent.

7. For me, the dotted-line tiles make it easy to visualize that the rope is behind a spar.  Similarly, I suspect that dotted-line tiles would make it easy to visualize that the rope is behind another section of rope.  The current "crossing" tiles cause me to do some extra mental processing since they look different than the "spar crossing" tiles.

   This would be my #1 choice.  I'd like to see this in order to compare it with the current implementation of the "rope crossing" tiles.  This will bring consistency between the "spar crossing" tiles and the "rope crossing" tiles, and I think it will be visually clearer than the current implementation of the "rope crossing" tiles.

8. Sometimes it might be useful if multiple vertical spars (or multiple horizontal spars) can be placed in the drawing area.

   Not urgent.

9. File/New should remove all of the spars in the drawing area.

   You've already addressed this.

10. When we close the application, it would nice to be prompted if we have made some modifications which we haven't yet saved.

   Not urgent.

11. It would be useful to be able to write comments in the drawing area (and be able to draw an arrow from a comment to the particular section of the knot).

   Not urgent.

12. To help make the knots easier to visualize in FCB, what if you use actual photographs of rope for your pallet tiles instead of line drawings?!  

   You've already addressed this with some interesting points in favor of line drawings.

13. Would it be possible to make the drawing area essentially an "infinite plane" with scroll bars?

   This would be my #4 choice.

14. Would it be possible to allow zooming in/out in the drawing area?  I.e. the ability to make the grid squares larger or smaller.  This way we can "zoom out" to get a bird's eye view of the knot, or "zoom in" to see the details of a section of the knot.

   Not urgent.

15. How about the ability to rotate a diagram (or a section of a diagram)?  For example, I might draw a loop knot with the loop at the top, and then decide that I want the loop at the bottom.

   Not urgent.

   
Dave