Jigsaw CAD Math - Jigsaw curve mathematics, Jigsaw curve images, or CAD macros

[Keith Mc]

I need some advice, from any of the graphical math wizards out there.
I am seeking the Curve Math Equations for Jigsaw Puzzle Curves...

Using CAD, I wish to add a jigsaw style "mushroom bump" in
a given line, or curve on my drawing. (In essence, turn
a part of my drawing into a "jigsaw puzzle"... :-) To do that,
I wish to better understand the Curve Equations for Jigsaw Puzzle tabs.

1) Does anyone know of a resource for understanding the various
curve mathematics of "jigsaw puzzle line curves"? (Equations of curve.)
Sure, anything that jams would WORK as an interlock, but there
appears to be a fairly well defined "jigsaw standard" that everyone
seems to use - namely that recognizable mushroom-bubble headed
gentle tab curve. I wish to find the MATHEMATICS of those curves,
which will generate that cool curve family used by jigsaw puzzle makers.

2) Images - Alternatively, does anyone know of any image resource
for a number of DIFFERENT jigsaw "bumps" that can be simply
cut-n-pasted onto a drawing to, create unique jigsaw interlocks?
Puzzle makers seem to simply "tilt" or "offset" the heads on the
"standard tab", or stagger their position, to make unique interlocks.
A simple set of jigsaw bump images may work for me for now,
until I can find the equations.

3) Lastly, does anyone know of any CAD Macros already existing
now for any standard CAD package? IOW, given a selected straight
line or curve segment, hitting the macro will change it INTO a
"Jigsaw-style Line", with one or more of those mushroom "jogs" in
it as parameters. Does such a macro exist now, for any standard CAD,
to make Jigsaw Puzzles out of your image or drawing?

Thanks!
- Keith Mc.

[Keith Mc]

> Keith wrote:
>> I need some advice, from any of the graphical math wizards out there.
>> I am seeking the Curve Math Equations for Jigsaw Puzzle Curves...

This is turning out to be a pretty cool exploration into math... Some progress to report:

Ron Hartikka from Go-Tech and Nate B from i3detroit both pointed at:
> http://forum.make-the-cut.com/discussion/16199/jigsaw-puzzle-generator-beta

Hey Ron & Nate,
That's a pretty cool package, for making puzzles. Thanks for the link!

However, I am not JUST trying to create a puzzle from scratch on a new rectangle.
Instead, I am seeking a fundamental understanding for the math behind the curves,
or a way to add "jigsaw-ish geometry" to *existing* CAD data.

For example - if I wished to cut an already CADded oval into a jigsaw (or one
where the pieces are only, say, a certain area average size or dimension),
without creating any "orphan slice" tiny pieces, how would one go about doing that?

Based on my research so far, there seems to be three distinct phases to this problem:
1) Subdivide the geometry into desired areas of interest (what will become the
pieces). This can probably be done with simple grid overlays, Voronoi Diagrams,
various tessellation algorithms, etc. Given the complexity of the original outer
curves, Voronoi seems to be better than gridding, so you won't get tiny orphan pieces:
... http://en.wikipedia.org/wiki/Voronoi_diagram

2) Perturb the boundary or grid lines to make them mesh better - for example:
convert a basic grid line into a pseudo sine wave curve, so the pieces better
"nest" into each other and won't "slide" around along the lines. This should include
intersecting angle control, so there are no "sharp points" to the pieces to break off.

3) Finally, perturb specific places on the segments by "growing" jigsaw lock
curve "warts" on them (edit replace lines or curves with other curves).
But quantity per segment, position selection, and direction control are
all important, so for example you won't try to add a lock bump or a dimple
on an outer boundary line, or a dimple will not intersect with one on another
edge on the same part (or even come TOO close and create a "thin" piece.).
That's a lot of controls and checks to implement!

The first phase shouldn't be tough. Adding grids within CAD isn't hard,
and I can envision some sort of a Voronoi algorithm where you drop
a cornfield of dots, migrate them around a bit to equidistant them,
then generate the segments. Even doing it manually with line drawing,
you should be able to easily subdivide it into a pleasing, orphan-less grid.

But the next two steps are rather challenging. Changing each of those
grid lines into perturbed curves (eg fit a sine wave-ish spline of controlled
frequency and amplitude over the length of the line) especially while
still controlling the intersection angles, doesn't look easy at all.
Growing jigsaw lock curve "warts" at specified points on each line or
curve don't look tough, but controlling them to insure the pieces as a
whole are still nice and thick everywhere is a whole 'nother ball
game! A number of substeps, checks, and even iterations may be
required to finely control this, to generate the required geometry.

Edward Vielmetti wrote:
> Keith, I was able to find this on designing jigsaw puzzles with Mathematica:
> http://blog.wolfram.com/2012/06/28/designing-jigsaw-puzzles-with-mathematica/
> which describes the knob-and-neck in terms of splines.

Now THAT is pretty darn cool.
Too bad for individuals it's ~$300 plus $150/year for maintenance, or I'd play with that!
But, that does proves there are some interesting math formulas out there for this!
That alone is encouraging...

Here's a cool puzzle animation, generated by playing with a Voronoi Diagram
... http://blog.wolfram.com/data/uploads/2012/06/move.gif
But if you look closely at the generated picture, there are a LOT of "minor bumps"
generated, that aren't good puzzle interlocks. (That is CLOSE, but no cigar...)

But using Mathematica also raises another issue - How can you take its output,
and tie it into existing CAD/CAM data?

Image generation files, and CAD files are two different worlds. You often don't
have a lot of deep control access of CAD internal files, to convert a drawn
image into CAD lines and curves, without overdrawing it manually.
That's not a good solution for this complex of a drawing.
BTW, I spent several YEARS full time doing just those kind of internal database
modification of CAD file systems for industry, just to invent Automatic Clearance
Detection for Mechanical Simulation Systems (ACD/MSS). It was NOT a simple
thing to do. It also required a LOT of cooperation from all of the CAD/CAM
vendors, to tweak their internal file systems and add additional database hooks
to get the level of access to the database that I required.

Better ways would be to either find an import method, or generate drawing macro(s)
to make the CAD system itself make the required database edits in its own language.

Other Progress - Some interesting links I've found:
... http://mathematica.stackexchange.com/questions/6706/how-can-i-calculate-a-jigsaw-puzzle-cut-path

The Shmuzzle Puzzles - REALLY cool... (I must keep them in mind for presents! :-)
... http://shmuzzles.com/shmuzzle_resources.htm
... http://shmuzzles.com/formula.htm

I also played around with a Bezier curve generator. I was able to create something
that looked like a darn good Jigsaw bump curve. But it needed at LEAST 10
control points, with several strongly crossed at insane distances to do
the "waist cinch" on the bulge.

On one hand, this implies that trying to do it with pure Bezier, Spline, or
NURB curve fitting IS possible! BUT... it may also require a pretty
sophisticated macro to pull it off. You'd have to first divvy up line segments
into a LOT of control points, migrate the control points around in just
the right way relative to the original line for a curve fitting, etc.
That's an awful lot of work, for a hobby effort.

Bottom line - I still believe the goal should still be to create some kind
of CAD macros. But to prove whether this will be EASY or not,
I must first better understand the math behind the jigsaw curves.
Hence this thread, to give me a better grounding on the math required,
(vs just finding a "make your own jigsaw puzzle" standalone package,
that only uses rectangles, or rounded corner rectangles).

Anyone else have any good ideas on how to "jigsaw-ify" already
created CAD geometry, from WITHIN the CAD software?
THAT would allow me to then walk it through the CAD/CAM tool chain,
and let the CAM system fab the pieces for me.

Cool thoughts all. Please keep them up!
- Keith Mc.