Here is a different script from the previous post.

The BraidedBands script creates knots that are commonly known as "Turk's-Head knots"
References with more information:
http://www.mi.sanu.ac.rs/vismath/pennock1/index.html
http://www.mi.sanu.ac.rs/vismath/pennock/index.html

Cylindrical equations are used to model two types, Disk, and Cylindrical (Tubular), which may be physically manipulated into each other.
Definitions:
A Bight is a scallop or curve of the rope or cord, at the rim of the knot,
where the cord changes direction. The count of the Bights can be made at
the top rim, or the bottom rim of a cylinder, or the outer or inner rim of a disk.
Bights is a column count, or a radial column count, and is >= 2.

A Lead is the number of revolutions the cord makes around the center of the disk or cylinder, for one Ply only.
The Lead is a ""row count."" Leads is >= 2, for a knot, or no knot will occur, or an unknot.
Per the reference, if a line is drawn from the origin, the number of times the curve crosses the line, is the Lead,
provided Bight and Lead have no Common Divisors.

The Ply count is the number of "parallel" strands of the single cord in the knot.

The GCD (Greatest Common Divisor) of (Bights, Leads) must be equal to 1, or multiple cords will be needed for the knot.

The mathematical knot is a closed curve, and can be a pattern for a physically braided knot.
The physically braided knot is one rope or cord, woven in "parallel" Ply's, with two ends.

Cylindrical coordinates are converted to Cartesian coordinates.

Pipe is handy.
I would like to reconcile these knot formulas with the previous rope mat formulas.

- Brian

Brian,

Another non-virtual product of another of your scripts: http://www.thingiverse.com/thing:1955972

Mike.

Linearizing a Baided Band.

The Braided Band script produces a class of knots, commonly known as "Turk's head knots", (There is no cultural stigma intended.) There is a "disk" version, and a "cylindrical" version, which are topologically the same, as a flexible physical model can be manipulated to either form. The inside center edge and outside edge of the disk version are equivalent to the top and bottom edge of the cylinder version. The disk version can be thin, and the cylindrical version can be thin walled. See the link to Skip Pennock's paper, for detailed descriptions.
http://www.mi.sanu.ac.rs/vismath/pennock1/index.html
Leads are the number of trips around a cicle. (= 2*PI*Leads)

The Braided Bands are defined parametrically in cylindrical coordinates.
It comes to mind that a Braided Band could be unrolled, to give a straight braid or weave. Better yet, rather than circularize the cylindrical coordinates and then unroll them, just present the cylindrical coordinates as linearized. This was accomplished using the equations for the cylindrical version, in the BraidedBands3 script.
z coordinates are the same.
y coordinates are the thickness.
x coordinates are the arcLength, MidrangeRadius * theta2.
For a 7 Bight, 4 Lead Linear Band, the result is ONE curve, with sine and cosine bends, rather than 4 curves. There are 21 cycles.
For relatively prime Bights and Leads, The number of cycles is q * ( p-1 ). = Bights * ( Leads - 1 ).
The curve can be divided into 4 pieces.
Place a point at the start of the curve, and ArrayCurve the point on the curve, using a value of 5. (5 = (Leads+1)).
Trim the curve with these points, resulting in 4 curves.
Move the 3 curves on the right, to the left, back to the origin.
The result is a 4 curve weave, which can be swept with a circle, usually the
diameter of the circle to use is the cord diameter.
An end view of the curves looks like a "race-track".
This could be scripted, or the 4 curves formed initially at 2PI intervals, either by script, or by nodeeditor.
BraidedBands3Alpha is a quick kludge modification. The menu should be changed to have a Build menu. For Disk, checkbox Disk. For Cyl, do not checbox Disk, nor Straight. For Straight, checkbox Straight, but NOT Disk.

The weave unit can be chained, end to end.
The chain should be flowable to some other curve, kind of like chain script.

A circle sweep bug was found. The bug was the same in V4 and V5Beta.
For some weaves, a circle will not sweep two selected curves correctly. A circle will sweep ONE selected curve at a time.
Some other weaves sweep multiply selected curves jus fine.

Multiple Ply's should be do-able, with more scripting.



Here is the "race-track" end view:



- Brian

Edit, Version 3 alpha deleted. See Version 4 in a later post.

A BraidedBands script with a Build menu has been created.
Getting the unrolled type with pLeads curves is being difficult.

Anyway, there is some discrepancy in the formulas??? re division by pLeads???

- Brian

Success with unwrap of Tube.
Division by (pLeads-1) was the key.

Version BraidedBands4 almost done.

- Brian

Version BraidedBands4 should have all 6 Build types correct.
The different Build types required some variations of the code.

The unwrapCyl and unwrapDisk results are the same as each other, just rotated 90 degrees. (Produces pLeads number of woven curves.)

The two Linear versions are the same as each other, just rotated 90 degrees. (Each version only produces one long wavy curve.)

More than one Ply could be done by re-running script with different, appropriate, Midline Radius.

Sweep can be done manually.

- Brian

A chaining script could be done?
A node version?

Hi Brian,

You left some ambiguous questions on your post. Is this something your are working on, or would like someone else to forge onward from your prior work? I would be interested in creating a node project in collaboration with you, if you are interested.

Wayne

Hi Wayne,

I'll be very busy at work for the next few months, so have not done anything else.
So anything that you want to do is "fair game" :-)

The last thing I was looking into was an end portion of a braid, with a 50% reduction in taper, like the end of a hair braid.
So basically, the scale variable "a", in the braid equation, would not be constant, but would have a "taper" value.

My unfinished, preliminary notes:

Initial problem: Taper a braid by 50%, using arithmetic progression in nodeeditor.

The following link explains taper well, with drawing and formulas:
https://www.mechical.com/2021/02/types-of-taper.html
(They should have put D-d in brackets (D-d))

D is the beginning diameter of the object to be tapered.
d is the ending diameter of the object.
L is the length over which the taper occurs.
K is the Taper Ratio.
DraftAngle is the taper angle. (One side in 2D).
The taper is constant. taper is concerned with diameters.
The +/- sign is + for now.
The taper could be taper up or taper down in diameter size.

In MoI Extrude taper mode, the taper angle is represented by the Draft Angle. (recent James node...)

Taper ratio = K = ( D - d ) / L

theta = DraftAngle = arctan(( D - d ) / 2*L)
The 2 divisor is because the angle is measured on one "side" only.
The taper "revolves" around the axis of symmetry.
*****

using _ for subscript:

The braid equation uses a_i for scale values.
For a reducing braid cross section, (~ diameter),
the scale decreases, and the decrease will be at a constant rate.


The arithmetic progression formula, , is:
a_n = a_1 + (n-1) * factor.
Let a_n = d
Let a_1 = D
For integer values of L: Let n = L+1
Let factor = K = taper ratio, adjusted by step size??? (+/-)
THE ABOVE SECTION IS NOT FINALIZED... (Might be in error, or need modifications?)
(I started to get confused by Integer values, versus ? values...

- Brian

Hi Brian,

That is quite a laundry list of potential improvements. Reducing the braid taper like a hair braid is something to figure out.
Perhaps using Michael's pipe script for a tapered braid.

http://moi3d.com/forum/index.php?webtag=MOI&msg=1463.48

This should keep me busy for a while.

Thank you,

Wayne

Yes to tapering diameter of each tube, but also progressively tightening up the interleave of the strands.

- Brian