Thanks :-)

Hi James,

Is the TriGrid node (or macro?) available?

- Brian

Brian,

Please find attached.

It a macro: one of the first I made, & so consequently not well resolved.
The problem with it is that the individual triangles are not easily addressable.
I've been planning to redo, but have never gotten 'round to it.

James
https://www.instagram.com/nodeology/

PS Brian do you have Instagram. I was looking to link you on limaBean animation post

Thank you James.

I do not have Instagram, but will consider signing up.

- Brian

How to convert a 2D limaBean curve into a 6_3_1 link knot, in MoI

MoI V5 beta dark theme was used, but MoI V4 will work as well.
The dark theme is easier on the eyes, IMHO.

The knotplot drawing of a 6_3_1 link knot is here.
https://www.scirp.org/pdf/am_2020060915362814.pdf
The link knot can be magnified and printed, for a visual reference.
It is not really necessary to load it as a MoI image.

The 2D limaBean curve comes from the nodeeditor limaBean node, and
is a good approximation of one of the 3 links. Segments of the curve provide the curves needed for the 4 blends used.

The first steps are to find the axis of symmetry, and initial center point on said axis, and perform circular array 3, to match the visual reference. This involves trial and error to locate a center point which causes the circular array to have a nice overlap, matching the visual reference, in the Top frame. Note that only one limaBean curve must be modified. The other two curves are equivalent. Save the resulting centerpoint. In the 2D Top view, select a separation distance between two links, to avoid future pipe tube overlap. The "eye" hole at top center of the link-knot provides an appropriate number. Half of this separationn is the Amplitude +A value.

To convert to 3D, it is necessary to add z-axis offsets.
MoI Blend will be used to accomplish this.
To understand how this is done, it is beneficial to use the analogy of using cosine curves to modify the z values. (Variable wavelength cosine curves could have been used, but that would involve quite a bit of math. Blend is easier to use.) (Z values of the ultimate Blend curve resemble a cosine wave. )
Examining one limaBean curve, and how it crosses over the other two curves, running clockwise, there is a peak Z value of amplitude A, (z=A), then z=0 then z=-A, then z=0, then z=A, then z=0, then z=-A, then z=0, then back to the start z=A. This is two wavelengths of cosine wave, 4PI. At +A and -A, the tangent is zero, so only 4 Blends will be needed. Blending will take place between approprtiate curve segments, from +A to -A, and from -A to +A, repeat. Trim the limaBean curve at the 4 crossing points.
I. Copy the longest (last) curve to z=+A.
II. Copy the second curve to z=-A.
III. Copy the third curve to z = +A.
IV. Copy the longest curve (last) curve to z=-A.

Locate the Midpoint of the 4 trim curves, and place location points there, to assist with Bulge selection.

Tangent Blends may be used. (Did not try other varieties of Blend.)
Blend tail end of curve I. to start end of curve II.
In Top View, adjust the Bulge to be very close to the first Midpoint.
Use the slider, and fine tune with decimal numbers.

Blend tail end of first Blend, to start end of curve III.
In Top View, adjust the Bulge to be very close to the second Midpoint.

Blend tail end of second Blend, to start end of curve IV.
In Top View, adjust the Bulge to be very close to the third Midpoint.

Blend tail end of third Blend, to start end of first Blend.
In Top View, adjust the Bulge to be very close to the fourth Midpoint.

Join the 4 Blend curves. The resulting 3D NURBS curve shows very efficient number of ShowPoints. The old limaBean curves and curve segments may be deleted. The new link may be circular arrayed, with the previously saved center point.
Pipe, with an appropriate value for outer diameter, may be used.

Well, I think that I got the steps correct. Common sense is helpful.
A .3dm of the resulting curves is attached. Pipe may be applied. I used 0.15 units for the outer radius.

Brian

Here is a picture of the curve segments, blends, and important points.
It is after the fact, after the Blends are done. The midpoints are different from the initial midpoints, but are nearly essentially the same, very nearly. (Two of these final midpoints are slightly off of z=0.)
(Well, the segment labels are for the (deleted) 2D planar segments, but look, very nearly, the same as the Blends in 2D, Top View)
(In this 2D view, everything looks planar, but does have z offsets.) (Initial segments Elevation to z=+A, or Lowering to -A, could be discernable in a side view.)

Blend with Bulge passing through a specific point would be handy. Some sort of optimization routine, to narrow down the Bulge decimals, to nearly match the point?
A little trial and error is easy enough.
A second stage fine-tuning slider might be nice? To capture more decimals. Display could show more decimals?

So that is the algorithm. I wonder how hard it would be to script it?

- Brian

Link Knot 6_3_2 took only a few minutes to add z offsets with Blend.
Used G2 Blend on longest segment, G1 on the first 3.

Apply Pipe with radius about 1.8 to thicken lines.
Change colors per taste.

From a 3D slant view, the rings look partly square.

- Brian

The 8_18 knot, a single curve.

A pipe outer radius of 1.5 units is suggested.

- Brian

Here is the 12 crossing knot, re: SVG post, slightly modified. A knot that preserves the crossings, etc, is topologically equivalent. Used the amplitude blend method.
Pipe2 used also.

A crude attempt at 3 dragon pendant, with a knot modification. Pipe2, Taper from about 4 at head, to 0.5 at tail. Blendcap, G3, Head Bulge about 4, Tail Bulge about 5.

Note that Blendcap slider appears as the Blend slider, since Blendcap uses Blend. (Used a Bulge slider mod.)
Note that the Bulge number is not restricted to the slider range. The Bulge number can be manually entered, and can be out of the slider range.

Have yet to make mouth, teeth, ears, eyes, nose, fins/wings, etc.

The heads could start at different points of the knot. The tail end was just roughtly point dragged in Top view.

- Brian

Hi Brian,

Very good information. A good source for a future project in the queue.

Thank you,
Wayne

Here is a node for a Lemniscate curve.

Also, after puzzling over how to use an arithmetic progression to duplicate the "t" parameter for parametric equations, as is often done in scripts,
incorporated in the Lemniscate node is a nodeeditor Macro, to make fractional "t" values, from 0 to 2PI.
First tried with sinewave node.

From the loaded Lemniscate node in the nodeeditor canvas,
the Macro can be right clicked, and Exported to the Appdate>nodeeditor>macros directory, for future use.

Thought I'd apply an "amplitude" Blend to the center line of the "infinity" symbol (FatLines) Lemniscate.

- Brian

I may try to modify (Max's?) FatLines script, with a separate line for the slider.

The BeanSlider is a MoI type of script, not a node. (.js and .htm)
Selecting the Blue question mark has some documentation, including creating other shapes, with numpoints set to 3, 4, 5 or 6, then running MoI command Shoe pts, and dragging them.

In 3D view, Dragging a control point to the Z access seems to require an existing geometry point there, (or e.g. line segment).


Motivation was using the shape for a Template cut away, but using offset of an arc, adding a tangent line segment, and then using Blend, produced a nice cut out completing a Scroll tracing template.

- Brian

The script can be placed in CustomUI, ...%appdata%...>commands folder. (Windows 10 or 11.

- Brian

ps, the old LimaBean.js node file seems to do fine if added to NodeEditorPlus, extensions folder. (Curves2 type of nodes.)

This script is for a "CashewNut" profile curve.

https://math.stackexchange.com/questions/4642098/parametric-curve-resembling-a-bean

It is supposed to be some sort of conformal mapping?

Control point dragging can also be done for low numpoints versions.

I will try the Bean curve from Arc curves next.

- Brian

Here is another "Bean" curve, created with 4 arcs.
The link was given previously, and is also in the .js file.
The math is simple, yet fairly intricate.
The x and y coordinates (lengths) are based upon a right triangle, with the other two angles at 45 degrees.
Radical r = square root of 2.
If the two legs of the triangle are 1, the hypotenuse is r.
If the two legs are r, the hypotenuse is 2.
Pointpicker is used to pick local origin in desired View.
This version moved the cPlane origin from point "O" to point S, which is on the largest arc.
This required adjusting the math for each point.
Point S is offset from point O.
3 point arc factory was used, because center arc factory does not always draw the arc on the desired side.
The center arc factory seems to work "counter clockwise", but as the points move around, some arcs may be on the "wrong" side.
Coordinate Frames, and Local coordinates versus World coordinates are now slightly less confusing.
I was thinking of doing an essay on Coordinate Frames. MoI5beta API, and numerous past posts on the forum were helpful.
Help from Michael from past years was utilized, and Max script techniques were also helpful.
Added some reset pushbuttons.
This version may be considered an "alpha".
It seems to work in MoI4 and MoI5beta.
There is brief documentation by using the Blue ? button.

- Brian

Added an angle rotate parameter and slider to the Bean3ptArc script. (Bean3ptArcR)

Description of scroll spiral template

The scroll TEMPLATE has a red spiral, which has been cut into segments with eraser script. These gaps represent support TABS for the template. (The red curved segments are not quite arcs, but close.) The six blue regions approximate cutouts which enable pencil tracing of the scroll template, onto a block of wood. The desire is to replace the six blue cutouts with more precise curves. On the top right is a Bean arc curve, rotated and sized, into position. (An angle rotate slider was added to the BeanArc script.)
( Script Sliders do not do a smooth or good enough placement. A bit crude. ) On the top left is the remains of a bean curve, cut out by the cyan circle. It matches a portion of the blue curve fairly well. The black piece of bean curve is a much better match than an offset of the red curve. The two green line segments are perpendicular to the scroll spiral segment, and are Blended with the black bean segment, forming the orange curves. The orange/green corner could be filleted a little. A join of the red, orange, black, and orange segments, is one of the desired cutouts.



- Brian