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
