SplinesFeb 24, 2025
Sometimes you get the most unexpected design ideas from the most unexpected source.

Here, I took the top portion of the #braid strand sliced horizontally in https://pixelfed.social/p/Splines/798252244743520392 and extracted the surface borders of each "bean" shaped segment of the strand.

The extracted curves shown in the top portion of the diagram look like 4 pairs of tiny bathroom slippers, each with a single strap near the front where the toes would be.

Fun activity for the entire family: Use #uniformScaling or #nonuniformScaling to fit every feet and be creative with the straps.
Splines (@[email protected])

#Braids #3StrandBraids After we #sweepOneRail with the blue #sweepingCurve on the orange #railCurve for the #braid strand, we cap #planarHoles to get a single #airtight strand. As described in https://pixelfed.social/p/Splines/798015349727305297, the radius of the blue circle is 8 units, and the total height of a strand swept by that circle is 24 units. Half of that is above the #tectonic surface, which is still twice of what #Vignola documented in #RegolaArchitettura. We will apply #nonuniformScaling in the Z direction to reduce the height in half while keeping the length (X) and width (Y) the same. But before we do that we split and discard some of the bottom portion of the strand that is below the tectonic surface and is not needed. The top half of the figure shows the front view of a single strand. The bottom shows the perspective view of the same strand. Note that the orange braid rail starts at the center of the blue sweeping circle which looks like an ellipse in the front view. The orange rail itself looks like a flat sinusoid in the front view, but its beautiful meandering shape is really apparent in the perspective view. The orange rail curve is centered on the "ground" or XY plane, which is also where the #tectonicSurface for the braid is. The amplitude of the orange curve (maximum or minimum from axis of the curve) is 4 units. We need to preserve the geometry of the strand at least up to the bottom of the orange rail. If we split the strand exactly at 4 units below the ground plane, we hit a limit that not all #CAD tools are able to handle. To get around, we extend it past that limit by an arbitrarily small fraction, like 0.2, and discard the portions below. We could have extended it by 0.1 or 0.3 but it wouldn't have mattered. Next, we draw a profile curve for the braid channel which is 32 units wide and its groove is 12 units tall for now. The rims on both sides of the channel are each 8 units wide. Total height of rim and channel is 16.2 units

Pixelfed
SplinesFeb 7, 2025
Refer to https://pixelfed.social/p/Splines/793554853964898442 for the backstory on this post.

This is a perspective view showing #scroll #scaffolding surfaces #extruded from #primaryCurves F1-R1, F2-R2,…. Be sure to follow these blue curves from back to front and then back again.

Metaphorically speaking, we want to use these blue curves as #walkingSticks to find the curves that meet at the front tangent points T1 through T6 and corresponding tangent point in the rear (not shown), WHILE remaining faithful to the original curves we extracted from #Vignola's original sketches.

This means that the point F1 should somehow move toward T1, F2 toward T2, and so on, with corresponding movements on the rear rectangle, yielding us 6 new #secondaryCurves.

Being "faithful" to the original means that when secondary curves derived from horizontal #primaryCurves are viewed from the top, corresponding curves are indistinguishable from each other, even though they are clearly different curves with distinct trajectories. Secondary curves derived from vertical primary curves must be indistinguishable from each other when viewed from a side.

In order to accomplish this feat, we need the remaining #volute tangent points on rectangles P, Q, and R. That means we first need the corresponding volutes that are used to #modulate the scroll surface.

In https://pixelfed.social/p/Splines/792616677005177924, we used the #scale operation to scale the original volute down from 3x to match the scale of our model using a #uniformScaling factor of 0.33333333.

In https://pixelfed.social/p/Splines/792966507797633558, we can see that the frame rectangles Q and R have independent scaling factors that are different in the X and Y direction. So here we use #nonuniformScaling.

To get the modulating volute for Q, scale the original volute by 56/112 in X direction and 80/128 in Y direction. Scale factors for the modulating volute at R are 28/112 in X direction and 48/128 in Y direction. P is same as Q.
Splines (@[email protected])

#SeeFeelTouchHug In both #art and #engineering, one must be able to both #see and #feel things that might not be there (yet). We were able to "see" the outlines of the #scroll surface from #imageScans of #Vignola's sketches in https://pixelfed.social/p/Splines/793169876757012827 and https://pixelfed.social/p/Splines/793215298082967733. Vignola's images are on a 2-dimensional surface, as are the outlines we extracted from them. We believe the scroll surface also exists, but it is not yet manifest in 3-dimensional space. So, like a visually impaired person, we try to "feel" our way to the scroll surface using the outlines as our #walkingStick. This diagram is identical to that in https://pixelfed.social/p/Splines/793493316852849994 but with the rear ends of the horizontal #primaryCurves marked with R1, R5, and R3, which are paired with F1, F5, and F3, respectively. We know that the scroll surface must #touch the tangent points T1, T2, and so on in front, as well corresponding tangent points in the rear (not shown here to reduce clutter). In https://pixelfed.social/p/Splines/792906324854792619, I mentioned that a scroll starts with a volute in front and is #modulated by as many as six volutes of different shapes and sizes as it reaches the back, with the scroll surface tightly hugging the volutes at EACH contact point in ALL 3 dimensions. In other words, it is not sufficient for the scroll surface to "touch" the #volute #spirals just in the front and rear. It must also "hug" the intermediate #modulatingSpirals. I will first show this technique with 4 modulating spirals using rectangles M, N, P, Q, and R as their frame, and add more later on. Intuitively, we know that if we use curve F3-R3 as our walking stick on the straight vertical extrusion of that curve, we will feel the scroll surface *somewhere* on that extrusion along every point from front to back. We can narrow it down further by excluding portions above and below as we approach rectangle R in the rear.

Pixelfed