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.
#nonuniformScaling
#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
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
#Braids #3StrandBraids
With the #rail curve for a #braid strand in https://pixelfed.social/p/Splines/797916882329430160, we can start the construction of a braid that matches the sketch in #Vignola's #RegolaArchitettura shown in https://pixelfed.social/p/Splines/793215298082967733.
The strands for this braid have a radius of 1 part or 8 units. So their diameter is 16 units, and the braid itself is 32 units wide.
To create one strand, we start with a circle of radius of 8 units perpendicular to the tip of the orange curve. We use the orange curve as a #railCurve and the blue circle as the #sweepingCurve in the #sweepOneRail operation.
Note that the circle appears distorted like an oval in all views — front view is on top-left, top view on top-right, right view in bottom-left, and perspective view in bottom-right. That is because the circle is perpendicular to the rail curve, not to any of the "world" coordinate planes. If the sweeping curve does not appear like an angled line in the top view, something is wrong.
After the sweep, close #planarHoles to get a solid strand and, as always, check for #nakedEdges and #nonManifoldEdges to ensure an #airtight object.
Refer again to the middle portion of the top diagram in https://pixelfed.social/p/Splines/793215298082967733 between the two bell shapes of the scrolls. The total width of the #tectonic surface on which the braid will be laid is 4 parts or 32 units wide. The braid has a rim 8 units wide on both sides that rises 6 units above the tectonic surface. Half of the braid should be above the tectonic surface, meaning that the total height of the braid should be 12 units.
If you check the bounding box of this strand, you find that it is indeed 32 units wide as needed, but the height is 24 units. So we have to apply #nonuniformScaling to keep X and Y scale intact but halve the scale in the Z direction. This will reduce the roundness of the strand and cut its height in half to 12 as needed.
With the #rail curve for a #braid strand in https://pixelfed.social/p/Splines/797916882329430160, we can start the construction of a braid that matches the sketch in #Vignola's #RegolaArchitettura shown in https://pixelfed.social/p/Splines/793215298082967733.
The strands for this braid have a radius of 1 part or 8 units. So their diameter is 16 units, and the braid itself is 32 units wide.
To create one strand, we start with a circle of radius of 8 units perpendicular to the tip of the orange curve. We use the orange curve as a #railCurve and the blue circle as the #sweepingCurve in the #sweepOneRail operation.
Note that the circle appears distorted like an oval in all views — front view is on top-left, top view on top-right, right view in bottom-left, and perspective view in bottom-right. That is because the circle is perpendicular to the rail curve, not to any of the "world" coordinate planes. If the sweeping curve does not appear like an angled line in the top view, something is wrong.
After the sweep, close #planarHoles to get a solid strand and, as always, check for #nakedEdges and #nonManifoldEdges to ensure an #airtight object.
Refer again to the middle portion of the top diagram in https://pixelfed.social/p/Splines/793215298082967733 between the two bell shapes of the scrolls. The total width of the #tectonic surface on which the braid will be laid is 4 parts or 32 units wide. The braid has a rim 8 units wide on both sides that rises 6 units above the tectonic surface. Half of the braid should be above the tectonic surface, meaning that the total height of the braid should be 12 units.
If you check the bounding box of this strand, you find that it is indeed 32 units wide as needed, but the height is 24 units. So we have to apply #nonuniformScaling to keep X and Y scale intact but halve the scale in the Z direction. This will reduce the roundness of the strand and cut its height in half to 12 as needed.
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.
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.
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