Mastodawn

Splines Feb 24, 2025

Splines (@[email protected])

#Braids #3StrandBraids After creating the two #helix curves as described in https://pixelfed.social/p/Splines/797732962403957263, switch to the front view and #project the smaller blue helix on the vertical "wall" of the XZ plane. Hide the original helix. Then switch to the top view and project the larger magenta helix on the "ground" or XY plane and hide the original helix. Now compare the figure in this post with that in the previous post. Both curves have now been #flattened from 3D helix to 2D #sinusoid. When viewed from the front (top-left portion of the diagram), the blue curve is still visible as a #sinusoidal waveform but the magenta appears as a straight line flattened on the ground. When viewed from the top, the magenta curve is still visible as a sinusoid but the blue appears as a straight line clinging to the vertical wall. In the view from side (bottom-left portion of diagram), neither waveform is apparent, and both curves appear as perpendicular straight lines. Only in the perspective view you can see both waveforms, but even here it is clear that they are both flat 2D curves oriented perpendicular to each other in 3D space. Our goal is to convert these two flat sinusoids back into a single composite 3D curve that shows the smaller waveform in the front view and larger one in the top view. In acoustics, a sinusoid represents a pure tone with a single frequency. The tone varies with frequency and its perceptibility varies with amplitude. Musicians and people familiar with acoustic physics will immediately recognize that the blue curve has twice the frequency (or pitch) of the magenta curve, while the magenta curve has twice the amplitude (loudness) of the blue curve. We can divide the period or wavelength into phases. For the blue one, we divide the wavelength into 4 phases of 6 units each and shift the magenta curve left by that amount. Later, we will divide the magenta one into 3 phases — one for each strand, and shift each rightward by that.

Pixelfed

Splines Feb 24, 2025

#3StrandBraids #FlowOnSurface

In the top-left, the highlighted magenta portion shows the interface between the #braids assembly and the #IonicScroll from https://pixelfed.social/p/Splines/795276076797088402.

Extract the #profileCurve shown as ABC in the top-right where the interface meets the scroll.

In https://pixelfed.social/p/Splines/794199123072358090, we rebuilt curves from 2nd-degree arcs to 3rd-degree NURBS for smoothness and swept the scroll surface one set of arcs at a time.

Now we have to flow braids on a single surface in one operation. So we need to combine the separate segments into a single NURBS curve. To do that, #explode the profile curve into individual segments, discard the straight portion, and join the curved portions.

Curves and surfaces have a #direction that you can change in the #CAD tool. Check that the direction of the joined curve is A to C, not C to A, and flip it if necessary. Then divide the curve at 120 units starting at A. This is marked by point B. Split the curve AC at B so that AB is 120 units long.

At this point AB is still made up of 5 segments, and exploding it would again decompose the curve into separate segments. So #rebuild AB as a single NURBS curve with 32 sections.

#Extrude AB to get a 48 units wide surface shown in magenta in the top-right. Point D is at the midpoint of AE and lies on the XZ plane.

Slice the channel assembly so that it is 8 units tall, 6 of which will be above the #tectonicSurface for the braid and 2 below. The tectonic surface is shown in the bottom-left as the flat magenta surface on the channel and the curved magenta surface for the scroll neck.

Flow the entire braid and channel assembly along the curved surface lining up points A, D, and E. For the vertical part on the side of the capital, just use the 33 unit tall block from https://pixelfed.social/p/Splines/799340150182400358 and bury 1 unit inside the #ovolo.

This concludes 3-strand braids. Only the non-essential #column #flutes remain.

Splines (@[email protected])

After ensuring that the object in https://pixelfed.social/p/Splines/795271266191779399 is #airtight, extrude the front and rear planar surfaces by 1 part (8 units) on each end. At the end of this step, you should have a solid #scroll object with a smooth surface except for the flat parts that will butt against the head of the unadorned #capital. We have now concluded the #tectonic portions of the entire #IonicOrder. All that remains are decorative #eggsAndDarts that go on the #ovolo of the capital and the #3StrandBraid that goes on the scroll.

Pixelfed

Splines Feb 17, 2025

#EggsAndDarts

To transfer the egg and dart in https://pixelfed.social/p/Splines/797038670230603707 to the #doublyCurved surface of an #ovolo is a multistep process.

There are 24 eggs and darts around the entire Ovolo. So each egg and dart nominally occupies 360°/24 = 15°.

#Revolve arc AD in https://pixelfed.social/p/Splines/792124787573855518 about the #columnAxis to get the virtual surface. Points A and D are same in both figures. Then use cutting planes as described in https://pixelfed.social/p/Splines/790645054230337543 to get a wedge-shaped segment whose angle is 15° and one side of which is marked with A and D.

This is a doubly curved surface. Unlike a cylinder, where one side is straight while the other is curved, the surface of an Ovolo is curved in both horizontal and vertical directions.

To transfer the egg and dart to a doubly curved surface, we need a new operation called #UnrollSurface, which unrolls the wedge shape into a flat surface whose top-left corner is marked Q. Note the top is wider than the bottom.

Place the flattened wedge between the rim and the flat slab and align the top of the flat portion of the slab with the center of the top edge of the flattened wedge. Temporarily move the dart to align its center too, but only move it in the horizontal direction

If the flattened wedge were flexible and if the eggs and darts were flexible, we could flip all of these over, flex and squeeze the solid shapes, and line up Q with D so that the rim and dart appear "outside" the original wedge while the slab remains "inside" (or toward the center of the Ovolo).

Fortunately, in a #CAD tools, solids don't always have to be treated as rigid. Now we will use another new operation called #FlowOnSurface to flow the egg and dart on the wedge.

We then slice off the top of the egg and dart at the location PQ in https://pixelfed.social/p/Splines/796958366767133979.

Finally, we separate the dart, which we had aligned with the egg to minimize distortion

Splines (@[email protected])

#EggsAndDarts continuation from https://pixelfed.social/p/Splines/796961505955555432 The slab height depends on the roundness of the egg and whether we have a concave design or not. If we are using a concave base, then top half of the egg is eliminated. For a fully round egg, that means the concave variant must scoop out up to 16 units deep. The dart slab will match the egg slab in depth. To create the 3-dimensional shape of the dart, first #rotate the fin profile 90° in 3D space along the straight line at the bottom of the fin so that the rotated profile is perpendicular to the two #sweepRails for the dart. Using the two sweep rails and the perpendicular fin profile, #sweepTwoRails to develop the surface of the dart. Remember to close the planar hole at the end of the fin to get a solid #airtight object. As always, check for #nakedEdges and #nonmanifoldEdges to stave off problems later. #Extrude the bottom of the dart until it is flush with the bottom of the oval slab. Two details worth noting in the dart design are: 1. The most exaggerated portions of the dart fin are sliced off when the eggs are sliced. After slicing, the size of the fin is roughly in proportion to the rims of the eggs on both sides. 2. There is a gap between the dart arrow and the oval slab. See the gap between points R and T in https://pixelfed.social/p/Splines/796961505955555432. This gap is necessary and will automatically close when we transfer the egg and dart to the #doublyCurved surface of the #ovolo on the #capital of the #IonicColumn. That is because the Ovolo is shaped like a bowl whose top has a bigger radius than the bottom. As a result, the motif will be warped, and its bottom will be condensed to fit the smaller radius at the bottom, closing the gap in the process. If you plan to use the eggs and darts motif on a linear surface where there is no warping, experiment with the arrow and tip for a pleasing result.

Pixelfed