#cymaRecta

2025-02-13
#EggsAndDarts is a common classical design motif with endless variations, two of which are shown here — the top-left variant has convex eggs and the bottom-left variant has concave eggs. The sketch on the right shows the bottom view of the concave variant.

This motif is neither specific to the #IonicOrder, nor limited to the #ovolo of the capital. It is common to find it laid on linear #moldings like #cymaRecta or #cymaReversa of a #cornice.

The egg shape, the dart shape, the degree of convexity or concavity, and so on, are infinitely variable from subtle to pronounced. Designers are not limited to convex or concave, and it is possible to combine both in a single design. Also, it is not necessary to use the eggs and dart motif at all. There are infinite possibilities. However, when the eggs and darts motif is used, it is almost invariably sliced off at the top, as the bottom view of concave variant on the right reveals.

The concave version here is quite subtle, but a more pronounced version can be really eye-popping. I will show how to construct one using just straight lines and circular/elliptical arcs exclusively as I originally promised in https://pixelfed.social/p/Splines/789956327130679640.

As usual, we start with a flat 2-dimensional plan with lines and ovals to use as #sweepingRails. Then, we add circles and arcs as #sweepingCurves to define the cross-sections. After sweeping the cross section curves on the rails, we create the eggs.

Simply #revolve an ellipse on its major axis to get the convex version of an egg. To get the concave version of an egg, simply create a flat slab and perform a #booleanDifference on that slab using a convex egg.

Once we have all of this preparatory work done, we have to transfer the 3-dimensional design from the flat surface it was originally created on to the #doublyCurved surface of the Ovolo. This requires some elementary calculations using circle geometry.

Previous— https://pixelfed.social/p/Splines/795361973789834465
2025-01-31
Classic variation of the #IonicEntablature. Left side shows the macro-level plan. Right side shows the detailed plan for the #moldings.

All moldings should be familiar from the #IonicPedestal, except the #fascia — flat bands, of which there are 3, at the bottom of the #Architrave. The fascias grow progressively taller, starting at 36 units at the bottom, to 48 in the middle, and 60 at the top, with each successive one offset horizontally by 6 units from the previous one.

Above the fascia, we have a #cymaReversa which is 24 units tall and 20 units wide. Here we are using half turn of a helix with a vertical axis. Either helix or elliptical arcs are acceptable, but the choice must be consistent across the entire order. You cannot use ellipses in the pedestal and helices in the entablature, for example.

If you do use a helix, remember that it is a 3D shape like a round coil. To use it as a #profileCurve, it must be flattened to a 2D shape by #projecting it to the #constructionPlane. I will describe this technique in detail later.

The #frieze is a flat surface with no moldings. It is meant as a blank space on which to put custom decorative 3D #ReliefCarvings or sculptures.

There are no new moldings in the #cornice. Note that the order of #cymaRecta and cyma reversa are reversed from that of the #pedestal, with cyma recta at the top and cyma reversa at the bottom. The only difference between the two is that the recta has a horizontal axis and reversa has a vertical axis. Rotating either one 90° yields the other.

So, the bottom of the cornice starts with a cyma reversa 32 units tall and 34 wide. This is followed by a fascia 56 units tall on which the #dentils will appear later on. The 36 unit horizontal offset for the 4-unit thick fillet above it is to leave room for the dentils.

This is followed by a #reed (8 units), #ovolo (32 units), #corona (48 units), cyma reversa (16 x 12 units), fillet (4 units), cyma recta (40 x 44 units), and fillet 12 units
2025-01-30
From https://pixelfed.social/p/Splines/790645054230337543, we now have an open surface for the #IonicPedestal. To finish this, close the #planarHoles at the top and bottom with flat caps and join everything. And just like that, we have finished 1/3 of the complete #IonicOrder with very little work.

To ensure that your finished object is amenable to #3DPrinting or #CNCMilling, always check the edges of your object after all surfaces have been joined. Do this EVERY time you join surfaces to create a closed object.

Most CAD programs will offer edge analysis tools that let you detect #nakedEdges or #nonManifoldEdges. If you have either of those, your object is not #airtight, and you will not be able to physically realize it.

This version of the pedestal uses the classic variation of #CymaRecta and #CymaReversa. If you want to remain faithful to the original, then you are done.

However, designs are rarely static and they continue to evolve. There is an opportunity for a slight refinement at the top and bottom of the pedestal without compromising the integrity of the order, but it requires the introduction of a new kind of curve — a #helix, which is a coil-shaped 3D curve.

I will discuss the #helixVariation later. For now, look closely at the #basement and notice how pronounced the turns of the cyma recta are. Instead of using elliptical arcs in the #primaryProfileCurves of the cyma recta and cyma reversa, it is possible to substitute a half-turn of a helix that has been flattened to a 2D shape. The result is a softer, more gradual profile curve that produces a very refined shape.
2025-01-30
This shows the detailed measurements of the top and bottom portions of the #IonicPedestal. For macro-level measurements, see https://pixelfed.social/p/Splines/790571135473463588

Each of the blue curve segments (lines and arcs) that are marked with a yellow bubble is the #profileCurve for a #molding whose name is inside the bubble.

Starting at the bottom, we have a #plinth, a #fillet, a #cymaRecta, and a #reed as part of the #basement of an Ionic pedestal.

Next up, we have a #fillet and a #cavetto at the bottom of the #dado, and another cavetto and fillet at the top of the dado.

Moving higher up, a reed, an #ovolo, a #corona, a #cymaReversa, and a final fillet top off the cap of the pedestal.

They are called profile curves because each is the outline or silhouette of a 3D molding as seen from one side or in a cross section. In the case of a pedestal, these curves can be used directly to recreate the 3D shape of the pedestal. For this reason and in this case, I call them #primaryProfileCurves.

This is not always the case. For more complex shapes, such as the #scroll surface of an #IonicCapital shown in https://pixelfed.social/p/Splines/789956327130679640, the profile curves recovered by #reverseEngineering the image scans in #Vignola's book cannot be used directly to sweep the scroll surface because the scroll shape is not cylindrical. Like the inside of a rose, the scroll surface follows the outlines of spiral #volutes in the front and back, neither of which are circular. So, additional steps are necessary to derive the curves that we can actually use to reconstruct the surface.

In the case of the scroll surface, the derivation of these curves is not trivial and not obvious, but it is not difficult to understand, and no math is involved. There are multiple sets of curves, and each successive set is derived from a previous set. I call them secondary, tertiary, and quaternary curves.

For now, we stick with the primary profile curves for the pedestal.
2024-12-31

⬆️ #IonicPedestal #3DModeling #ProfileCurves

Bottom-right portion of profile curves for #Ionic Pedestal without dimensions, but still showing the centers for all arcs.

Notice that the center for the #reed is 2 units to the right of the center for the top arc of #CymaRecta.

The two gaps to the left of the reed MUST be closed if you want your models to be airtight for #3DPrinting. If you forget, the model might render fine but the model will have #nakedEdges.

Bottom-right portion of profile curves for Ionic Pedestal without dimensions, but still showing the centers for all arcs.
2024-12-31

⬆️ #IonicPedestal #3DModeling #ProfileCurves

Bottom-right portion of profile curves for Ionic Pedestal. Details in Alt text.

Starting from bottom, the #plinth, #fillet, #CymaRecta, and #reed belong to the #Basement.

The fillet and #cavetto (circular arc) above those belong to the #Dado.

The cyma recta is 40 units wide and 24 tall. So the arcs for that are cut from ellipses.

A refined variation of cyma recta uses half turn of a helix instead of 2 elliptical arcs and will be presented later.

Zoomed-in view of bottom-right portion of the profile curve for Ionic pedestal showing curve shapes and measurement details.

The cavetto (concave circular arc) and fillet near the top belong to the #Dado. The reed below the fillet belongs to the basement.

A detail that is easy to miss is that the right edge of the reed is 8 units farther away from the column axis than the right edge of the fillet above it. But, the radius of the reed profile is 6 units (for reed height of 12). 

So there is a small gap of 2 units at the foot of the fillet and top of the reed that must be closed with a straight horizontal line of 2 units.

There's a similar gap of 2 units at the bottom of the reed and top of the cyma recta that must also be closed with a straight horizontal line.

All of the individual shapes must be joined to form a single profile curve that is flat on the XY plane (Front view).

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