#Arcade #Intercolumniation without #Pedestal

In https://pixelfed.social/p/Splines/803089629244302486, we saw #simpleIntercolumniation, also known as #Architravato.

Roman architects combined columns with walls thick enough to bury half of the column width inside the walls and added arches to them for better load distribution. An arcade (multiple arches) can be run in series along a single wall, or in parallel to form a walkway. They can also be combined in both series and parallel configurations, perhaps the most famous of which is the #Colosseum in Rome.

In the Colosseum, the outer walls follow an elliptical curve (even though it looks circular from the outside), and it has multiple tiers of arches in series. The interior has arches in concentric passageways in the lower tiers giving it a lattice-like design.

Because arches distribute the load from above, they allow for wider intercolumniation. The rules for #ArcadeIntercolumniation differ depending on whether the columns have pedestals or not.

Besides the arch itself, which is part of the wall, the figure shows some new architectural elements.

The narrow part of the wall immediately behind a column is known as a #pier. The visible face of a pier between a column and the opening under the arch is known as #alette. The base of the pier has a molding, the flat part of which has the same height as the column base (µ) while the rest follows the #fillet and #cavetto or #conge of the #shaft.

As we move up the pier, there is a horizontal molding known as #impost just below where the arc of the arch starts. The impost wraps around on the sides of the pier.

Around the arc is a circular molding known as #archivolt, the bottom portion of which has a #fascia that is aligned with the face of the wall.

The wall itself extends all the way to the top of the #entablature. It is worth noting that the entablature is repeated on the wall. It doesn't end at the columns and has two "outside" corners and one "inside" corner.

#IonicColumn #Flutes

This diagram shows the 2D geometry of an #Ionic #flute. The larger blue circle shows the flute outline near the #base of the #column. The smaller blue circle shows the flute outline near the #neck of the #shaft. Both subtend a 12° angle at the center of the column.

Like an egg in the #EggsAndDarts motif, a flute must be centered on the column axis when viewed directly from the front, back, or the sides. This is why the 12° are split into 6° on either side of the X axis. The center of the larger circle is µ = 144 units from the origin on the X axis. The center of the smaller circle is 5/6 of µ, or 120 units from the origin.

In https://pixelfed.social/p/Splines/799340150182400358, I mentioned working at sub-micron precision, and you might wonder where that came from when we have been using abstract units like µ without specifying any physical units. My apologies for not making it clear that I had assumed 1 unit was equal to 1 mm. If that assumption holds, then µ = 144 mm gives a total order height of 4104 mm, that is 13.46 ft. At smaller scales, the precision is even higher than 1/10 of a micron.

With that said, here the radius of the larger circle is 15.0728 units and that of the smaller circle is 12.5606 units, with sub-micron precision if 1 unit = 1 mm.

Refer to https://pixelfed.social/p/Splines/791399680747885646 and place the center of the smaller circle exactly at point J on the neck line. Later we will draw a sphere at the same location with the same radius.

If you want a flat bottom for flutes, place the center of the larger circle at exactly 28 units (12 for the #fillet and 16 for the #cavetto or #conge) above point A in that figure. If you want a round bottom, then further move the larger circle up by the size of its radius.

Nobody would quibble if you used a radius of 15 units instead of 15.0728 units, but it would make it easier to switch from flat to round bottom or vice-versa by simply moving the circle up or down 15 units.

The bottom 1/3 of the #columnShaft for an #IonicColumn is a perfect cylinder. So the line below point B is a straight line.

In https://pixelfed.social/p/Splines/791723063470910081, we blended the bottom end of the 60° arc and the top end of the long interpolated curve between points J and K. Now blend the bottom end of the interpolated curve and the top end of the straight line between points B and C to obtain the 3rd and final #NURBS segment for the #primaryProfileCurve of the shaft.

Just like there's a #cavetto and #fillet near the #neck of the shaft, there is a fillet and cavetto near the foot of the shaft. However, there is a subtle difference between the two. The cavetto near the neck is tangential to the blended #NURBS curve that is not a straight line. The profile curve for the cavetto near the foot is tangential to a straight line.

There is a special name for a cavetto that is tangential to a straight line or flat surface, like the two cavetto moldings in the #dado of the #pedestal. It's called a #conge. Another alternate name for the cavetto molding is #cove, which is evocative of "cave" because of its concave profile curve.

Above the neck is a fillet 8 units tall and an #astragal 16 units tall that #Scarlata puts in braces in the column shaft section within his tables of #VignolaProportions, with a note saying they are not counted as part of the shaft but are accounted for as part of the #capital.

I decided to include the top fillet as part of the shaft and keep the astragal with the capital. It does not change the design or alter the proportions in any way, but the inclusion of the fillet makes it more practical for #3DPrinting and #CNCMilling of the neck. This concludes the profile curve for the shaft with a height of 291 parts or 2328 units + 8 for fillet.

The column shaft is tapered in the upper 2/3 due to #entasis whose purpose is to make optical corrections to the shape of the column which, without correction, appeared concave near the top.

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.