#ReverseEngineer #ImageScans

We now dig into the archives and resurface old sketches for #restoration. This one is from #Vignola's #RegolaArchitettura at https://archive.org/details/gri_33125008229458/page/n39/mode/2up. This lavishly illustrated book with copious notes that also flaunt his #calligraphy was written (in Italian) when America was still a British colony. The book went out of copyright a long time ago.

Straighten the image as much as you can in an image editor and crop it before bringing it into a #CAD tool.

Then, stare at the image for a while and squint occasionally until you "see" crucial features and patterns emerge, while ignoring the "noise."

Finally, try #curveFitting with the simplest of curves — straight lines, circular arcs, ellipse, and so on to get as close an approximation as possible. Remember that with hand-drawn sketches, the fit will rarely be perfect. So use some structure as a guide or #scaffolding as I laid out in https://pixelfed.social/p/Splines/792966507797633558.

In the top left of the diagram, I show the measurements that I was satisfied with after a lengthy process of trial and error because the numbers comport with my understanding of the proportions the original designers intended — many, but not all of which are documented in #Scarlata's #PracticalArchitecture with #VignolaProportions in tabular form.

For measurements that are missing, use plausible heuristics to fill in the blanks and try to justify your choices using simple rules. In this case, the bedrock rules are:

1. The entire #volute is exactly µ = 144 units wide, including #ArcZero, which extends 32 units beyond the portion of the volute that is actually used in the design.

2. The portion of the volute that is actually used in the design is 112 units wide, same as the height of the unadorned #capital.

3. Width of the #scroll bell shape as seen from the bottom is 112 units in front, 56 units in the middle and 28 units in the rear — all in #geometricSequence.

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.

Arcs and lines toil for #splines

2500 years ago, when they had neither computers nor #CAD tools, designers and architects relied on knowledge of algebra, geometry, and trigonometry for their daily work. It was a mere 350 years ago that Leibniz and Newton brought calculus as a new mathematical tool for design and engineering.

Before computers arrived, artists, designers, and architects toiled with manual drafting tools to engineer breathtaking masterpieces. "Toil" is not an exaggeration to describe that endeavor, even though I suspect some of them really enjoyed what they were doing.

#Scarlata compiled an entire book on #VignolaProportions with painstaking accuracy and high precision before there were calculators and spreadsheets, making it "easy" to convert from µ to physical units in both English and Metric systems, but the world has moved on, his work is forgotten, and nobody is thankful for his contributions.

If you have a CAD tool, you need not toil. Simply draw an arc of radius µ = 144 that is centered on the #columnAxis and passes through point B. Then draw a vertical line parallel to the column axis at x = µ * 5/6, or 120 units. Use this line to split the arc and trim away the left portion of the arc. Next, divide the length of the remaining portion of the arc into 8 equal portions using your CAD tool to mark points 1 through 8 as shown. If your CAD tool is able to divide the leftover arc this way, you can just ignore the angular lines radiating from the center. Otherwise, I will show you how to use them as a fallback.

Now look at point C, which seems like it is vertically above point B, but it is not. It is actually vertically above point 1.

Draw 7 more vertical lines starting with point 1, then point 2, and so on. Mark point C at 192 units vertically above on line 1, D at 192*2 on line 2, E at 192*3 on line 3, and so on until you reach point J.

Select these 8 points and use your CAD program to interpolate a free-form NURBS curve to fit these points.

There are two variations of the #IonicEntablature. The classic variation has #dentils, which are teeth-like structures shown here above the #frieze. The modern version has #modillions, which are projecting brackets under the #corona of the #cornice. Well, "modern" is a relative term. For designs that are more than 2000 years old, even an alteration 1000 years ago would qualify as modern.

Although the sketch shows the #entablature with a square footprint, in practice, it runs the entire length of a #colonnade (multiple columns) or an #arcade (multiple arches).

#CAD construction of the entablature is very similar to that of a #pedestal.

The first step is to consult #Vignola's #RegolaArchitettura for the visual appearance, and then consult #Scarlata's #PracticalArchitecture for #VignolaProportions in tabular form.

It is convenient to create a spreadsheet to convert the measurements given in Scarlata's book from module "parts" to your own model units based on your choice of value for the module parameter µ.

Armed with these measurements, it is time to plot the points and draw the #primaryProfileCurves on our standard 2D grid with minor grid lines 8 units apart and major grid lines 32 units apart.

In the first pass, skip the dentils and draw the profile curves for the rest of the moldings. Just as with the pedestal, I will show the macro-level plan as well as the detail plan. So, you don't have to go to Scarlata's book, but you know it's there if you want to.

I will show the dentil arrangement in a subsequent post.

Based on µ = 144, the classic Ionic entablature is 648 units (36 parts, or 4.5*µ) tall. Of this, the #architrave at the bottom is 180 units (10 parts, or 1.25*µ) tall, the frieze in the middle is 216 units (12 parts, or 1.5*µ) tall, and the cornice at the top is 252 units (14 parts, or 1.75*µ) tall.

This shows macro-level measurements for the #IonicPedestal.

The key to #effectiveModeling is to simplify a complex shape into elementary components. Sometimes, this involves mentally flattening and reducing 3D shapes to 2D shapes, extracting elementary curves from them, and then recreating the 3D shapes from the extracted 2D curves.

This is not always easy for organic shapes (which can still be approximated by Bézier curves). I extracted the #primaryCurves for the #IonicScroll surface in https://pixelfed.social/p/Splines/789956327130679640 after a lengthy trial-and-error process that involved #curveFitting images from #Vignola’s book, #RegolaArchitettura. I had to #reverseEngineer the details because the measurements have either been lost, or are locked away in some library. Web search yields no details on these measurements.

Fortunately, for geometrical shapes like pedestals, this is very easy. Because of its square footprint, mentally slicing it through the middle from top to bottom, it is easy to “see” the outline. Another way to think about #curveExtraction is to shine an imaginary bright light on an object from behind in a dark room to reveal its silhouette.

For the pedestal, even this silhouette or outline can be further reduced because the shape is symmetrical about the #columnAxis. With this realization, we only need to focus on one half of the outline, and methodically proceed from bottom to top, marking every kink and inflection point on the outline.

Fortunately, the other authoritative book, #Scarlata’s #PracticalArchitecture, I mentioned in my introductory post already documents #VignolaProportions in tabular form. So we can skip everything else and go directly to that.

Total height of #IonicPedestal is 864 units (108 parts, or 6*µ) of which the #PedestalBasement and #PedestalCap are each 72 units (9 parts, or µ/2) and the #Dado is 720 units (90 parts, or µ*5) tall.