SplinesIn both #art and #engineering, one must be able to both #see and #feel things that might not be there (yet).
We were able to "see" the outlines of the #scroll surface from #imageScans of #Vignola's sketches in https://pixelfed.social/p/Splines/793169876757012827 and https://pixelfed.social/p/Splines/793215298082967733.
Vignola's images are on a 2-dimensional surface, as are the outlines we extracted from them. We believe the scroll surface also exists, but it is not yet manifest in 3-dimensional space. So, like a visually impaired person, we try to "feel" our way to the scroll surface using the outlines as our #walkingStick.
This diagram is identical to that in https://pixelfed.social/p/Splines/793493316852849994 but with the rear ends of the horizontal #primaryCurves marked with R1, R5, and R3, which are paired with F1, F5, and F3, respectively.
We know that the scroll surface must #touch the tangent points T1, T2, and so on in front, as well corresponding tangent points in the rear (not shown here to reduce clutter).
In https://pixelfed.social/p/Splines/792906324854792619, I mentioned that a scroll starts with a volute in front and is #modulated by as many as six volutes of different shapes and sizes as it reaches the back, with the scroll surface tightly hugging the volutes at EACH contact point in ALL 3 dimensions. In other words, it is not sufficient for the scroll surface to "touch" the #volute #spirals just in the front and rear. It must also "hug" the intermediate #modulatingSpirals. I will first show this technique with 4 modulating spirals using rectangles M, N, P, Q, and R as their frame, and add more later on.
Intuitively, we know that if we use curve F3-R3 as our walking stick on the straight vertical extrusion of that curve, we will feel the scroll surface *somewhere* on that extrusion along every point from front to back. We can narrow it down further by excluding portions above and below as we approach rectangle R in the rear.
Splines (@[email protected])
#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.