#SeeFeelTouchHug
In 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.
