Refer to https://pixelfed.social/p/Splines/793554853964898442 for the backstory on this post.

This is a perspective view showing #scroll #scaffolding surfaces #extruded from #primaryCurves F1-R1, F2-R2,…. Be sure to follow these blue curves from back to front and then back again.

Metaphorically speaking, we want to use these blue curves as #walkingSticks to find the curves that meet at the front tangent points T1 through T6 and corresponding tangent point in the rear (not shown), WHILE remaining faithful to the original curves we extracted from #Vignola's original sketches.

This means that the point F1 should somehow move toward T1, F2 toward T2, and so on, with corresponding movements on the rear rectangle, yielding us 6 new #secondaryCurves.

Being "faithful" to the original means that when secondary curves derived from horizontal #primaryCurves are viewed from the top, corresponding curves are indistinguishable from each other, even though they are clearly different curves with distinct trajectories. Secondary curves derived from vertical primary curves must be indistinguishable from each other when viewed from a side.

In order to accomplish this feat, we need the remaining #volute tangent points on rectangles P, Q, and R. That means we first need the corresponding volutes that are used to #modulate the scroll surface.

In https://pixelfed.social/p/Splines/792616677005177924, we used the #scale operation to scale the original volute down from 3x to match the scale of our model using a #uniformScaling factor of 0.33333333.

In https://pixelfed.social/p/Splines/792966507797633558, we can see that the frame rectangles Q and R have independent scaling factors that are different in the X and Y direction. So here we use #nonuniformScaling.

To get the modulating volute for Q, scale the original volute by 56/112 in X direction and 80/128 in Y direction. Scale factors for the modulating volute at R are 28/112 in X direction and 48/128 in Y direction. P is same as Q.