SplinesCreate a 270° circular arc of radius 1 part (24 units at 3x scale), spanning quadrants 2, 3, and 4 as shown in orange for the #eye of the volute. The arc for the eye intersects arc 12 of both inner spiral (shown in green) and outer spiral (shown in magenta).
Outer arc 12 makes a kink where it meets the orange arc as seen in the left diagram. Discard the magenta arc 12. We can do better.
Trim both the inner arc 12 (green) and the orange arc for the eye where they meet and discard the right portions of both.
Finally, perform an #arcBlend between points A and B as shown in the right diagram. Arc blend is a new operation we are seeing for the first time. Previously we used #tangencyBlend to blend various sections of the #primaryProfileCurves for the #shaft of an #IonicColumn [https://pixelfed.social/p/Splines/791723063470910081]. Arc blend also maintains tangency, but instead of generating freeform #NURBS curves for blending, it exclusively uses one or more sections of circular arcs to blend the ends.
Join all segments of the inner spiral, outer spiral, eye, blended arcs, and straight lines near the top-left of the volute to create a single #closedCurve.
Mark the center of the eye as the origin or base point for #moving, #scaling, and other #transformations, and don't forget to scale the entire design to 1/3 using a scaling factor that has a high degree of precision, e.g., 0.33333333.
I mentioned that Dürer's approximation of a #logarithmicSpiral is close, but doesn't fit perfectly. So far there's nothing that doesn't fit. The fit issue only comes up during #scroll construction.
This concludes the task of volute construction. Next, we will look at creating the 3-dimensional volute slab using this closed volute curve and adapt it for the recessed #channelGroove in the slab.
Splines (@[email protected])
Refer to https://pixelfed.social/p/Splines/792499765146596723 for Step 1 in #IonicVolute construction. Remember that we are temporarily scaling up by 3x. Connect points 4 and 5 and extend it to a length approximately 144 units. Connect points 8 and 9 and extend to approximately 72 units. These are #transitionRays that we use to transition from one turn of the spiral to the next. The volute is constructed in discrete steps using 13 arcs numbered zero to 12 in the figure. The arcs are centered on numbered vertices of the 3 concentric squares we made in the previous post. The center for arc 1 is point 1, for arc 2 it is point 2, and so on. Point 4 is special — I call it #groundZero. It is the center for Arc Zero as well as the circle that encloses the volute. Neither the circle, nor arc zero are used in the volute, but a portion of arc zero is used in constructing the #scroll surface. You can throw away the outer circle or use it to create a #medallion around the volute in other unrelated designs. With 3x measurements, start arc 0 at ground zero, or point 4 with radius 228 and stop after 90° clockwise sweep. Start arc 1 where previous arc ended using point 1 as center and sweep clockwise 90°. Start and end arcs 2 and 3 similarly using points 2 and 3 as their centers. For arc 4, continue sweeping past 90° until it meets the first transition ray. For the 2nd turn of the spiral, start at point 5 with a radius equal to the distance between point 5 and point 13. The angles for arc 4 and arc 5 are #supplementary — They total 180°. So arc 5 ends on a horizontal axis. Continue with 90° arcs centered at points 6 and 7. For arc 8, continue sweeping past 90° until it meets the second transition ray. For the final turn of the spiral, start at point 9 with a radius equal to the distance between point 9 and point 14. The angles for arc 8 and arc 9 are #supplementary — They total 180°. So arc 9 ends on a horizontal axis. Continue with 90° arcs centered at points 10, 11, and 12.