We completed the #primaryProfileCurves for the classical flat #IonicVolute in https://pixelfed.social/p/Splines/792616677005177924.

To create a 3-dimensional slab with a recessed #channelGroove for the volute, you will need an outline of the volute without the inner #spiral arms.

To create the outline, make a copy of the spiral curves and work on the copy so that you don't destroy the originals. Drop a straight vertical line from the start point of outer Arc 1 of the spiral to the maxima or horizontal tangent of outer Arc 5. Trim away all other interior spiral lines and close the curve as shown in the left figure.

#Extrude the closed outline curve front to back by 1 part or 8 units in the side view. Extrude the #closedCurve of the inner and outer spirals by 2 parts or 16 units in the side view (but without the 6 unit extention on the top, which is only used when integrating the volute face with the #capital). Perform a #booleanUnion of both solids, and remember to check for #nakedEdges and #nonManifoldEdges.

The #volute design can be used outside of the #IonicColumn, such as in a #medallion. For a medallion, you have two options regarding the size of the enclosing circle.

You can either use the circle that Arc Zero lies on, or you can use the circle that Arc 1 lies on. Obviously, the latter is more compact. Just remember that the center for the larger circle is #groundZero or point 4 and the center for the smaller circle is point 1. In either case, inset the chosen circle with a concentric circle whose radius is 1 part or 8 units less.

The figure on the right shows the outlines of the enclosing circles based on the size of Arc 1 with center at Point 1.

We saw how to create the #outerSpiral for the #IonicVolute in https://pixelfed.social/p/Splines/792511464365923534 and the #innerSpiral in https://pixelfed.social/p/Splines/792561721929860260.

Create 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.