SplinesFeb 5, 2025
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.
Splines (@[email protected])

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.

Pixelfed
SplinesFeb 5, 2025
In https://pixelfed.social/p/Splines/792511464365923534 we created the outer spiral for the #IonicVolute. We will now create the inner spiral. Remember that we're temporarily using a 3x scale.

Recall from https://pixelfed.social/p/Splines/792124787573855518 that the vertical gap between the maxima for the 2nd turn of the spiral and that of the first turn shown by points G and H, respectively, is exactly 4 parts or 32 units when µ = 144. With a 3x scale, it is still 4 parts, but each part is 24 units, so gap is 96 units now. Of these 4 parts, 1 part (now 24 units) is the #fillet that gets progressively narrower as it follows the spiral, and 3 parts (now 72 units) form the #channelGroove.

The volute extends 3/4 parts or 18 units now (6 units in the original) to the left of the point of horizontal tangency of the outer spiral. This is the amount needed to accommodate the ribbon and 3-strand #braid on the flat vertical wall on the sides of the #capital as shown earlier in the same post by points F and H.

Now focus on the square where we have the 12 numbered vertices. At present scale, the square is 1 part or 24 units wide. Now inset each concentric square by 1 unit on each side, effectively dividing the gap between the original squares in a 1:3 ratio — same as the fillet height to groove size ratio.

Move to new point 1 in the square and sweep arc 1 of the inner spiral using a radius of 181 units so that the vertical gap between the top of original arc 1 and new arc one is exactly 1 part (24 units at present scale), and the vertical gap between the start of new arc 1 and the old arc 5 is exactly 3 parts or 72 units.

Continue with inner arc 2 from where inner arc 1 ended, using new point 2 as the center. Finish drawing the entire inner spiral exactly as described earlier for the outer spiral. This time you don't need arc zero or the outer circle unless you are contemplating a #medallion.

All that's left now is to complete the #eye of the volute, which I show next.
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.

Pixelfed