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.