SplinesSuccessive egg slabs are 1/2 part or 4 units away from each other. So the thinnest part of the dart is 4 units. The tip of the dart is at point P, which is 22 units from the major axis for the egg slab.
The inside egg is 4 parts or 32 units wide, and we will start with the dart fin also at 32 units wide, even though a portion of the fin will be sliced off when the egg is sliced by the red cutting plane.
To find the start and end of the fin, draw a vertical line from P of length µ/2 or 72 units. Offset that line on both sides by 16 units. Mark the intersections of these lines with the outline of the largest ellipse with points A and B.
Draw an oval centered on AB with major radius 16 units and minor radius 2/5 of that, or 6.4 units. Divide each arc of the oval into 5 equal parts and mark AD and EF at 2/5 of that. Copy the oval from its top point C to D as well as E. Mark the intersection of the copies at F.
Trim the 3 ovals to get 4 arcs AD, DF, FE, and EB. Join them and close the curve with a straight line from A to B.
Draw a square 12 x 12 whose bottom edge is centered on P. From each of the top 2 corners, draw a circle with diameter 4 units, shown by RS.
#Reflect point P using the major egg axis as a #mirror to get point Q. Draw a #circleThrough3Points P, Q, and R. The origin of the circle, point O should be 3.5 units directly below the base of the large oval.
Draw a vertical line up from S to where it is tangent to the side of the large oval. Trim the straight line and arcs to get the left profile of the dart starting with A, passing through S and T, and ending at P. Join all 4 segments and reflect them using the line PC as the mirror.
These mirrored copies are the #rails along which we will sweep the fin of the dart with #sweepTwoRails operation to create the dart, but we must orient the fin to be perpendicular to the rails first
Splines (@[email protected])
#Ovolo #TectonicSurfaces for #EggsAndDarts Reconcile this figure with the figure in https://pixelfed.social/p/Splines/792124787573855518. The points A, B, C, and D are the same in both. The arc from A to D is the profile for the invisible virtual surface that encloses the decorative elements. Arc BC is start of the #tectonic surface where decorative elements rest. Points K, L, M, P, and Q are new here. Point K is the center of the Ovolo and lies on the #columnAxis. KD is the upper radius of the Ovolo including the decorative elements, and its length is 22 parts of 176. The horizontal distance between C and D is 1 part (8 units), and the vertical distance between C and P is also the same. PQ is where the eggs and darts are sliced off exposing the wall CP around the entire Ovolo and making it visible. Normally, the decorative elements rest on the surface swept by arc BCP. In the case of https://pixelfed.social/p/Splines/796786779066451143, we have a fully round egg shape half of which is buried behind the arc AQD. In the concave variant, this means part of the tectonic surface is also carved out. In other words, instead of limiting ourself to the range CD (or PQ), we are taking liberty to gouge out portions as far back as arc LM, and that's OK. Neither #Vignola nor #Scarlata mention tectonic surfaces or how far back they should or would be. The actual depth will depend on the choice of our decorative elements — in this case, concave eggs that are fully round like natural eggs. The concave version of the egg is shown on the bottom right and it lines up with the orange wireframe with the round hole on top. The wireframe of the convex egg is superimposed on the concave portion to show where it will be placed if both are used together. The depth of the dart slab is the same as that of the slab of the concave egg. The slab of the convex egg is thinner because we have to leave room for the bulge of the egg.