Arcs and lines toil for #splines
2500 years ago, when they had neither computers nor #CAD tools, designers and architects relied on knowledge of algebra, geometry, and trigonometry for their daily work. It was a mere 350 years ago that Leibniz and Newton brought calculus as a new mathematical tool for design and engineering.
Before computers arrived, artists, designers, and architects toiled with manual drafting tools to engineer breathtaking masterpieces. "Toil" is not an exaggeration to describe that endeavor, even though I suspect some of them really enjoyed what they were doing.
#Scarlata compiled an entire book of #VignolaProportions in painstaking accuracy and high precision before there were calculators and spreadsheets, making it "easy" to convert from µ to physical units in both English and Metric systems, but the world has moved on, his work is forgotten, and nobody is thankful for his contributions.
If you have a CAD tool, you need not toil. Simply draw an arc of radius µ = 144 that is centered on the #columnAxis and passes through point B. Then draw a vertical line parallel to the column axis at x = µ * 5/6, or 120 units. Use this line to split the arc and trim away the left portion of the arc. Next, divide the length of the remaining portion of the arc into 8 equal portions using your CAD tool to mark the points 1 through 8 as shown. If your CAD tool is able to divide the leftover arc this way, you can just ignore the angular lines radiating from the center. Otherwise, I will show you how to use them as a fallback.
Now look at point C, which seems like it is vertically above point B, but it is not. It is actually collinear with point 1.
Draw 7 more vertical lines starting with point 1, then point 2, and so on, Mark point C at 192 units vertically on line 1, D at 192*2 on line 2, E at 192*3 on line 3, and so on until you reach point J.
Select these 8 points and use your CAD program to interpolate a free-form NURBS curve to fit these points.
2500 years ago, when they had neither computers nor #CAD tools, designers and architects relied on knowledge of algebra, geometry, and trigonometry for their daily work. It was a mere 350 years ago that Leibniz and Newton brought calculus as a new mathematical tool for design and engineering.
Before computers arrived, artists, designers, and architects toiled with manual drafting tools to engineer breathtaking masterpieces. "Toil" is not an exaggeration to describe that endeavor, even though I suspect some of them really enjoyed what they were doing.
#Scarlata compiled an entire book of #VignolaProportions in painstaking accuracy and high precision before there were calculators and spreadsheets, making it "easy" to convert from µ to physical units in both English and Metric systems, but the world has moved on, his work is forgotten, and nobody is thankful for his contributions.
If you have a CAD tool, you need not toil. Simply draw an arc of radius µ = 144 that is centered on the #columnAxis and passes through point B. Then draw a vertical line parallel to the column axis at x = µ * 5/6, or 120 units. Use this line to split the arc and trim away the left portion of the arc. Next, divide the length of the remaining portion of the arc into 8 equal portions using your CAD tool to mark the points 1 through 8 as shown. If your CAD tool is able to divide the leftover arc this way, you can just ignore the angular lines radiating from the center. Otherwise, I will show you how to use them as a fallback.
Now look at point C, which seems like it is vertically above point B, but it is not. It is actually collinear with point 1.
Draw 7 more vertical lines starting with point 1, then point 2, and so on, Mark point C at 192 units vertically on line 1, D at 192*2 on line 2, E at 192*3 on line 3, and so on until you reach point J.
Select these 8 points and use your CAD program to interpolate a free-form NURBS curve to fit these points.
Vertices mapped by distance to splines & their pose + twist. Useful for other procedural generation like tree branches, flowers, plants 
etc. And it's really fun ^^ 
![TypeScript source code snippet to generate the 2D curve shown in the first image:
// convert polygon to path of Hobby spline segments
asPath(
// iteratively subdivide poly
subdivCurve(
// create square and convert to polygon
asPolygon(rect(100))[0],
[
// 1st & 2nd pass: cubic spline subdiv
SUBDIV_CUBIC_CLOSED,
SUBDIV_CUBIC_CLOSED,
// 3rd & 4th pass: subdivide w/ displacements
// (inserts 2 new points per edge, each point
// is displaced along the edge normal...)
SUBDIV_DISPLACE([0, -0.4, 0.4], true),
SUBDIV_DISPLACE([0, -0.4, 0.4], true)
]
),
{ mode: "hobby" }
)](https://mastothing.files.fedi.monster/media_attachments/files/112/531/654/841/781/119/original/a080a43eb115f0de.png "TypeScript source code snippet to generate the 2D curve shown in the first image:
// convert polygon to path of Hobby spline segments
asPath(
// iteratively subdivide poly
subdivCurve(
// create square and convert to polygon
asPolygon(rect(100))[0],
[
// 1st & 2nd pass: cubic spline subdiv
SUBDIV_CUBIC_CLOSED,
SUBDIV_CUBIC_CLOSED,
// 3rd & 4th pass: subdivide w/ displacements
// (inserts 2 new points per edge, each point
// is displaced along the edge normal...)
SUBDIV_DISPLACE([0, -0.4, 0.4], true),
SUBDIV_DISPLACE([0, -0.4, 0.4], true)
]
),
{ mode: \"hobby\" }
)")
