Hi Fellows,
… as promised, this is the follow-up from:
https://forums.autodesk.com/t5/fusion-360-design-validate/constraint-bug/m-p/12247715
What are the siblings of a standard curve offset?
First, we must realize there are two standard curve offsets. One is generated by progressing along a curve, a circle of offset radius and finding the developed figure's outlines (internal and/or external). The second is finding points at the offset distance on normal vectors to the 2D curve.
They produce different results! Which method does F360 use?
Using symbolic math notation, both functions can be presented as:
- Offset(s)= ± 𝓒 (s)
- Offset(s)= ± 𝓟 (n(s), k(s)=const, 𝓣(s))
- Where: (s) – a parameter along a curve, n(s) – a normal vector at point, k(s) – a constant offset value
Both methods develop the resulting offset curve progressively moving along the curve while making very weak assumptions about the curve shape:
- the coinciding circle method – no assumption
- the parallel line method exploits the curve's tangential property 𝓣(s) at a point.
It should be stated here that:
- The congruent circles offset method is not additive (i.e. offsetBy_2+offsetBy_2=?)
- The parallel lines offset is additive (i.e. offsetBy_2+offsetBy_2=offsetBy_4)
Can the offset be generated differently, and in particular, can the curve shape be considered?
Yes, it can. One such method is called Curve-Shortening-Flow, in short, CSF. It introduces a curve curvature as a parametric function into an offset equation, giving it a form:
- Offset(s,t)= ±CSF(N(s)|t, 𝓕(K(s)|t))
- Where K(s) is a curve's curvature at point, and 𝓕() well-behaving function over K(s)
- N(s) and K(s) should be viewed as integrals of n(s) and k(s) over (t) parameter denoting offset time progress
- Important … n(s) switches here from a normal vector to a curvature vector where k(s) is its norm/length, hence local offset value,
- thus allowing performing offset on 3D curves, while a classical offset of the 3D curve in 3D space is ambiguous! There are a limitless number of normal vectors to such a curve at a point, … but only one curvature vector!
So, a seemingly slight modification of standard offset(s), where its local value varies with a curve curvature, has more than subtle repercussions. Another significance is that the CSF process might resemble many physical phenomena… and as those are primarily smooth, CSF can generate organically looking and visually pleasing outcomes ( given proper 𝓕() also opposites).
It is especially pronounced when an additional function (call it 𝓩()) maps (t) parameter, generating together with Offset(s) a 3D-manifold.
- 𝓜(s,t, 𝓩(t)) = 𝓩(Offset(s,t))
The subject is exciting and worth exploring the basics in free media, or even its more profound intricacy, in not always widely accessible scientific publications/archives.
Answering, I think, questions many would like to ask … Will it be available in F360?
Well, it is a complex function, expensive in terms of computation. It might also be overwhelming by its convolutedness and obviously very hard to parametrize straightforwardly.
Thus, … I direct the question to … TF360!
Over the following days, I will present some visuals showing the CADtential of the CSFs family of offsets.
Below is the first, almost genetic/classic one.
- Offset(s,t)= ±CSF(N(s)|t, (K(s)|t)^0.5)
- and, as a cherry on the cake …
- 𝓜(s,t, 𝓩(t)) = 𝓩(Offset(s,t))
What does power (^0.5) modification of K(s) values mean?
It means the offset's fronts with k(s) curvature values greater than 1.0 will progress slower than the k(s)<1 curvature curve's segments, as when the power exponent is 1.
In simpler words, which might be pretty cryptic now … the sword tip will disappear at a slower pace as its arch curve's segment has a small radius, thus a large curvature value.
As a testing object, I will use … what I have caught (with some effort) in the Pacific/Indian Oceans!
It is a beautiful but intricate and CADangerous species. I attached it as a plain sketch… in case you would like to check it yourself. Do it at your own risk, though!
Below are visuals of the first CSF species' training session called Mode: Shorten.
Some other lessons will follow over the coming days/weeks.
The title of the next one: CSF – CurveShorteningFlow Mode: ?????
Attached files:
..//Shorten//..
CurveShorteningFlow_template.f3d F3D ( 58 KB) https://a360.co/3Q075B3
CurveShorteningFlow_A_mono.png 4K_mono (1.6 MB) https://a360.co/46xkHcu
CurveShorteningFlow_C_mono.png 4K_mono (1.4 MB) https://a360.co/469a6nX
CurveShorteningFlow_A_arcd.png 4K_stereo (2.6 MB) https://a360.co/3Pm0hvK
CurveShorteningFlow_C_arcd.png 4K_stereo (2.1 MB) https://a360.co/3PAOCJN
CurveShorteningFlow_mono.mp4 4K_mono ( 67 MB) https://a360.co/48BpE5O
CurveShorteningFlow_arcd.mp4 4K_stereo (110 MB) https://a360.co/48C4lkk
..//ZMapping//..
CurveShorteningFlow_B_mono.png 4K_mono ( 6 MB) https://a360.co/3tjXifY
CurveShorteningFlow_B_arcd.png 4K_stereo ( 8 MB) https://a360.co/3RG2k0I
CurveShorteningFlow_mono.mp4 4K_mono ( 41 MB) https://a360.co/3RGRXtj
CurveShorteningFlow_arcd.mp4 4K_stereo ( 97 MB) https://a360.co/3tlKR3h
To be viewed on 4K media devices (monitors, UHD TVs, projectors...) of reasonable performance. For the best experience, use stand-alone media applications (WMP, VLC) and the native resolution 3840x2160 - full screen. The '_arcd' files require an anaglyph red/cyan glasses, while '_al' is for active shutter glasses 3D hardware (~30 deg viewing angle is recommended). Download the files over a network, where the cost of doing so is not a concern. The files are to be used for private, non-commercial purposes only.
Regards
MichaelT
