Cineversity

Hello

What is a good way to create the attached shape?  Would it be a formula spline? 

thanks!

Hi Alex,

I have no idea about the formula that would be needed to describe this Mathematical Knot in the Formula Spline.

Each of those shapes follows a clear idea. This is what I have set up with some Primitives, replaced all of it with Splines, and connected them. (I have placed all of that in a Layer [prep]. Just in case you have any interest at all. A previous idea was based on an animated Null and trace its path. However, I preferred the building of the path more.

Scene file
https://www.amazon.com/clouddrive/share/FtDkCXGBa4g0MvCvvv0qTnOcmx5YjgtZbsVPdM4dcQA

Let me know if there anything else; I’m happy to look into it.

Cheers

P.S.: Working through those examples gives me more about their “inner logic”.

Gotten a little bit more in play mode, I got this prep set up:
https://www.amazon.com/clouddrive/share/Sf7wUPvstL5w7Onn0qQyZKZpWyeqFM7BdwKofjHUVRC

(I can see now how I would animate this to “draw” this shape. If needed.)

Hey Sassi!
Thanks for this.  I was looking at your “Prep” layer, that seems quite complex.  Is the idea to just create the shape with a series of partial Torii? Also curious how you setup the second file, was the spline just “hand drawn”?

Thanks so much! I look forward to understanding your process more!

Hi Alex,

The Tori was just an exploration to see how the “Knot” was set up in the provided image. In the picture, it looks like that it is not based on repetition (triple). Edit: the stripes threw me off. /edit. Hence my exploration. It seems more like two half Tori, and another half Torus creates the base, and then they were connected. Those connections are more organic than geometrical.
When I had the shape, I turned the radius down and replaced it with Spline fragments. After they were in the right sequence, I placed them in a Tracer (Connect), then in a MoSpline. The MoSpline allowed me to produce a new Spline with a freely adjustable vertex amount. That was my target to get as little point amounts as possible. With these few points, I was able to get the shape in “flow”.

In the “P.S.” post, I shared something similar but equal in its three parts. So no patchwork as in the image. When you switch off the Cloner, you can see two Splines only. One half-circle and a “connection” element. I find this cleaner. But you decide which one to use.

I “traced the last shape with a null rig (animation) and saw what motion it needed, which helped to understand it a bit more. I might check later if I can get it into a formula. But no promises.

I hope the results so far helped.

My best wishes for your project.

P.S.:
Here is the animated version to create that spline, via a Null-Rig and a Tracer. Let it run from 0-240 frames, and while the Tracer is active, use the Object Manager> Object> Current State to Object to “harvest” the spline.
Project file
https://www.amazon.com/clouddrive/share/LILjMDwWY8k5eKMxPGetOKarHHB9rRE2tSsoGbR6AQq

Which lead me also to bake that motion of the Null to see the F-Curve. These curves show me what the formula would need to express per “track” to create that constructions. It doesn’t look like a simple sine or cosine but a mixture out of it. (see image below).
In the Timeline>F-Curve> Functions is an option to create out of the Positions Tracks a spline, which showed to be congruent with the drawing of it.
I mention this only to be more complete with this thread and to point to some more or less hidden features.

Cheers

Thanks Sassi!  This is great.  I did a bit of research as well as I had a hunch it had to have some kind of mathematical formula.  I found out that it is called a “Trefoil”, and I saw that 3d Meier had a formula for it on his website.
This is what it becomes with a formula spline
https://www.dropbox.com/s/xtxq9ibmxm9nlg7/Trefoil.c4d?dl=0

Your methods are really useful though, as I’m embarking on a project where I’d have to create a series of objects based on mathematical formula and I might not always find the right ones, so knowing how to build them manually is extremely valuable.

Thanks again!!

Hi Alex,

Great that gives me something to learn. I think I had used Meier’s site around two decades ago, so glad it has not vanished. Thanks for pointing this out!

Yes, that is an exquisite formula. I can’t find that formula, not that short, on his site. I checked all “Knoten” (knots). I found only more complex formulas with the same (?) result.

I think the way with MoGraph was the cleanest manual way.

Thanks for your post!

Cheers

Hey Sassi

Yes so the shorter version comes from Wikipedia : https://en.wikipedia.org/wiki/Trefoil_knot

And yes, Mograph approach is great!

Thanks again!

You’re very welcome, Alex, thanks as well from my side.

I’m undoubtedly enjoying it. For me, it is always to feel that there is is an interest and a need to dive more deeply into it. My target is mostly to find something that pulls it out of the abstract formula and gives some clues to adapt it. Like the graphs I showed above was the first step toward it.

The shorter Trefoil was a huge help, as it is so clear to take it apart or alter it, to see what happens.

sin(t)+2*sin(2*t)
cos(t)-2*cos(2*t)
-sin(3*t)

My current favorite short tip would be to count the back and fore movements (cycles) in each direction (X, Y, Z) To get a rough idea about the formula that is needed. The formula above has two full movements in x and y but three in Z.

sin(t)+2*sin(3*t)
cos(t)-2*cos(3*t)
-sin(4*t)
x and y has 3 and z 4 of such back and fore movements.

Or:
sin(t)
cos(t)
1 Has only one for x and y and zero for z, a circle.

(I’m not convinced that this will help master a vast majority of “shapes,” but it is a start. Thanks for the interest, which gives me the motivation to break those things down.)

My best wishes.

Sassi, I have one more question…
Your texture seems to be creating some odd aliasing…See image.
What’s odd is that even if I set the anti-aliasing pretty high it doesn’t go away.
This seems to be a product of the combination of both checkerboard layers, which I understand you created to make thicker black lines.
What do you think we can do to remove the aliasing?

Thanks!

Hi Alex,

Please have a look here:
https://www.amazon.com/clouddrive/share/fGmm6zbxOLk3x9YiQ2MqvL3oav8neDNfssL2BHim8RW

There are two areas where it can fail.

The first is given where the stripes small and large meet, there is a slight overlap that causes the thin lines. This can’t be healed with AA.
I have addressed this with a texture.

The second problem with this is the “borders”, where the camera “rays” touches perpendicular the surface. Which means the white will be infinitely thin.
This is address with a Fresnel.

I hope this will work for you. Sorry about the trouble, I slapped this Checkerboard shader on it to show the animation of it.

All the best

That works perfectly!  Thanks Sassi!

You’re very welcome, Alex. Thanks for the reply.

Let me know if there is anything else. I’m happy to look into it.
https://www.amazon.com/clouddrive/share/PqIW37Rk4iewAytMikrrZdmhzRPaDmQdzT25t2mCa1x

Enjoy.

P.S.: Image below is a Gif