generative modeling for Rhino
As the title say - when you use the "4 point surface" component, you get the a surface governed by the four boundary lines that span between the points you feed the component.
But what kind of surface is it? especially, is it the minimal surface?
its not a minimal surface, think of it as a loft between 2 lines.
It is either a non-planar or planar untrimmed surface. But it is definitely not a minimal surface.
Ok, so its not a minimal surface ;) And I know its just a planar surface if the four points are in the same plane.
But my question is related to the mathematical expression behind the component. The reason why I thought it was a minimal surface is that I can't find any information on the subject and it was the only logical explanation I could think of... Also, I did some visual tests of the component-output and tried to create some simple and well known minimal surfaces and it looked like a match.
Afterwards a friend told me to test the mean curvature, and naturally you're right its not a minimal surface.
@ Michael - If it's just a loft between two lines, why not just use the loft component instead? And the loft feature in Rhino has settings, which causes different output...? (can't remember if the Loft component has the safe settings - I'm on computer without RH...)
@ Danny - you say its a non-planar untrimmed surface, but what kind of surface is that? Again, what's the mathematical expression behind the surface?
I discussed the issue with a friend who uses GH every day at work, and they (him & his co-workers) had often had issues with not knowing the math behind the components... Is there somewhere to find this information?
I mean, I use GH for parametric structural design, which also apply results from structural analysis software; and I must be able to study the math behind the FEM computation - otherwise it's no good - and I can't use computations I can't verify! And the same goes for the geometric output I get from Rhino and GH!
Why not use loft instead? well this depends how your data is structure, most obviously maybe you have 4 points or you have 2 lines. In this case all loft settings will have the same results as a surface from 4 points will always have straight edges from point to point, so it is similar to lofting a straight line to another straight line, this is true for 2 straight lines. See here a surface made from surface 4 point (left) or lofting two edges (right) are exactly the same.
The comment was related to - why have the component "4 point surface" if it delivers the same result as "loft" when governed by the same boundary conditions, i.e.:
"4 point surface" = 4 points -> 2 straight lines -> lofted surface
Nevermind - i get the difference between the two components...
But my issue is still - what is the mathematical condition, which governs the output from "4 point surface", or "loft" for that matter?
"This is the difference with minimal surface as the path between points of the surface edge most likely will not be straight." quote from Michael
Yes, the minimal surface governed by four points will, naturally, not have straight edges; however, the component may create the straight lines between the points and the create the area-minimizing surface between these lines, i.e. a soap-film-surface???
I was under the impression that a soap-film-surface is also a minimal surface, but maybe that's where I'm wrong??? A soap-film-surface is the area-minimizing-surface governed by the boundary conditions, which may have straight edges... Well, actually the soap film creates the surface, which have a minimized internal work, which equal the minimized area.
So my (new) question is: does the loft and "4 point surface" create the a surface with a minimized area?
Thank you for taking the time to answer my questions, but I find it a bit curious that nobody seems to know the math behind the features many people use every day! Or is there a well known math-expression behind the term "lofted surface", which I haven't realized???
I removed that quote as I was not thinking when I wrote it. It is false
I'm not really sure how one would define a minimal surface from 4 points only without also setting the edge curves. If you think of a soap film (which does approximate a minimal surface), attaching to only points doesn't work to make a surface.
There does exist a minimal surface with a boundary of 4 straight edges, but it is not actually a hyperbolic paraboloid.
Many years ago I got into a discussion with Chuck Hoberman about this, and ended up writing this blog post:
which describes this in a bit more detail
No, that's true - you need some kind of boundary condition
I think the technical definition of a surface like this is that it is a portion of a hyperbolic paraboloid. Or maybe a parabolic hyperboloid, I can never keep those apart.
I did a test yesterday - and "4 point surface" on the points (0,0,0) (1,0,1) (1,1,0) and (1,0,1) - thats a square in xy plan, where the 1st and 3rd point has z=0 and 2nd and 4th point has z=1 - and that did create a hyperbolic paraboloid.
Again, what's the mathematical expression behind the surface?
Again? I have not seen you ask this question before. The answer is NURBS - non-uniform rational B-splines Surfaces
The horrible maths is a tensor product that looks like this:
Do you apply your FEA to Surfaces or Meshes made from Surfaces?