reverse-way to make a hyperbolic paraboloid surface

hallo,

i create a boundry of hyperbolic paraboloid with expression below:

(r^2/b^2*(sin(z*2*pi/y))^2)-(r^2/a^2*(cos(z*2*pi/y))^2)

It get exact math definition of hyperbolic paraboloid, I just give it a boundary and Delta Z.

in order to get a hyperbolic paraboloid surface we are used using 'project-method', that mean we create a hyperbolic paraboloid surface and then we just project a curve onto this surface.

I am very curious if anyone has been tried to create a boundary-limited hyperbolic paraboloid surface without 'projcting' or 'cutting' a surface, but creating it by 'reverse' way.

I am mean just use a undulation-boundry and reconstruct it into hyperbolic paraboloid surface!!

Attachments:

Hyperbolic Paraboloid.gh, 6 KB

Replies to This Discussion

Permalink Reply by Hannes Löschke on April 2, 2013 at 2:01pm

NURBS surfaces will always have a quad UV grid underneath. The base surface can simply be built by a loft between two straight lines. This will always give a well defined HypPar surface in between. You just need to trim the bounds.

On the other hand by matching the UV quad edges to the boundary, you will distort the whole grid. NURBS surface points rarely are on the surface but above or below. While the points on the edge will easily coincide with your border, you'd need to interpolate/adjust the points inside the boundary. A lot of work for a task that already has a very simple solution.

May I ask for your motive for this reverse approach?

Permalink Reply by Jon on April 3, 2013 at 2:53am

thank you hannes,

i understand that there is a very simple solution already. i am just wondering that rhino/gh dosent provide this most used geometry as one of 'primitive'.

i think we could simplify the question: the approache we define a boundary on a hypPar/ freeform surface.

i beliebe most of people use projection/trimming approach, no doubt it make sense and it is the most of easy way to deal with.

the idea was: becoz there is only one math expression of hypPar. it means there is only one surface-curvature to be create, even when we change all of parameter of length, width od hieght, but the curvature must always follows the math expression. So if we could creat his boundary(like what i created before) by math expression, why cant we create his surface as well?

Permalink Reply by Hannes Löschke on April 3, 2013 at 1:33pm

A NURBS hyppar needs 4 CPs to define the surface. Spacing and vertical displacement of diagonally opposing points define curvature in "hyp" and "par" direction. Now you can project any boundary, you like and get a perfect hyppar.

Actually, the formula is the result of a procection of the XY plane coordinates onto an infinite Hyppar surface definded by a and b. You may realize that NURBs control points only coincide with the curve(or surface) at the ends (edges).

Now if you want to fit the quad structure of the NURBs surface you will need at least 8 points for a flat circle, more if you want to fit a 3D boundary. Divide that number by 4 and square the result to get the number of CPs you need at least to create CP grid for the surface. I may be missing a clever way to fit the CPs inside the boundary to get hyppar surface but since the CPs need to kind of pull the surface, they will need to be positioned above or below the final surface.

Except for the boundary, your formula cannot be directly used to find the CP positions.

You may be talking about a mesh primitive to create a hyppar. Then of course, every vertex location is a solution to the formula. But meshes are not the core concept of Rhino modelling. Precision depends on the number of vertices created. On the other hand, the NURBs is precise to whatever precision you want...

Permalink Reply by Jon on April 4, 2013 at 1:59am

thx for explanation, its pretty clear.

as i said, i am trying to get a new way to approache a hyppar, except cutting/projection-way. i just discussed with a engineer, maybe there is still a smart way deal with...

well, wish me luck. so far i get something i let u know!

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