Determining whether surface is true hyperbolic paraboloid

Hi guys, I’m stumped with a problem and thought this is the best place to come for guidance, especially with the likes of @DanielPiker on the forum :sweat_smile:

I have a script that produces a Candela surface that seems to the eye to be a perfect hyperbolic paraboloid surface. I need to determine whether the underlying surface is truly, mathematically, a hyperbolic parabola (or behaves like one). Full disclaimer: I did not write the script myself. In fact, I found it on this forum around a month ago but have been unable to find the original author after searching for a long time. If you recognize the script and know the original author, please let me know. I have slightly modified the script. (original author found: @doolittles source )

I really like this script because it allows me to control the exact parameters I need for my particular application (I will model variations and run structural analysis on them and hopefully turn this into a master’s thesis)

I appreciate any guidance in determining whether the surface produced is really part of a hyperbolic parabola.

hypar_alp_3.gh (23.9 KB)

EDIT: short animation showing the formula for the curves forming the shape. The curves are lofted in order from nearest to furthest.

I do not understand how the formula for the curves (which I derived from the script) fits into the general formula for hyperbolic paraboloids. I would appreciate help in making the connection.

I also want to mention: I understand how the script works and how it produces the surface, however I cannot wrap my mind around how the method in the script relates to the general formula for hyperbolic paraboloids: z=(x²/a²)-(y²/b²)

I have tried several methods to determine whether points on the surface satisfy this formula but have not had success.

Bumping for visibility :slightly_smiling_face:

EDIT:


hypar_alp_3b.gh (17.0 KB)
using your surface and trying to push a and b values to reach that shape with galapagos… it almost reach it… an average error of 0.006 units…

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A hyperbolic parabaloid has negative Gaussian curvature at all points along the surface.
You could sample the surface to your desired level of resolution and ensure that the curvature at each of these sampled points are all negative.


hypar_alp_3.gh (25.8 KB)

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A hyperbolic paraboloid is not unique in having everywhere-negative curvature. This test can rule out a surface if it has zero or positive curvature, but it isn’t conclusive.

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Eek–good point.
It’s a necessary but insufficient condition :slight_smile:

Great analysis, it confirms my suspicion that the shape is indeed a hyperbolic paraboloid, however, it doesn’t help me understand WHY, mathematically, the script produces one :sweat_smile: I guess my problem is more of a math problem than modelling problem. I have added a small animation at the end of the post to show where I am stuck, I do not understand how the curves fit to hyperbolic paraboloid formula. Thank you for the valuable analysis.

Yes, I read about the negative Gaussian curvature. Although this helps me confirm that the shape is indeed a hyperbolic paraboloid, it doesn’t help me understand mathematically how the script produces the hyperbolic paraboloid. I’m not able to make the connection between the script and the general formula for hyperbolic paraboloids. Thank you for your input, it showed me that I can cause a positive gaussian curvature when I play with the parameters so it means it is not hyperbolic for every value.

bumping with the hope that I might be able to get helpful direction :slightly_smiling_face:

First, the source does not make sense, there is no script mentinoed.

A known way of constructing HP surfaces is the translation of a parabola y(x) = ax^2 along another parabola with negatively signed curvature.

While the open edge of the provided surface seem to be indeed very close to a parabola, I am very skeptical about the way the surface has been created with a IMO problematic loft, which inherently creates a singularity as all loft profiles share the same point and also I am not sure what the Tight Loft option is exactly doing.
Per definition, it should be easy to check if your surface is a true HP surface checking the following geometric property:

All sections along the z-axis should result in parabolas, with 2 exceptions along the asymptotes.

A simple vertical section red shows that this is not true:

I constructed the blue parabola using the mid-point of the section and using 2x midpoint height as the second control point of a degree 2 Nurbs curve. In contrast to that, the outer inclined curve seem to fit a parabola pretty closely.

My suggestion would be to just create your own custom HP surface script, then you know what you are creating exactly. Not a mathematician, so take everything with a grain of salt.