Bug: Catenary fails to go trough apex or through point

[encephalon]

i am not sure if that is a bug or my lack of understanding how this curve should work. when i create 3 points that i want to have the catenary go through, the result ends slightly above. i tried method through and apex with the same result. it helps when i go into ridiculous amounts of points but i never quite reach it.

to be fair the discrepancy is really small but i still would like to have an exact result due to modelling purposes. anything i can do or is it a bug?

catenary discrepancy.3dm (17.3 KB)

[jeremy5 (Jeremy)]

My understanding is that a catenary can only be approximated by a NURBS curve. That’s why the Rhino function returns a tolerance value. As you’ve found, you need to increase the number of points to get a closer approximation.

Using the default 20 points you are within a tolerance of approx. 0.00661828. With 40 points you would get 0.000314187, which is better than the commonly used 0.001 Rhino modelling tolerance. This catenary is 0.0000493 units above your target apex point. If this bothers you, I imagine that were you to scale it to a drop of exactly 70 units the catenary would not be visibly inaccurate.

HTH
Jeremy

[davidcockey]

Use an odd number of points. Then a knot will be at the lowest point for a symmetric catenary and for this example will be at the mid-point within 0.00000 or better. With an even number of points the lowest point is at the middle of a span of the NURBS curve.

[davidcockey]

From a previous thread:

@davidcockey

A catenary curve can be approximated using Catenary command to any desired level of accuracy, up to a relative accuracy of e-11 or so based on my testing. That level of relative accuracy is more than sufficient for any design or engineering use of a catenary curve.

Catenary curves are used in civil engineering and elsewhere as an approximation of a cable hanging between supports. For a cable to hang in an exact catenary curve the cable would have to have zero bending stiffness and uniform weight/load along the entire length. No cable in a real application satisfies those requirements exactly. However a catenary curve is a useful approximation for many engineering purposes. There is no need to calculate the catenary curve to multiple orders of accuracy better than the inherent deviation is using the catenary curve to model the physical situation.

More accurate calculations of real cables in real applications can be made, if needed, using various methods such as finite element modeling. But even those calculations will be approximations to the “real world” for various reasons.

@davidcockey

A relative accuracy of e-11 or 0.0000000001 means a catenary 1 km long would have a maximum deviation of 0.00001 mm.

Fabrication accuracy, material variation, enviromental effects, load not being exactly uniform and cable bending stiffness of an actual structure application will cause a real cable to deviate from an exact mathematical catenary curve by far more than a relative accuracy of 0.00000000001.

The accuracy of any structural analysis compared to reality will be much, much less than a relative accuracy of 0.0000000001.

[encephalon]

@davidcockey

Use an odd number of points.

thank you @davidcockey that worked.

@jeremy5

My understanding is that a catenary can only be approximated by a NURBS curve.

yes nurbs are always an approximation mathematical.. i did not think about that. davids solution solves it actually.

but i am thinking should it not be more accurate and choose the 2 points aside of it so that one still can hit the apex?

also generally, since it seems to be a control point curve maybe an interpolated would be better here if that is possible at all.

[davidcockey]

@encephalon

also generally, since it seems to be a control point curve maybe an interpolated would be better here if that is possible at all.

All curves in Rhino are NURBS curves. There are not “control point curves” and “interpolated” curves. The coefficients of a NURBS curve are the control points. The coefficients/control points can be determined in multiple ways including directly specifying control point locations or interpolation.

My guess of how the Catenary command works:
The equation of the exact catenary curve which passes through the three specified points with the specified direction is determined.
A set of points along this exact catenary curve is determined. The number of these points is as specified. Except in special situations none of these points will exactly coincide with the specified interior point.
A NURBS curve is interpolated through the set of calculated points.

[encephalon]

when this set of points is determined, it should just respect that one pick point as part of the value through which the calculation has to go through. i have no idea how that works here in Rhino but theoretically i assume that should be possible?

so i guess the command is developed with some lack of control when the pick point or apex is just approximated. it most certainly is not ideal having to revert to some work around or some peculiarity in mind.

i have marked it as Bug for now.. so its not a real solution though your solution works for now.