Wire Tension Formula: Shape Expression w/ 3 Pivots

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TL;DR
If the wire is bent by three pivots, I want to find an expression that represents its shape.
If the wire is bent by three pivots, I want to find an expression that represents its shape.
There will be tension of wire and other physical factors.
How can it be expressed?
 
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That does not sound like the elastica problem at all to me. The elastica is the shape of a post-buckled column, but OP is talking about a bending problem.
 
Dr.D said:
That does not sound like the elastica problem at all to me. The elastica is the shape of a post-buckled column, but OP is talking about a bending problem.
The elastica is a more general field. You are considering only your experience and should also consider other things like meanders, splines and Bezier curves.

The "pivots" are control points or boundary conditions, while the elastic deformation is distributed through the elastic bending of the rod between the pivots.

Because the elastica problem can become complex, the OP needs to present a more complete description of the question. Maybe define "pivot" and "wire".
 
Baluncore said:
The elastica is a more general field. You are considering only your experience
No, you are mistaken. The term elastica is well defined as the shape of the post-buckled column. For all the other items you mention, we have their own names to identify them.
 
Dr.D said:
No, you are mistaken.
The Elastica is now a multidisciplinary field of applied mathematics. While it can obviously be applied to the shape of a slender column, it is not restricted to buckling failures in structural engineering.

As a narrow minded structural engineer you have no right to censor the OP, nor to deny the use of Elastica, for example, to biological structural membranes and their equilibrium positions.

I suggest you find and study a copy of “The Many Faces of Elastica” by Ivaïlo M. Mladenov and Mariana Hadzhilazova. 2017.
 
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