It is the site of the article "Catenary = Physic + Math + IT" (color picture + Mathcad sheet)
We can have thru 5 points a hyperbola (D<0), an ellipse (D>0) or (good chance D=0) a parabola:
The solution with the Find Function: the partial vectorization!
And second - we can add more points, change the Find function to MinErr function and get not the interpolation but approximation!
With lsolve function! We do not need guess values! But how can we add more points (6, 7, 9 etc) without change the Matrix M and the vector v?
One more experiment (see please the attached Mathcad 15 file)