My 2 cents on the parabolic/catenary.
"Just an opinion"
A cable must be completley flexable to hang in a true catenary. Line conductors a very stiff and do not hang in a true catenary. A chain would probably be very close to a true catenary. I believe catenary is latin for chain? So...
IEEE proceedings
1979
"TECHNIQUE TO INCLUDE ELEVATED TEMPERATURE CREEP IN CONDUCTOR SAG-TENSION CALCULATIONS"
Authors Harvey and Larson
I was unable to reproduce the example problems I am sure I am missing something simple, If you figure it out let me know.