I think the first part of the curve (the part between your hand and the stationary "node" where the curve crosses back over the vertical axis) would be described by the Troposkein Curve, or at least be extremely similar.

This curve would, I imagine, either fully describe, or at least be related to, any other segments between nodes if you had more than one.

The last "tail" bit at the bottom of the rope would be different, obviously, but you could probably get the shape using a similar method.

Shape of rotating rope (lasso problem?)
The Pact Magic paragraph on page 164 of the Player's Handbook multiclassing section reads: 


  you can use the spell slots you gain from the Spellcasting class feature to cast warlock spells you know.


If I were to multiclass as a Warlock 5 / Rogue 3 Arcane Trickster and cast a Warlock spell using my Arcane Trickster's 1st level spell slot, would it be cast as a 3rd level spell?

From the Warlock Spell Slots section on page 107 of the Player's Handbook: 


  The table also shows what the level of these slots is: all of your spell slots are the same level.


The proposed duplicate question appears to only address the issue of spell slot recovery, not casting.
