I developed this math I thought would be cool to have in inkscape... Stuff like text along a quadratic curve and the cage transform are special cases of this more general formula... http://benpaulthurstonblog.blogspot.com/2015/12/parameterized-quadratic-conformal.html

Oh, ok I guess I was thinking it didn't show the url in the preview so I thought I was doing it wrong... Someone also pointed out the red dots in my drawing don't quite match up that's because I was manually drawing the dots, but for example 520,20 is a point given in the program and it is on the cu...

You were right about the endpoints, the first and last affect the curve a little but can be put whereever.
http://benpaulthurstonblog.blogspot.com/2013/07/smooth-curve-generator-implemented-in.html