I would like to define a polynomial that passes through two points and has a particular gradient at those two points. I would also like the curve to increase in gradient as smoothly/progressively as possible between the two points. That is to say, the simplest possible curve with no inflections between the points.
Given the four constraints I have tried to define a cubic polynomial but in doing so I have ended up with an inflection between the two points.
What can I do to force the solution to avoid the inflection? Do I need to add more constraints? Would this require me to raise the order of the curve? Can I translate the problem elsewhere to avoid the inflections, find the solution and then translate it back?
My parameters are:
$y(0)=0$
$y(0.077655)=18354.465$
$y'(0)=54000$
$y'(0.077655)=13500000$
I have just used a simple matrix inversion method in Excel (using the built in functions) and the coefficients returned are:
$a=2169259231$
$b=-166105501.7$
$c=54000$
$d=0$
This satsifies my constraints but unfortunately also puts an inflection between the two points.
I would be grateful for any guidance. Regards, Simon.
Thanks for the comments. The value of y' at the second point doesn't need to be exact but varying its value shows that it needs to be down around 500,000 before the curve behaves as I would like, so that is not an option.
Two further thoughts are: if I raise the order by one so that I can add a further constraint, how do I state that there should be no inflections between the points?
Secondly, perhaps I can change tack and use an exponential curve or some other alternative to get a smoothly rising curve between the points. Any advice on using my four constraints in that way?