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A can be written as... $y = a$

B can be written as... $y = bx + a$

C can be written as... $y = cx^2 + bx + a$

1) How can I write D? I was looking at implementing bezier curves into some code but is there something similar which could be computed much quicker?

2) C is a quadratic... what is the name for D?

Edit: enter image description here

Perhaps if I explain what I'm doing.... I am trying to build a decent servo controller. I have a servo which will move from servo position 30 to 0. I want it to move in a more elegant way than B (above)... ideally like in the picture. I will come up with a way of defining the movement between 2 positions when I send a command to the controller. Perhaps this is a quadratic equation which could be calculated easily for each x (time) between y (servo angles)... this would mean I would only need to pass a, b and c to the controller along with the time (length) and both positions.

I figured it would be better to have it able to be more irregular... but still super quick to compute. I considered bezier curves as a definition for the line, but they seem overcomplicated and slow to compute... if there anything else I could use?

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It's hard to tell. There are a lot of curves that locally look like that. If you want it to look like a rotated parabola, for example, that's pretty easy to do. –  Potato Aug 21 '12 at 4:14
    
If you fix the two endpoints of curve $D$, and you know the slope of the curve at those endpoints, you can build the Hermite interpolating polynomial that more or less looks like curve $D$. –  J. M. Aug 21 '12 at 4:14
    
I googled hermite interpolation. looks promising. seems fairly heavy on computation, though. –  Beakie Aug 21 '12 at 4:41
    
@J.M. i think u answered my question.... please add it as answer and i will mark it as such –  Beakie Aug 21 '12 at 4:44
    
For a rotated parabola, $y$ would not be a function of $x$ (globally). –  Robert Israel Aug 21 '12 at 4:45
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2 Answers

One can find a cubic, $y=ax^3+bx^2+cx+d$, looking like $D$.

EDIT: If you want a curve that looks like the blue curve in the edited version of the question, you can pick a few points you want the curve to go through and then join them up with cubic splines. There is a lot of info about these on the web and in the more applied Linear Algebra textbooks and elsewhere, and while some of the formulas may look complicated, there's really nothing there that a computer can't handle quickly and efficiently.

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please see above edit –  Beakie Aug 21 '12 at 4:30
    
Well, part of it will look like $D$. Then somewhere outside the picture, the graph will turn down. –  Robert Israel Aug 21 '12 at 4:47
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To me, $D$ looks more like a branch of a hyperbola, which could have an equation $y = c x + \sqrt{a^2 x^2 + b}$ with $b > 0$, $|c| < a$. This is asymptotic to $y = (a+c) x$ as $x \to +\infty$ and $y = (-a+c) x$ as $x \to -\infty$.

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