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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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    "seems fairly heavy on computation" - not if you restrict yourself to **cubic** polynomials, as Gerry says. You have four conditions in the previous comment I gave; that is sufficient to compute the values of $a,b,c,d$ in Gerry's answer.2012-08-21

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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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    Well, part of it will look like $D$. Then somewhere outside the picture, the graph will turn down.2012-08-21
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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 $x \to -\infty$.