# What is the name of this curve (figure inside)?

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:

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

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