# B How to find a curve function

#### thetexan

is it possible to draw a random curve on a piece of graph paper and find the function that defines that curve? Assuming smooth curves.

And if so,is it possible to do so with complex curves?

tex

#### fresh_42

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2018 Award
is it possible to draw a random curve on a piece of graph paper and find the function that defines that curve? Assuming smooth curves.
Approximately, yes. You can choose as many points $(x_i,y_i)$ you like - the more the better - say $n+1$ many, then set $p(x)=a_0x^n+a_1x^{n-1}+\ldots +a_{n-1}x+a_n$ and solve $p(x_i)=y_i$ for the coefficients $a_i$. That doesn't give you the correct answer in case your function is defined otherwise and you only drew a certain part of the graph, but it is a good approximation for what you have drawn.
And if so,is it possible to do so with complex curves?
How do you sketch a four dimensional graph, $(Re(x_i)+i\cdot Im(x_i)\; , \;Re(y_i)+i\cdot Im(y_i))\,?$

#### RPinPA

In general the most common approach, the one that is used in computer graphics to go through an arbitrary set of points, is cubic splines. Between each pair of points is a different cubic polynomial $y = a_0 + a_1 x + a_2 x^2 + a_3 x^3$ with different coefficients. There are four free coefficients on each segment which are chosen so that the curves pass through the points and also meet smoothly.

For a complex curve, you'd use separate splines for the real and imaginary parts. I've done that on a number of occasions in fact.

Similarly, for a curve that doubles back on itself like a circle or something more complicated, you would use separate cubics for $x$ and $y$.

There are infinitely many smooth curves that go through a given set of points, since you aren't restricting what happens between those points. But cubic splines usually give a natural looking curve, one that follows the points in a way you would expect.

#### Miharu Edogawa

Quiet helpful. thank you!

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