Can You Find the Function of a Random Curve on Graph Paper?

[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

 

[YoungPhysicist]

The answer should be yes according to Fourier transform.
You can watch a video by SmarterEveryDay to get my point
https://www.physicsforums.com/media/fourier-series-visualized-by-drawing-circles.6111/

 

[fresh_42]

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!