B How to find a curve function

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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?

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fresh_42

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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))\,?##
 
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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.
 
Quiet helpful. thank you!
 

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