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

In summary, it is possible to draw a random curve on a piece of graph paper and find the function that defines that curve, assuming smooth curves. This can be done by choosing a sufficient number of points and solving for the coefficients of a polynomial. It is also possible to do so with complex curves by using separate cubic splines for the real and imaginary parts. This method gives a natural looking curve that follows the points in a expected manner.
  • #1
thetexan
266
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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?

tex
 
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  • #3
thetexan said:
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))\,?##
 
  • #4
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.
 
  • #5
Quiet helpful. thank you!
 

1. What is a curve function?

A curve function is a mathematical equation that describes the relationship between two or more variables in a graph or plot. It is often used to model real-world phenomena and can be represented by a line, parabola, sine wave, or other mathematical shape.

2. How do I find a curve function given a set of data points?

There are several methods for finding a curve function from a set of data points. One common approach is to use regression analysis, which involves fitting a curve to the data using a mathematical function that best represents the relationship between the variables. Another method is to use interpolation, where the curve function is constructed to pass through each data point.

3. Can a curve function accurately predict future data points?

In most cases, a curve function can provide a good estimate of future data points. However, it is important to note that the accuracy of the prediction depends on the quality and quantity of the data used to create the curve function. Additionally, if the underlying relationship between the variables changes, the curve function may no longer accurately predict future data points.

4. Are there different types of curve functions?

Yes, there are many different types of curve functions, each with its own unique shape and mathematical formula. Some common types include linear, quadratic, exponential, logarithmic, and trigonometric functions. The choice of which curve function to use depends on the nature of the data and the relationship between the variables.

5. How can I use a curve function in my research or experiments?

Curve functions can be useful for understanding and analyzing data, making predictions, and modeling real-world phenomena. They are commonly used in scientific research and experiments to describe and explain the relationship between variables and to make predictions about future outcomes. Curve functions can also be used to develop mathematical models for various systems and processes.

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