The discussion explores the feasibility of determining a mathematical function that defines a random curve drawn on graph paper, with a focus on smooth curves and the potential for complex curves. Participants consider various methods and approaches for approximating these curves.
Participants express differing views on the methods for approximating curves, with some advocating for polynomial interpolation and others for cubic splines. No consensus is reached on the best approach or the feasibility of finding a definitive function for arbitrary curves.
Participants acknowledge limitations in their methods, such as the dependency on the number of points chosen and the nature of the curves being approximated. There is also an implicit assumption that the curves are smooth.
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.thetexan said:
How do you sketch a four dimensional graph, ##(Re(x_i)+i\cdot Im(x_i)\; , \;Re(y_i)+i\cdot Im(y_i))\,?##