I literally want to draw a curve on a paper, and get Mathcad to predict the equation!!
That is not possible. Not for Mathcad, not for any software. If you have data that represents a curve and you already know the functional form of the curve, but not the parameters, then in principle you can find the parameters. For example, you might know it's quadratic, in which case you can fit a quadratic to the data to find the coefficients. There is no software that can tell you it's a quadratic though. You could also fit a general interpolating function to the curve, for example a cubic spline. That's not "the" equation of the curve though, it's just an equation that passes through the known points.
i know a software named 1stOpt can do this,but the data must have only one independentvariable，that means f(x),not f(x,y),f(x,y,z).... you may ask how it works,the theory is simple.it has storaged many functions，exponential, logarithmic, trigonometric...then you enter your data，it fit the data using more than 3000 functions，at last,you can pick up the function that fit the data best.
I have never tried this but often wondered .. what if I drew a curve on a piece of paper and scanned it, then brought the scan in as a bit map. Now use the imaging function binarize, turning the scanned image into a white uniform grid with black data points representing the curve. In line programming could allow me to loop thru rows/cols, when image/matrix value is 0 (black) I get x,y from row.col. Then I just use regression/curve fitting to get the equation.
Then I just use regression/curve fitting to get the equation.
What am I missing? Why will this not work?
You don't know the functional form of "the" equation. You can guess a form, and fit that to the curve. If your guessed function has fewer parameters than there are data points you will not get an exact fit though, unkess by some small miracle you actually did guess "the" equation. Or you could use interpolation, for example a cubic spline, which will pass through every point, but is that "the" equation?
And if the curve is multivalued in both x and y then a cubic spline will not work. A parametric spline will of course, but then you don't even have an equation, you have two equations in an independent variable.
And what if I draw both a vertical line and a horizontal line as part of my curve? Then it is undefined at one point in both dimesions.
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