Need to find the equation of a curve from a Hand drawing

Question

My uncle makes hand drawings of furniture on a large piece of chart paper at 1:1 scale. He has recently passed away. I have the task of converting those hand drawings into Autocad drawings. The straight lines are easy to draw. But the curves of the furniture are very hard to replicate exactly. I need to be as precise as possible.

Currently, I mark points on the curve and measure the distance from a reference line and then recreate that on the computer. This is extremely time-consuming.

Is there a mathematical way in which I can solve this problem? Either by getting the equation of the curve or something?

Answer 1

If your CAD software does not support freeform curves (which I doubt), go for natural cubic splines. https://en.wikipedia.org/wiki/Spline_(mathematics)#Algorithm_for_computing_natural_cubic_splines

You will define the curves from a set of interpolating points. When there are angular points, start a new spline (which they didn't do in the example below).

Note that cubic splines can be converted to cubic Beziers, which are quite commonplace. (You can also try Beziers straight away, but fitting to an existing curve can be more difficult.)

Answer 2

If You can guess the curve could be approximately the graph of a rational function of say degree $n$ i.e. the graph of a function of the form $$f(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0$$ where $a_j\in\mathbb{R}$ for $j=0,...,n$ and $a_n\neq 0$ and in Your case $n\geq 2$ and You find $n+1$ points on the curve $P_1(x_1,y_1),...,P_{n+1}(x_{n+1},y_{n+1})$ after a suitable choice of Your coordiante system, You get a system of linear equations $$f(x_j)=a_nx_j^n+a_{n-1}x_j^{n-1}+...+a_0=y_j,j=1,...,n+1$$ that You can solve ( if it is solvable ) by Gaussian elimination.

Answer 3

This is the Freedom Tower in $Iran-Tehran$.

And this is how I drew it with 202 equations:

Go to www.desmos.com and paste your picture in the background and try to trace the equation over that. At the beginning it may take some times but it will be easier as you practice more.

And these are the first 58 equations that I wrote, if you need the rest of them I can send them to you: