Finding the equation of a curved line

[LT72884]

Ok, so i have done many math classes, as i am an engineer, however, a theory class and proof class i h ave not done, except maybe one haha.

here is my question, if i draw some sort of curved line by hand, how do i find an equation for that?

for example, i draw half of a tear drop but along the horizontal (like the posted image). How would i find the equation of that line, preferably without a computer? im just wanting to see how this was done back in the past.

thanks

 

[erobz]

Ok, so i have done many math classes, as i am an engineer, however, a theory class and proof class i h ave not done, except maybe one haha.

here is my question, if i draw some sort of curved line by hand, how do i find an equation for that?

for example, i draw half of a tear drop but along the horizontal (like the posted image). How would i find the equation of that line, preferably without a computer? im just wanting to see how this was done back in the past.

thanks

View attachment 332931

For this I would just guess a functional shape, with some constants to tweak it to fit data if you have it.

$$ y(x) = \sqrt{x} - x $$

Plot it?

Otherwise, you could use the data from the drawing to do Fourier Transform. They can draw anything with those.

 

[jedishrfu]

Could do something like Bézier curve fitting.

https://raphlinus.github.io/curves/2021/03/11/bezier-fitting.html

https://en.wikipedia.org/wiki/Bézier_curve

 

[LT72884]

For this I would just guess a functional shape, with some constants to tweak it to fit data if you have it.

$$ y(x) = \sqrt{x} - x $$

Plot it?

Otherwise, you could use the data from the drawing to do Fourier Transform. They can draw anything with those.

yeah, its coming up with the functional shape thats tough to do haha.

i mean, how did engineers do this in the past before computers did curve fitting?

thanks

 

[erobz]

yeah, its coming up with the functional shape thats tough to do haha.

i mean, how did engineers do this in the past before computers did curve fitting?

thanks

Computer programs that "curve fit" a data set still assume a functional form to do so. Are you asking what the computer program (like Excel) is doing under the hood? What people would have done manually at the time of its inception?

If so, that's likely the Method of Least Squares, If not I'm having trouble figuring out what is being asked exactly.

 

[LT72884]

Computer programs that "curve fit" a data set still assume a functional form to do so. Are you asking what the computer program (like Excel) is doing under the hood? What people would have done manually at the time of its inception?

If so, that's likely the Method of Least Squares, If not I'm having trouble figuring out what is being asked exactly.

yes, how is this done manually with no computer help.

also what is this least squares method?

thanks

 

[erobz]

yes, how is this done manually with no computer help.

also what is this least squares method?

thanks

The basic idea for the Method of Least Squares:

You have some data points $( x_1,y_1),(x_2,y_2),(x_3,y_3)$

Say you want to fit a linear function of the form $y = mx+b$ to the data.

You have 2 parameter to adjust $m$, and $b$ in that form ( If you want to try to fit a different form, like $y = ax^2 + bx + c$, you have more free parameters that can be adjusted, etc..).

You begin by forming the total squared error. That is some measure of "how far"(just magnitude -hence the squaring) the $y$ value of the function is off from the data when it is evaluated at the independent coordinate of a particular point:

We would have above:

$$ f(m,b) = ( y_1 - (mx_1+ b))^2 + ( y_2 - (mx_2+ b))^2 + ( y_3 - (mx_3+ b))^2 $$

From there you optimize $f(m,b)$ using standard techniques from multivariable Calculus, namely:

$$ \frac{\partial f }{\partial m } = 0 $$

$$ \frac{\partial f }{\partial b } = 0 $$

Are solved simultaneously for the parameters $m$ and $b$ that minimize the sum of squared error i.e. $f(m,b)$.

For a full explanation you'll be best to search the term "The Method of Least Squares", I just wanted to give you "the flavor" with a basic example.

 

[Vanadium 50]

yeah, its coming up with the functional shape thats tough to do haha.

ha.

This is something that people unfamiliar with the problem you are trying to solve are least able to help you with. An understanding of the system may tell you that you are looking for a polynomial, or a trig function, or something else. All we can say is "Sure looks like a parabola."

 

[FranzS]

View attachment 332963
The basic idea for the Method of Least Squares:

You have some data points $( x_1,y_1),(x_2,y_2),(x_3,y_3)$

Say you want to fit a linear function of the form $y = mx+b$ to the data.

You have 2 parameter to adjust $m$, and $b$ in that form ( If you want to try to fit a different form, like $y = ax^2 + bx + c$, you have more free parameters that can be adjusted, etc..).

You begin by forming the total squared error. That is some measure of "how far"(just magnitude -hence the squaring) the $y$ value of the function is off from the data when it is evaluated at the independent coordinate of a particular point:

We would have above:

$$ f(m,b) = ( y_1 - (mx_1+ b))^2 + ( y_2 - (mx_2+ b))^2 + ( y_3 - (mx_3+ b))^2 $$

From there you optimize $f(m,b)$ using standard techniques from multivariable Calculus, namely:

$$ \frac{\partial f }{\partial m } = 0 $$

$$ \frac{\partial f }{\partial b } = 0 $$

Are solved simultaneously for the parameters $m$ and $b$ that minimize the sum of squared error i.e. $f(m,b)$.

For a full explanation you'll be best to search the term "The Method of Least Squares", I just wanted to give you "the flavor" with a basic example.

Yep, this can always be done by hand, but the calculations (i.e. solving the linear system of $n$ equations) get exponentially more time consuming. I did that with $n=2$ (quadratic fit) and it took me four to five pages of writing.
The drop-like shape will probably require a quartic polynomial fit at least if you want a decent approximation.
Beware of Excel, as it shows very few sig figs in the fit formula. I had to wrote a vba macro (which uses matrix calculations) to get the coefficients with all the 15 sig figs manageable by Excel.

 

[FranzS]

Also, choose the sampling points wisely (i.e. denser where your shape is steeper).

 

[erobz]

Beware of Excel, as it shows very few sig figs in the fit formula. I had to wrote a vba macro (which uses matrix calculations) to get the coefficients with all the 15 sig figs manageable by Excel.

I don't know what version you are (were)running, but you can format the trendline label to show however many sig figs you want in the current Office 365 version.

 

[OmCheeto]

The drop-like shape will probably require a quartic polynomial fit at least if you want a decent approximation.

It looked like a parabola to me so I digitized the curve and rotated the points until my spreadsheet maximized the R² value for a parabola. Almost perfect @ 127°.

 

[FranzS]

I don't know what version you are (were)running, but you can format the trendline label to show however many sig figs you want in the current Office 365 version.

Nice to know! I'll try that.

 

[FranzS]

It looked like a parabola to me so I digitized the curve and rotated the points until my spreadsheet maximized the R² value for a parabola. Almost perfect @ 127°.

View attachment 333007

Nice intuition!

 

[Bystander]

here is my question, if i draw some sort of curved line by hand, how do i find an equation f

https://en.wikiversity.org/wiki/Cubic_Spline_Interpolation

 

[OmCheeto]

Nice intuition!

Thanks!
I suppose if I were stuck on a deserted island without a laptop, I could solve for 3 points picked from the parabolish curve, and solve with simultaneous equations:

given​	given​
x​	y​	ax²​	bx​	c​	y​
-0.057​	-0.017​	0.0030​	-0.057​	1​	-0.017​
-0.41​	0.32​	0.17​	-0.41​	1​	0.320​
-0.60​	0.80​	0.36​	-0.60​	1​	0.798​
					matrix math​	curve fit​
a​		1​			2.77​	2.68​
b​			1​		0.330​	0.295​
c​				1​	-0.00724​	-0.00571​
				R² =​	0.9996​	0.9999​

Mathematically a bit less accurate, but visually identical.

 

[LT72884]

i have been messing around with desmos and started to add functions together. y=sqrt(x) -x -x^2 and things like this. but y=sqrt(x)-x produces a nice curve, then i went all fancy from an equation found in my airfoil design book. How the heck did they discover this equation back in the 50's before computers could do this? thats what i want to know? i feel so lazy when i use a PC haha