Calaculating surface area of a bottle

[rocky jay]

How would I use desmos to calculate the surfrace area of this bottle? because as u can see, the green section is not exactly straight so im not sure if that will affect the surfrace aprea

[candid pawn]

saw this and thought you were building some type of bong

[rocky jay]

sooo, do you have any ideas how to deal with the optical illusion?

or could i just assume that from x=0, it doesnt matter

for more context, im seperating the clear part from the wrapper part

[proven condor]

I would make two functions

[rocky jay]

and what would that be for?

[proven condor]

https://www.desmos.com/calculator/i9tetlxpmj

Could be something like this

The more details you add, the more functions you would add

[rocky jay]

oh, I see what you mean

like here, would i also calculate the area of that plastic and subtract it of

or just consider that an politcal illusion

[proven condor]

You could do n points, and always find a polynomial of n-1-th degree that interpolates these points

but the more you do, the more equations you would have to solve for

but you could let a computer do the work

it really depends what you want and need

[rocky jay]

i mean dont mind the amount of equations i need to solve

[rocky jay]

could you elaborate on this?

[proven condor]

https://www.youtube.com/watch?app=desktop&v=Xj129kA3Ci0

[rocky jay]

i apologise

i dont rly understand

my maths knowledge is only at a HS students

[proven condor]

interpolation here just means to find a function passing through the points

[rocky jay]

ohhh

yea that make sense

[rocky jay]

then let me reprhase, what did u mean by these points?

[proven condor]

say you have two points

[rocky jay]

mhm

[proven condor]

if you want a function to pass throught these two points

that can do

a straight line

aka a linear function

which has degree 1

so for 2 points you would need a polynomial of 1 degree

so for n points you would need a polynomial of (n-1)th degree

[rocky jay]

ooooh

yes i understand

[proven condor]

so for 3 points

you would need a parabola

at the very least

[rocky jay]

x^2

[proven condor]

yup

ax^2+bx+c

[languid magnet]

bacc (unhelpful)

[proven condor]

If we reduce it down to a formula

For one point, you would need at least a constant function

for example you want a function that goes through (2,1)

well y = 1 does that obviously

you can also take a polynomial of higher degree

but notice then you would get a system with infinite solutions

for example take a linear function

if it is supposed to just go through one specific point

then the other point can be anything

it still would go throught the intial point that you gave it

https://www.desmos.com/calculator/sm6bkelg1w

No matter how you choose A_1 there are infinite many ways with a linear function to pass one point

[rocky jay]

ohhh

yea, i understand that

i think ik what to do but im leaving this open to come back to it

in case i get stuck again

[proven condor]

yup do that!

[rocky jay]

https://www.desmos.com/calculator/vtv7deidwa

so what i ended up doing

is finding 4 functions that fit to the bottle

where, sections that are vertically stragiht can distorted with curve

and so i can ignore that curved sparts, such as the wrapper past 0

when calculating surfrace of the bottle

since the wrapper is a where its straight

and i plan to use formula of cylinder

to calculate that

[rocky jay]

how thos sound?

[proven condor]

looks amazing

but one notice

your image is off

you will notice it's a bit sides ways

[rocky jay]

sideways?

okay im rly blind when it comes to details like, could u highlight what u mean if possible or circle it?

[proven condor]

yea like distorted

[rocky jay]

ohh

[proven condor]

if you look on the left

and right

[rocky jay]

oh, i assumed

thats because

its a 3d object

and im tryna force it into a 2d

plane

[proven condor]

yyea

[rocky jay]

cuz this was the og one

so distortion is expected

[proven condor]

i guess you can change the fov of the camera or some

but shouldnt matter really for this project

[rocky jay]

yea

[proven condor]

did you know

[rocky jay]

mhm?

[proven condor]

you can also calculate integrals on desmos

[rocky jay]

fuck

off

[rocky jay]

actually??

[proven condor]

yup

[rocky jay]

i apologise if this came off rudely, it was more out of amazement

oh i got it

[proven condor]

didnt even read it

[proven condor]

oh nice

[languid magnet]

bacc (unhelpful)

[proven condor]

Now if you define your functions like f_1(x), f_2(x) etc

[rocky jay]

just testing but whats wrong here btw?

[rocky jay]

mhm?

[proven condor]

you can also calculate the derivative with d/dx(f_1(x))

can you send link

[rocky jay]

https://www.desmos.com/calculator/vtv7deidwa

oh nvm, i got it

[proven condor]

Do it like this

https://www.desmos.com/calculator/yzbpottmaw

you will have more order

[rocky jay]

did u get the same?

[proven condor]

nope

[proven condor]

taht's the formula for the surface

not area!

you rotate the function around the x-axis

and from that object you calculate the surface area

[rocky jay]

oh yea

that i understand

i was just testing the intergration section

[proven condor]

did you klick on the link

[rocky jay]

yes

[rocky jay]

yes this much more orgranised

i need to sleep, but i will be continuing this when i wake up, so i plan to leave thread up if i get stuck again in the future. Thank you so much bacc

also if i ask for help, may I ping you? if no, thats understandable and okay

[proven condor]

yea sure

[rocky jay]

thank you

[unique oracle]

I think this is much better suited to a program like blender

Which automatically tells you the surface area

[green crystal]

To deal with the optical ilusion use a chess section 4x4 and apply a bilineal transform. The section the bottle, for example the basic will be 3 secctions the top, the body an the bottom. Then divide with a line right in the middle, the for all sections approximate the outside curve ( the simpless method is putt a lot of points and the apply cubic splines) the you will have a function to describe the outside curve to all sections and then use the revolution surface theorem. thats is what I will do

Or you can download a scanner in your phone to get 3D model, upload tu cad and you will have it with a few clicks. But for me that is boring

[green crystal]

Cubic splines, a single polinomial is not enough

Omg, let I just read the preview msg. Lets do this complete. Well separate que the bottle in 3 sections. Top, body and bottom. Lets start for eliminate the distortion of the the camera. Please print a this cuadriculate.

Measure the with a ruler or print it with a definite side

Then put the bottle in this position

And then put the grid in one side like this

Off course the grid will not be perfect like the picture, but you know the real sides, so u can use bilineal transform. This can be apply with python, scilab or most simple matlab. This is an example

https://la.mathworks.com/help/images/ref/imwarp.html

Now you have a image which not depend of the angle of you take the picture because u use the grid to center an put in place, and more you have que equivalence of inch or cm in pixels !!!

Now lets back to our bottle

now we need to get the middle, so we have many alternatives, erase the background to a empy space, then binarize the image. with binarize image have many options but the easy one in export in png. then the other easy opcion is go to, idl canvas, make a square bigger enough to paste the png bottle and use the automatic center. its important to have to know the pixels number of the new square. the import the image in any format

the go back to our program like matlab o scilab(is the same)

you know the withd of the new square so you can draw a line just in the middle, and you will have the perfect division.

now I recomend to cut in yhe middle the image and rotate to horizontal (please put the outside of the bottle up)

and there you go, now yo now the equivalence of the pixel in inch on mm or cm, but the real size

now can cotinuate use matlab but but can use any program to put point on the border of you mid bottle

now lets divide the problem in two ( lets solve the bottom later)

cut you image in the top and the body of the bottle

for the top put how many points u want, and you will have the cordinate points (x,y), maybe is simple use a high degree polynomial but in not very acurate, in image analysis the simple method realible is cubic spline. (if u dont leave matlab you only need to use a function) but if u leave matlab u need to make a equation system where you have to for every two point write a third degree polynomial and make equal tu the next in the share point, the you have to derivate and do the same , and then use the second derivate and do the same, the you will need another equation to make the system square and I recomend use the thrid redivate or if u have a rectangular matrix, multiply with his tanspose in both side. with his u will get the coeficients of all equations. for n many points you will have n-2 equations. so for any of this equations you have to use the surface area revolution theorem and add all the results and that its all. for the body do the same

but the bottom

you will have to use the skill u learn in the previews things I comment you to get the surface of that irregular shape

I hope this help u

[rocky jay]

sure, but the point of doing this is to calculate the surface area using intergration

[rocky jay]

thank you fellow one piece enjoyer for the steps, but i dont think my math skills is high enough to do what you said, sorry

[green crystal]

Dear Nakama, you need to use integration, look
https://mathworld.wolfram.com/SurfaceofRevolution.html

[green crystal]

In 3 hours I will be able to help u, if u want we can make a call and I can guide u Nakana

its no complicate, dont understimate yourself, prepare your observation haki and I will show u the next level

and btw, I just give u the formal way to do it, there are more ways to do this

maybe, when I explain it I use all my coqueror haki

sorry

[rocky jay]

lmfao

i love one piece fans man

[rocky jay]

i get that i need to use intergration, like this is how ive gone abt it.

Ive split the bottle into 3 parts.
left side, i need to find the function for

middle is formula for cylinder

right side, ive found the functions for as seen on left of the picture

[unique oracle]

the problem is parrallax

so you will need to use measurements and areas of revolution

very interesting post with interesting solutions i must say

[green crystal]

for a undergraduated works! well done Nakama

[green crystal]

but I must said, never use the excuse you dont have the level, you always can train yourself and improve you toolbox of abilities. open to teach yourself new skills, keep your mind open and u will be surprice of yourself

[rocky jay]

is this the distortion?

[rocky jay]

thank youuu

[rocky jay]

thank you for stating that, that helps alot actually to explain distortion

[unique oracle]

yes as in if you have ever traced a referene it will be inconsistent because of perspective

so the best way to measure the bottle and then use a curve to approximate the outline

then find the answer by using areas of revolution

[rocky jay]

okay so I was looking at how the function looks when i flipped

and see how the flip function isnt at the very end, does that my bottle isnt perfectly symetrical?

[rocky jay]

im assuming yes since my knowledge is that full revolution be fliped vertrically without chaning anything too much

[rocky jay]

this formula is used when rotating against the x axis correct?

and another question is how does resizing works? as the actual bottle to the img on desmos is ratio of

26 : 8.761

so id assume ill multiply the result from intergration by 8.761

would i multiply there area by 8.761^2? im assuming the latter as that follows the area rule where it become be squared

[rocky jay]

also how complicated would

is

[rocky jay]

.close