# A Ideas for determining the volume of a rotating object

#### Lardos

Hello everybody,

I am currently working on an experiment investigating the formation of planets.

I have a vacuum chamber in which dust particles form bigger agglomerates through accretion (sticking together).
From the imagery I can see those agglomerates which are build up by smaller monomeres.

They look a bit like this (see picture).

Now in the imagery I can see those agglomerates rotating with a constant velocity
(you cant actually see the individual monomeres. Its more like a shadow of the agglomerate).

My goal is to find a way to approximate the volume of such polymeres. But I dont really know how to do this. My idea was
to somehow determine the volume by integrating over the "dark area" at every frame. Then I could take the mean of all
the integrations and would get something like the mean value of the visible area. Then I could calculate the solid of
revlution around the rotation axis.

But I am afraid the measurement error would be huge.

I am certain that there is a better way of doing it. Since I know the "2d shadow" of the object from every angle, there should be some kind of way to determine the volume right?

I am very thankful for every idea / link / or literature that someone would suggest give me.

Greetings from germany!

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#### .Scott

Homework Helper
There are two things I would note right away:

1) If the camera view is perpendicular to the rotation axis, then you can segment the volume into slices - each slice being perpendicular to the rotation axis. That turns your 3D estimate into the sum of many 2D estimates.
2) These agglomerates should be rotating about their center of gravity (CG). If you develop a model that substantially differs from the observed CG, then you should adjust it before using it to estimate the volume.

Certainly adjusting the camera angle to view the rotation axis from the side simplifies the problem a lot.
For any slice, you can start your model as a simple disc with the rotation axis in the middle. Then with each view, rotate the model the same amount and remove material where there is no shadow. After 180 degrees of turn in 30-degree steps, you should have a very good estimate.

In contrast, a camera angle that views the rotating object along the rotation axis would provide information about the volume only in the first frame. Every frame after that would be entirely predictable from the first frame - with no new information and never sufficient information to make anything more than a guess at the volume.

#### jbriggs444

Science Advisor
Homework Helper
Since I know the "2d shadow" of the object from every angle, there should be some kind of way to determine the volume right?
As a counter-example, consider a cubical shell with a hole in the center of one face as compared to a solid cube. The volumes of the two objects are very different, but their silhouettes are identical.

#### .Scott

Homework Helper
As a counter-example, consider a cubical shell with a hole in the center of one face as compared to a solid cube. The volumes of the two objects are very different, but their silhouettes are identical.
If you put the hole in the center of the cube instead of the center of a face, the problem is more difficult.
With the hole in the center of a face, you would suspect something to be wrong when you compared the rotation axis to your predicted center of gravity.

With that in mind, we can look at the actual objects he trying to measure and recognize that in most cases, he is facing a simpler problem.

In fact, it looks as though he is dealing with fixed-size dust particles where in most cases, every particle has a line of sight to the outside world. If that dust particle size can be presumed to be within a very limited range and shape, that offers methods of determining the volume that start out be counting those particles.

#### jbriggs444

Science Advisor
Homework Helper
With that in mind, we can look at the actual objects he trying to measure and recognize that in most cases, he is facing a simpler problem.
Yes, agreed. And I have no solution to offer.

The point of cutting the hole in the face of the cube was to make it more clear that we were talking about the volume of the shell rather than the volume of the enclosed space.

Note that a thin cubical dusty shell with a dusty tuft in the middle and a hole in one side still shares the same silhouette with a thin cubical dusty shell with a hole in one side and no dusty tuft in the middle. Even though every dusty particle may still have a direct straight-line path to infinity.

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#### .Scott

Homework Helper
I just did a bit of an analysis on your png images.
You need better images.
Most importantly, you need images with more consistent light levels. The images at the bottom are much darker than the ones at the top.
Also, aim for the best focus and greater contrast. You will be using these images as input to your algorithm, and you will need to be making decisions on each pixel. An 8-bit pixel value greater than say 250 will represent the background (completely unblocked). A value less than 10 will represent fully blocked. Anything in between will indicate a portion of the pixel area is blocked. So make the photography support that.

Again, the camera needs to be looking directly at the equator. In the example you gave, the particle is rotating more than 180 degrees. This is useful since it doesn't hit the 180 mark exactly. So you get degree rotations something like 0, 38, 76, 114, 152, 190 (mirror to 170), 228 (mirror to 132), 266 (mirror to 94).

So the rotation beyond 180 degrees gives you three pieces of info:
1) You have silhouettes that can be mirrored to fill in the 0 to 180 range.
2) As the object approaches 180, 360, 540, etc, it can be compared to the first image (at 0) to determine rotation rate.
3) Once you have an image at 0 and one close to some multiple of 360, you can estimate the drift. Then looking at the ones close to an odd multiple of 180, you can estimate the position of the rotation axis.

With that much information, you should be able to set up a model and whittle it down with each image. For simplicity, I would start with a model that is a 40x40x40 binary array that starts at as a cylinder of all 1's. Then use the white area of the image to determine pixels that need to be 0. When you're done, add up the pixel values.

#### LURCH

Science Advisor
This sounds too simple (Maybe I’m not understanding the question correctly), but do you know the volumes of the original monomers? Seems like knowing the volume of each building block and the total number of building blocks would give you an approximate volume, though it wouldn’t correct for any deformation that takes place during accretion, so maybe that’s not as accurate as you want?

Also, what is the scale of these experiments? When you say monomer, I’m pretty sure you’re not talking about individual molecules, so how small are they exactly?

Also, WELCOME!

#### Dadface

Perhaps you could look up the methods used to estimate the volumes of tumours from PET scans. That might give you some clues about devising a method that falls within your budget.

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