Modelling object w/ functions, how to find volume loss from indents?

[ChessGM]

TL;DR

Need a way to calculate a loss in volume if an object has indents of unknown shape.

I'm modelling an object with functions to get its volume. I've laid it along the x-axis lengthwise, to scale, and have functions that model its curvature well. I can calculate the gross volume with the solid of revolution integrals, but there's one section with 'indents' and I have no idea how to account for this. The section in question is the one modelled by the third function from the right in the graph image. I've taken a top-down image of the object, is that at all useful? It looks somewhat like a cylindrical indent, but the indent might not be completely parallel to the x-axis.

 

[erobz]

TL;DR Summary: Need a way to calculate a loss in volume if an object has indents of unknown shape.

I'm modelling an object with functions to get its volume. I've laid it along the x-axis lengthwise, to scale, and have functions that model its curvature well. I can calculate the gross volume with the solid of revolution integrals, but there's one section with 'indents' and I have no idea how to account for this. The section in question is the one modelled by the third function from the right in the graph image. I've taken a top-down image of the object, is that at all useful? It looks somewhat like a cylindrical indent, but the indent might not be completely parallel to the x-axis.

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The black curve at the tip appears to need some fine tuning.

As for the "nicks", it looks like something with a donut shaped profile intersecting with your red curve would create something like it.

 

[jedishrfu]

What about looking at how much water is displaced when you place it in a measuring flask?

Mathematically you could ignore the indents and model it via rotation about the vertical axis.

Next model just the indentations in a similar manner.

Another approach is to determine its mass and find what the density of the resin material used is

 

[erobz]

I created the part by tracing the profile and revolving.

These "nicks" were created in the 3D model you see here by sweep cutting a circle along an inclined path, and then pattern radially. Where it intersects with your surface of revolution the material is removed. I have severe doubt it would be trivial to compute this process algebraically. If you are trying to use mathematics, then you need to figure out how to describe this from the geometry... So intersection of an oblique cylinder with your solid of revolution.

Check...

 

[DaveC426913]

TL;DR Summary: Need a way to calculate a loss in volume if an object has indents of unknown shape.

I'm modelling an object with functions to get its volume. I've laid it along the x-axis lengthwise, to scale, and have functions that model its curvature well. I can calculate the gross volume with the solid of revolution integrals, but there's one section with 'indents' and I have no idea how to account for this. The section in question is the one modelled by the third function from the right in the graph image. I've taken a top-down image of the object, is that at all useful? It looks somewhat like a cylindrical indent, but the indent might not be completely parallel to the x-axis.

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Surely this is an intractable problem.

How are we supposed to know what shape those indents make? We'd need their function in all three dimensions. Is the indent cylindrical? spherical? Does the maximum radius of the cutout align with the maximum radius of the lip? There are innumerable parameters that have to be defined before you can even begin arranging them in formulae.

Your top view makes it very apparent that we have no idea the shape or position of the cutout. Cylinder? Sphere? Blob? Above equator? By how much?

I see @erobz is illustrating exactly this problem.

 

[erobz]

I see @erobz is illustrating exactly this problem.

I think it's likely just a drill bit inclined at some angle based on how it would actually be manufactured. So it need not be an oblique cylinder like shown, but still not simple.

 

[DaveC426913]

I think it's likely just a drill bit inclined at some angle based on how it would actually be manufactured. So it need not be an oblique cylinder like shown, but still not simple.

I don't understand why one would need to calculate the volume lost in any practical scenario. And it still seems intractable. Are all these cuts razor-edged? No rounding? Sanding?

 

[erobz]

I don't understand why one would need to calculate the volume lost in any practical scenario. And it still seems intractable. Are all these cuts razor-edged? No rounding? Sanding?

I don't think there is any practical reason. Like @jedishrfu says if you want to know the volume you just dunk it in some water. But if you want to think about how the modeling software I used generates the geometry maybe there is some practical knowledge to be gained there on mathematical modeling. At any rate, they are allowed to ponder for the sake of it. We have a mathematics forum just for such occasions.

 

[DaveC426913]

At any rate, they ar, allowed to ponder for the sake of it. We have a mathematics forum just for such occasions.

Yes, I wasn't suggesting it had to have a practical motive, simply that the motive would inform a number of questions currently unanswered, such as how accurate we are talking, how geometrically perfect (do we have to account for rounding off edges, such as in your 3D images? Or are we talking about an idealized geometrically-constructed object? Are those indents, in-actuality, saddle-shaped?), etc.

 

[erobz]

Yes, I wasn't suggesting it had to have a practical motive, simply that the motive would inform a number of questions currently unanswered, such as how accurate we are talking, how geometrically perfect (do we have to account for rounding off edges, such as in your 3D images? Or are we talking about an idealized geometrically-constructed object? Are those indents, in-actuality, saddle-shaped?), etc.

My default assumption when I see someone trying to generate a solid from a huge number of curves that intersect that this is just philosophical boredom or obsession!

Case in point:



We have too much time on our hands...

Maybe better if the OP used a Fourier Series!