Volume of partial ellipsoid cut by plane

In summary: You can also try asking on a math forum for help with setting up the integral.In summary, the conversation discusses the possibility of solving a problem involving a non-numerical approach and an arbitrary ellipsoid described by matrix A. The goal is to cut the ellipsoid with a plane at a given value and find the volume above or below that plane. One potential approach is to solve for the area of the ellipse generated by a cut at a specific value and then integrate it over a range. However, the specific method for carrying this out is currently unclear. The original poster has not yet found a solution and is seeking help or input from others.
  • #1
Chuck37
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I wanted to get opinions on whether solving this problem in a non-numerical way is realistic, or if someone has the answer, all the better. I have a totally arbitrary ellipsoid (not aligned with any axes) that I can describe by matrix A, like x'Ax=1 is the ellipsoid surface. I have the points describing the primary axes of the ellipse. What I want is to cut the ellipse by a plane at Z=(some value) and get the volume above/below that plane.

One approach that seems potentially doable is to solve for the area of the ellipse generated by a cut at Z=x and then integrate that over the range of interest. How exactly to carry that out is eluding me at the moment though. Thanks for any input.
 
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  • #2
hello ! Were you able to get an answer to your question ?If yes, could you please put it here because i have the same query.
Thank you !
 
  • #3
No, never did. Still would like to know though!
 
  • #4
Chuck37 said:
I wanted to get opinions on whether solving this problem in a non-numerical way is realistic, or if someone has the answer, all the better. I have a totally arbitrary ellipsoid (not aligned with any axes) that I can describe by matrix A, like x'Ax=1 is the ellipsoid surface. I have the points describing the primary axes of the ellipse. What I want is to cut the ellipse by a plane at Z=(some value) and get the volume above/below that plane.

One approach that seems potentially doable is to solve for the area of the ellipse generated by a cut at Z=x and then integrate that over the range of interest. How exactly to carry that out is eluding me at the moment though. Thanks for any input.

Have you attempted to set up an integral? The problem I see is getting the right limits for the integral, but it should be doable.
 
  • #5


I would say that solving this problem in a non-numerical way is possible, but it may not be the most efficient or accurate approach. While it may be tempting to use a purely mathematical approach, it is important to consider the limitations and assumptions that come with it.

In this case, using matrix A to describe the ellipsoid and solving for the area of the ellipse generated by a cut at a specific value of Z may not accurately reflect the true volume of the partial ellipsoid. This is because the shape of the ellipsoid may not be perfectly symmetrical and may have variations in its curvature that cannot be captured by a simple mathematical equation.

I would suggest using a numerical method, such as Monte Carlo simulations or finite element analysis, to accurately calculate the volume of the partial ellipsoid. These methods take into account the shape and curvature of the ellipsoid and can provide a more realistic estimation of the volume.

Additionally, it may be helpful to consider the physical properties of the material that makes up the ellipsoid, such as its density, in order to accurately calculate the volume. This would require a more detailed approach, such as using computer-aided design software, to accurately model the ellipsoid and its properties.

In summary, while a non-numerical approach may seem feasible, it may not provide the most accurate or realistic results. Using numerical methods and considering physical properties may be a more robust approach to solving this problem.
 

1. What is a partial ellipsoid?

A partial ellipsoid is a three-dimensional geometric shape that is obtained by cutting an ellipsoid with a plane. It is a curved surface that is a portion of an ellipsoid.

2. How is the volume of a partial ellipsoid cut by a plane calculated?

The volume of a partial ellipsoid can be calculated by using the formula V = (2/3)πabc, where a, b, and c are the semi-axes of the ellipsoid. The volume of the cut portion can then be calculated by multiplying this value by the fraction of the ellipsoid that remains after the cut.

3. What factors affect the volume of a partial ellipsoid cut by a plane?

The volume of a partial ellipsoid is affected by the angle and position of the cutting plane, as well as the size and shape of the ellipsoid.

4. Can the volume of a partial ellipsoid cut by a plane be negative?

No, the volume of a three-dimensional shape cannot be negative. However, if the cutting plane intersects the ellipsoid in a way that creates an irregular shape, the volume may be equal to zero.

5. How is the volume of a partial ellipsoid cut by a plane used in real-world applications?

The volume of a partial ellipsoid cut by a plane can be used in engineering and architecture to determine the volume of a partially cut object, such as a dome or arch. It is also used in mathematical modeling and computer graphics to create realistic three-dimensional shapes.

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