Sphere inside Ellipsoid

In summary, the conversation discusses finding the largest sphere that can be inscribed inside the ellipsoid with equation 3x^2 + 2y^2 + z^2 = 6. The conversation mentions using Lagrange multipliers to find the solution, but ultimately finds the answer by considering the geometric properties of the ellipsoid. The solution involves finding the extrema of the function x^2 + y^2 + z^2 subject to the constraint 3x^2 + 2y^2 + z^2 = 6.
  • #1
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I'm trying to find the largest sphere that be inscribed inside the ellipsoid with equation 3x^2 + 2y^2 + z^2 = 6.



Homework Equations



I know I will need at least 2 equations. One of them is the constraining equation (f(x) = a, where 'a' is a constant) and the other is the equation you want to maximize. I need to use Lagrange multipliers to find a lambda such that (del) F = lambda (del G).



The Attempt at a Solution



My first attempt was to maximize the function x^2 + y^2 + z^2 = f(x,y,z) and use the constraining equation for the ellipsoid (3x^2 + 2y^2 + z^2 = 6). However, there is no lambda that will made del (F) = L (del [G]). I think I'm using the wrong equations. Another idea is to find the point(s) on the ellipsoid that lies closest to the origin and use that as the radius for my sphere. However, that boils down to the same equations I already unsuccesfully used.

Any help appreciated.
 
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  • #2
You can do it by Lagrange multipliers if you really want to, but the easy way is to see what the equation of the ellipsoid means geometrically. What are its principal axes? What is the length of the shortest principal axis? Then you know enough to just write down the answer.
 
  • #3
yeah, but the problem makes you use lagrange multipliers =(
 
  • #4
Find the extrema of f(x,y,z) = x^2 + y^2 + z^2
subject to g(x,y,z) = 3x^2 + 2y^2 + z^2 - 6 = 0

The Lagrange equations are the components of the gradient of f(x) + L.g(x) = 0
2x + L.6x = 0 (1)
2y + L.4y = 0 (2)
2z + L.2z = 0 (3)
and 3x^2 + 2y^2 + z^2 = 6 (4)

(3) gives L = -1 or z = 0
if L = -1:
(1) and (2) give x = 0 and y = 0
(4) gives z^2 = 6 so the extreme radius = sqrt(6). But (by inspection) that's a maximum not a minumum.

If z = 0:
(1) and (2) both give L = 0
(4) gives 3x^2 + 2y^2 = 6

So I suppose you now have to find the extrema of x^2 + y^2 subject to 3x^2 + 2y^2 = 6. Obviously that will lead to the answer but I don't recall having this situation happen with Lagrange multipliers before.
 
  • #5
Sorry, I couldn't see what was in front of my nose in the previous post till after I logged off the forum!

2x + L.6x = 0 (1)
2y + L.4y = 0 (2)
2z + L.2z = 0 (3)

There are THREE sets of solutions of those equations with L non equal to zero:

L = -1/3 y = 0 z = 0
L = -1/2 x = 0 z = 0
L = -1 x = 0 y = 0

And using (4) you get the three extreme values of the function, corresponding to the three principal axes of the ellipse.
 

What is a sphere inside an ellipsoid?

A sphere inside an ellipsoid is a three-dimensional geometric shape where a sphere is completely enclosed within an ellipsoid, which is a stretched-out or squashed sphere.

How is a sphere inside an ellipsoid formed?

A sphere inside an ellipsoid is formed when a sphere is rotated around an axis and then stretched or compressed in one direction, resulting in an ellipsoid shape. Alternatively, a sphere can be cut into smaller pieces and reassembled into an ellipsoid.

What are the properties of a sphere inside an ellipsoid?

A sphere inside an ellipsoid has a center that aligns with the center of the ellipsoid, and the sphere's diameter is equal to the shortest diameter of the ellipsoid. The sphere also has the same volume as the ellipsoid, and its surface is tangent to the ellipsoid's surface at all points.

What are the real-life applications of a sphere inside an ellipsoid?

A sphere inside an ellipsoid is commonly used in geodesy, the science of measuring and mapping the Earth's surface, to approximate the shape of the Earth. It is also used in engineering and architecture to design structures that can withstand pressure from all directions evenly, such as domes and arches.

What is the difference between a sphere inside an ellipsoid and a spheroid?

A spheroid is another term for an ellipsoid with two equal semi-major axes, while a sphere inside an ellipsoid has a semi-major axis that is shorter than the other two. A sphere inside an ellipsoid is also always tangent to the ellipsoid's surface, while a spheroid may not be tangent at all points.

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