Re: Triangle inside an ellipsoid


matt271829-n...@xxxxxxxxxxx wrote:
> snapdragon31 wrote:
> > M.A.Fajjal wrote:
> > > What is the maximum area of a triangle inside an ellipsoid
> >
> > The equation for a ellipsolid is
> > x^2/a^2 + y^2/b^2 + z^2/c^2 = 1
> >
> > Assume a >= b >= c
> > The triangle lies in the x-y plane if c is the smallest.
>
> I agree, this seems most likely. In fact, I got exactly the same answer
> as you in exactly the same way. It slightly bothered me though that
> this step was unproved (and I couldn't be bothered to prove it the
> "hard way"!) Do you have a simple proof, or did you just assume it was
> "obvious"?
>
> >
> > The triangle will touch the ellipse defined by
> > x^2/a^2 + y^2/b^2 = 1
> >
> > We can transform the ellipse into a cirle by stretching the y
> > co-ordinates with a factor of a/b
> >
> > After the transformation, the ellipse would become
> > x^2/a^2 + y^2/a^2 = 1 or
> > x^2 + y^2 = a^2 (circle with radius r=a)
> >
> > The maximum area of the triangle inside this circle is a equilateral
> > triangle.
> > Height of the equilateral triangle = 3a/2
> > Bottom of the equilateral triangle = sqrt(3) * a
> >
> > Area of the equilateral triangle = 3 * sqrt(3) a^2/4
> >
> >
> > Transform the cirle back to the ellipse by mutiplying a factor of b/a.
> > Area the triangle becomes 3 * sqrt(3) * a * b / 4

Another way to solve this problem is to transform the ellipsoid into a
sphere first by stretching the y and z axis.
The equation of the sphere would be
x^2 + y^2 + z^2 = a^2

The maximum area of the triangle inside this sphere is
3 * sqrt(3) a^2/4 (Same area as the triangle in the circle above)

Now we have to transform the sphere back to the ellipsoid. It would
undergo 2 transformations.
One is along y-axis and the reduction factor is b/a.
One is along z-axis and the reduction factor is c/a.
Given a > b > c, so b/a > c/a

We select the triangle in the x-y plane so that the reduction factor
along z-axis would only affect its thickness not the area. I hope this
is a better explanation of why the triangle in the x-y plane is
selected.

Again the transformed area is 3 * sqrt(3) * a * b / 4

.

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