Finding new major axis of ellipse after stretching along arbitrary axis

[Keys]

If I stretch an ellipse with .5 eccentricity along an axis 45 degrees from its major axis, doubling its area. How do I find the angle of the major axis of the resulting ellipse? Is there a simple rule based on the amount you stretch and the angle and orig eccentricity? Thanks.

 

[tiny-tim]

If I stretch an ellipse with .5 eccentricity along an axis 45 degrees from its major axis, doubling its area. How do I find the angle of the major axis of the resulting ellipse? Is there a simple rule based on the amount you stretch and the angle and orig eccentricity? Thanks.

Hi Keys!

A circle is the intersection of a horizontal plane with a vertical circular cylinder.

Rotate the plane through an angle about an axis, you get an ellipse with that axis as its major axis.

Rotate the plane again about a different axis, you get your second ellipse.

The second ellipse could have been obtained from a single rotation of the plane about its major axis.

So the problem is reduced to … how do you combine two rotations into one rotation?

 

[Keys]

Thx for the reply Tiny-tim, but I don't think the way you're looking at it fits the problem I described because in your explanation the ellipse's minor axis can never be larger than the diameter of the cylinder, no matter how many arbitrary rotations you give the plane. In the problem I was pondering, starting with an ellipse which given will fit a sliced cylinder at some angle, once you do a transformation of that ellipse within its plane but at angle different from its major axis, the new transformed minor axis is definitely longer than the diameter of the cylinder being used to help visual things. So I do not think cross sections of the circular cylinder provide a useful analog after you start stretching things on axis other than the major axis. You could perhaps start manipulating the 2nd ellipse to get it to fit back on the cylinder but that would require the info we are trying to get at in the first place.

I guess my problem is if you have a vertical cylinder/rod with a horizontal elliptical cross-section (not a circle), then I slice a new ellipse using a plane through the rod at some arbitrary angle and orientation, how do I find the major axis of ellipses that results. I was hoping there was some equation were I just plug in the relevant data and it spits out the new angle. Thx again

 

[Keys]

I'm starting to think I'm only questioning this because I am trusting the software plot on my screen too much. For example I'd start with a circle and transform it 2x on the horizontal, then take the resulting ellipse and transform it 2x again along 45 degrees from horz, I was expecting the new major axis to be along 22.5 degrees but it was always something off, like there was more to the problem (this was all being judged visually). Now I'm starting to think it is just the software plotting slightly off or most likely my monitor has a slight squish in the X or Y axis so even though the software is producing the correct ellipse, my monitor is performing an unanticipated extra transform to mess with me.

 

[Keys]

Well, I'm back thinking about this. Looks like the example I gave in post #4 is not 22.5 degrees. Just to check for display problems I did a 2nd mirror image and then rotated it until it aligned with the other and it seems that the new major axis is around 19.5 degrees from horz, not 22.5 so there is something more to it.