Tilting a circle or ring over backwards creates an ellipse....

[Pogster]

What I would like to be able to calculate is the following:

Suppose a hoop or ring is held up perpendicular to the ground and you stood in front of it, it would look perfectly circular and knowing the radius you could calculate the area of this circle. Now if this hoop was tilted over backwards it would look elliptical to you the observer. I need to be able to calculate the area of the resulting ellipses for any angle of tilt of my hoop.

Would the following be correct: The horizontal axis length of all the ellipses that result from tilting my hoop over are going to be the same as that of the original circle but the vertical axis is going to get progressively less. Can I therefore ignore the horizontal axis and treat the vertical axis as a pole falling over backwards and as it falls it's height decreases by: (original height) x sin (tilt angle). Which means for different tilt angles I can calculate the minor axis and with this value and the constant major axis length calculate the area of the resulting ellipses?

 

[MarkFL]

Hello and welcome to MHB, Pogster! (Wave)

I agree with your reasoning, except I think we need to use:

(original height)·|sin (tilt angle)|

to avoid negative values when the tilt angle might be in the 3rd or 4th quadrants. :)