How to predict the shape of the circle from any point of view

[OFF_Smog]

As we know a circle view at an angle appears as an ellipse ,
as you see in the picture, the center of the camera aim to the center of the circle ,
the angle between the circle axis and the camera is ө,
the azimuth between mojor axis(a) and the camera is ∞,
the rotation of the camera is €,

1. How to predict an orientation of major and minor axis and the ratio of its axes(b/a) ,if we only know these ө,∞,€ angle, regardless of size of ellipses

2. Is there any theory to apply this example?

 

[pellman]

consider the simpler problem of a circle on a hinge along its horizontal diameter. The circle is oriented so that initially its plane is vertical. Let phi be the angle the plane of the circle makes with the vertical axis as we rotate it around the hinge.

now suppose we have a light source (at infinity) and look at the shadow cast by the circle on a nearby vertical plane. Initially (phi=0) the shadow is a circle of the same radius as the circle. As we rotate the circle on its hinge, the shadow changes to an ellipse in the same manner as the camera in your diagram viewing the circle from different angles.

The horizontal axis of the shadow corresponds to a in your diagram. It is constant and always equal to the radius of the circle. The height of the shadow (b) changes with phi and a simple diagram will show you that its value is

b=a\cos\phi

I will leave you to relate phi back to the angles in your diagram.

 

[OFF_Smog]

Hi,

We can apply the light source(at infinity) theory in case a camera rotating,
Thus the shadow of ellipses will rotate the same as the camera rotation(€) right?

One thing that i concern is how to find formula to apply this circumstance?


Thank you very much