A circle tranforming into ellipse

[vin300]

I don't know what category this question falls into. I have two parallel planes, on one I draw a circle and on the other I project it orthogonally. Now I incline the plane with the circle. The projection on the other plane will be an ellipse. I need to find out, the relationship between the angle of inclination and the eccentricity of the ellipse.

 

[vin300]

I've figured it out. If a is the semi-major axis, b is the semi-minor axis of the ellipse and r is the radius of the circle, then a= r and b= rcosθ (θ is the angle of inclination). Now it isn't difficult to find the eccentricity of the ellipse. But.. how to prove that an inclined projection of a circle is an ellipse in the first place?

 

[coolul007]

Look at the projection of a circle between parallel planes as a cylinder. then the angle of the second plane intersects the cylinder.

 

[vin300]

I was looking for a mathematical proof of "a non-uniform scaling of a circle changes its shape into an ellipse".