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".

 

[CellCoree]

I would first like to clarify that this question falls under the category of geometry and specifically, the study of conic sections. The transformation of a circle into an ellipse is a common phenomenon in mathematics and physics, and it can be explained using principles of geometry and trigonometry.

To understand the relationship between the angle of inclination and the eccentricity of the ellipse, we must first define these terms. The angle of inclination refers to the angle at which the plane with the circle is tilted or inclined, while the eccentricity of an ellipse is a measure of how "oval" or "stretched out" the ellipse is compared to a perfect circle. A circle has an eccentricity of 0, while an ellipse can have an eccentricity ranging from 0 to 1.

Now, as the plane with the circle is inclined, the projection of the circle on the other plane will change from a perfect circle to an ellipse. This happens because the projection of the circle on the inclined plane appears to be "squashed" due to the angle of inclination. The more the plane is inclined, the more "squashed" the projection will be, resulting in a higher eccentricity of the ellipse.

To find the exact relationship between the angle of inclination and the eccentricity of the ellipse, we can use trigonometric functions such as cosine and sine. By considering the angle of inclination as the angle of elevation, we can use the cosine function to calculate the ratio of the horizontal and vertical lengths of the projection. This ratio will be equal to the eccentricity of the ellipse.

In summary, the angle of inclination and the eccentricity of an ellipse have a direct relationship, where as the angle of inclination increases, the eccentricity of the ellipse also increases. This can be explained using principles of geometry and trigonometry, and it is a common phenomenon in the study of conic sections.