Consider the following wedge
cut from a cylinder of radius r.
The plane that cuts the wedge goes through the very bottom of the cylinder leading to an ellipse as the cross section of the wedge.
The long axis of the ellipse is $2R$ in length.
How can I prove that the minor axis of the ellipse is $2r$ in length where $r$ is the radius of the cylinder?
$\endgroup$2 Answers
$\begingroup$Set the usual rectangular coordinate system xyz with origin at the middle of the ellipse. The x axis is perpendicular to the cylinder and pointing towards the viewer. The y axis runs along the axis of the cylinder and the z axis is perpendicular to the y axis on the plane of the paper.
The equation of the cylinder is
$$x^{2}+z^{2}=r^{2}$$
Consider the vector from the origin to the bottom of the cylinder in the direction of the major axis of the ellipse. Its coordinates are
$(0,R\sin\theta,-R\cos\theta)$
Consider the vector from the origin to the back of the ellipse in the direction of the minor axis of the ellipse. Its coordinates are
$$(-a,0,0)$$
where $a$ is the minor axis of the ellipse.
A vector perpendicular to these two and therefore perpendicular to the ellipse is $$(0,Rr\cos\theta,Rr\sin\theta)$$. Therefore the equation of the plane on which the ellipse lies is
$$y\cos\theta + z\sin\theta=0$$
The ellipse is the intersection of this plane with the cylinder $x^{2}+z^{2}=r^{2}$. Thus the ellipse is described by these 2 equations in this set of rectangular coordinates. Notice that $\cos\theta=r/R, \sin\theta=\sqrt{R^{2}-r^{2}}/r$.
Consider the farthest back point along the minor axis of the eclipse. I mentioned before that it has coordinates $(-a,0,0)$ or $x=-a$.
If you replace this in the equations of the ellipse you get $a=r$ and therefore the minor axis of the ellipse has the same length as the radius of the cylinder.
$\endgroup$ $\begingroup$If I understand your question and the diagram (and this could be a long shot...), this seems to be pretty straightforward: no matter where that ellipse's center is, if you take the two extreme points of the minor axis they will be two antipode points on the cylinder, no matter what the angle $\;\theta\;$ is, and thus this minor axis's length is $\;2r\;$ .
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