Can any ellipse be represented as a diagonal half cross section of a cylinder? By "diagonal half cross section" I mean if a plane cuts the cylinder through these 3 points: a point from the top circle, the centre, and the diametrically opposite point of the first point. I know there are 2 cases of this, but right now I can't provide a picture of the case I want from the two. Sorry for that. I want the case where the plane cuts the curved surface of the cylinder, not the plane surfaces on both sides. I hope you understand.
If the answer is yes, that any ellipse can be represented like this, then can't we measure the perimeter of the ellipse from this? Because the distance between the point from the top circle and the diametrically opposite point should be measured by applying the pythagorean formula, right? Because the cylinder can be opened and can be made into a rectangle. Am I right? I think I am wrong somewhere. But I don't know where. I've searched and found out that there's no exact formula for calculating the perimeter of an ellipse.
Sorry that I can't make it more understandable :( If you have difficulty understanding what I am asking, you may go to the links below and maybe you will get the idea which I am talking about.
You can check the figures here. Though I don't want the cross section area, I want the perimeter.
And maybe this and this to get what I wanted by diagonal half cross section.
Thanks in advance...
Would someone please take a small portion of their valuable time and help?
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After unrolling the cylinder, the ellipse won't transform into a segment. Assuming that your cylinder stands on the circle side and goes up vertically, only helices, vertical lines and horizontal circles will become perfect line segments after the unrolling. Therefore, you are left not with perfect line segment on the plane, but with some complicated curve, so you are back where you started.
Here I use the fact that first quadratic form of the perfect right cylinder of radius 1 equals to
in the standard parametrization (φ, z) → (cos φ, sin φ, z), so lengths on the cylinder perfectly correspond to lengths of curves on the plane. Also, the classification of the geodesics on the cylinder is used.
You can see this fact yourself: roll up a cylinder out of an A4 paper sheet, draw any closed non-horizontal curve on it, unroll it and see how the curve goes up and down on the unrolled sheet.