Cylindrical section is an ellipse?

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Discussion Overview

The discussion centers around the geometric intersection of a plane and a cylinder, specifically whether the intersection of the plane defined by the equation x+y+z=1 and the cylinder defined by x^2+y^2=1 results in an ellipse. The scope includes mathematical reasoning and geometric proofs.

Discussion Character

  • Exploratory, Mathematical reasoning, Debate/contested

Main Points Raised

  • One participant requests a proof or disproof of the claim that the intersection is an ellipse.
  • Another participant suggests performing a change of variables to simplify the problem by transforming the plane into a new coordinate system.
  • A different participant proposes solving the two equations simultaneously to find the intersection points, specifically substituting y from the plane equation into the cylinder equation.
  • One participant mentions finding a more general proof in a geometry textbook, indicating that there may be established references supporting the claim.
  • Another participant asserts that the intersection must be an ellipse due to the angle between the normal vector of the plane and the axes, implying that the plane's orientation relative to the cylinder is crucial.

Areas of Agreement / Disagreement

Participants express differing views on the nature of the intersection, with some asserting it is an ellipse based on geometric reasoning, while others focus on the need for proof or further exploration of the equations involved. No consensus is reached.

Contextual Notes

The discussion involves assumptions about the geometric properties of the figures and the implications of the angle between the plane and the cylinder. The mathematical steps to fully resolve the intersection are not completed within the thread.

uman
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Prove or disprove: The intersection of the plane x+y+z=1 and the cylinder x^2+y^2=1 is an ellipse.
 
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If you perform a change of variables so that the x+y+z=1 plane is the new 'x-y plane' then do you not get a new equation for your cylinder? You can then work it out by setting the new 'z' to zero?
 
The intersection of the two figures is the values of (x, y, z) that satisfy both equations simutaneously so solve the two equations simultaneously: From the first equation, y= 1- x- z. What do you get when you replace y by that in your second equation?
 
Thanks for the help... I actually found a (more general) proof in a geometry textbook (Geometry and the Imagination by David Hilbert)
 
It would have to be an ellipse because the plane isn't parallel to the cylinder. The normal vector is i+j+k which makes an angle of cos\alpha =\frac{1}{\sqrt3} with all three axes.
 

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