Cylindrical section is an ellipse?

In summary, the intersection of the plane x+y+z=1 and the cylinder x^2+y^2=1 is an ellipse. By performing a change of variables, the x+y+z=1 plane becomes the new 'x-y plane' and a new equation for the cylinder can be obtained by setting the new 'z' to zero. The intersection is the values of (x, y, z) that satisfy both equations simultaneously. Replacing y with 1-x-z in the second equation results in an ellipse. This is confirmed by a proof found in a geometry textbook. The plane and cylinder are not parallel, with a normal vector of i+j+k at an angle of cos\alpha =\frac{1}{\sqrt
  • #1
uman
352
1
Prove or disprove: The intersection of the plane [tex]x+y+z=1[/tex] and the cylinder [tex]x^2+y^2=1[/tex] is an ellipse.
 
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  • #2
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?
 
  • #3
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?
 
  • #4
Thanks for the help... I actually found a (more general) proof in a geometry textbook (Geometry and the Imagination by David Hilbert)
 
  • #5
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 [tex]cos\alpha =\frac{1}{\sqrt3}[/tex] with all three axes.
 

1. What is a cylindrical section?

A cylindrical section is a shape that is formed by cutting a cylinder along its length. It is essentially a portion of a cylinder with a flat, circular base on one end and a curved surface on the other.

2. What is an ellipse?

An ellipse is a geometric shape that resembles a flattened circle. It is defined as the set of all points in a plane, the sum of whose distances from two fixed points is constant.

3. How is a cylindrical section related to an ellipse?

A cylindrical section is essentially an ellipse when viewed from a certain angle. This is because the curved surface of a cylinder can be seen as an ellipse when it is cut and laid flat.

4. What are the characteristics of a cylindrical section that make it an ellipse?

A cylindrical section has two main characteristics that make it an ellipse: a constant sum of distances from two fixed points (foci) and a curved edge that is symmetrical about its center.

5. What are some real-world applications of cylindrical sections as ellipses?

Cylindrical sections as ellipses have various applications in engineering and architecture. For example, they are commonly used in the design of bridges, tunnels, and pipes. They can also be seen in the design of satellite dishes and curved mirrors.

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