Question

Ellipse defined by cylinder and plane?

Answer 1

So say in 3 dimensional space i have 2 functions being plotted:
cylinder x^2 + y^2 = 1
and a plane 0 = x + y + z
the intersection between these two objects should form an ellipse if I'm not mistaken. However, I'm having trouble getting to that point. Any method of setting the equations equal will result in a surface, not a curved line. So I'm pretty sure I need some parametric equations in order to find my ellipse.
Can anyone help me solve the intersection of those two functions in terms of parametric equations?

Answer 2

let y = -x-z, and sub it in

Answer 3

Wouldn't that give the equation of a surface though? After the substitution you'd still end up with an equation in terms of x, y and z which would define a surface in 3 space. I want the curved line that defines all intersecting points between the cylinder and the plane.

Answer 4

http://www.wolframalpha.com/input/?i=x%5E2%2B%28-x-z%29%5E2%3D1

Answer 5

Err, i mean in terms of x and z. but i'm pretty sure that still defines a surface.

Answer 6

substitution is one of the most basics techniques for playing with systems of equations

Answer 7

Yes, i know how to substitute equations. The thing you linked is 2 dimensional. You're just plotting x vs z. I want an actual 3 dimensional curve.

Answer 8

yeah, a parametric concept would seem more like it then, like how a line is define by its separate x,y,z parts

Answer 9

If you imagine a cylinder being cut by a flat plane at some angle, you can picture in your head an ellipse being formed at intersection, I'm trying to find the 3 dimensional equation for that line.

Answer 10

Ok, I was thinking parametric. Do you know how to find the parametric equations for this case? Or the more general case of the line describing the intersection between 2 surfaces f(x,y) and h(x,y)?

Answer 11

i considered letting

x=t
y = sqrt(1-t^2)
z = -(t+sqrt(1-t^2)

gives us half of it since has to be split +-

http://www.wolframalpha.com/input/?i=plot+x%3Dt%2C+y%3Dsqrt%281-t%5E2%29%2C+z%3D-t-sqrt%281-t%5E2%29

Answer 12

maybe a polar representation would be better so that we can get a full sweep?

Answer 13

well, the 3d of polar is spherical

Answer 14

http://www.usciences.edu/~lvas/MathMethods/Surface_integrals.pdf

this seems to have something that can be useful.

Answer 15

http://www.wolframalpha.com/input/?i=plot+x%3Dcos%28t%29%2C+y%3Dsin%28t%29%2C+z%3D-cos%28t%29-sin%28t%29

that did it

Answer 16

by the cylindar: x^2 + y^2 = 1, this is the unit circle. x = cos(t), y = sin(t), and z=z due to the cylindars orientation.

by the plane: z = -x-y, z = -cos(t)-sin(t)

Answer 17

I'm reading through the paper you linked. Is that where you got the info you used to make the parametric plot?

Answer 18

it helped to jogged an idea :)

x = a cos(t)
y = a sin(t)
z = ....

it helped to me recall something ...

Answer 19

Just wondering: if the plane was a bit more complex, would a need to be a vector to express the orientation of the circle?

Answer 20

planes are pretty simple; if the cylindar was more complex then it could get a bit hairier.

i wouldnt say im a specialist at this or anything, but the the setup here was pretty basic :)

play with some different cylindars and other surfaces to get a better feel for it

Answer 21

alright, cool. Will do and thanks for your help.

Answer 22

good luck )