Results 1 to 3 of 3

Math Help - [SOLVED] Surface

  1. #1
    Senior Member Spec's Avatar
    Joined
    Aug 2007
    Posts
    318

    [SOLVED] Surface

    Which surface is created by the intersection of the cylinder x^2+y^2=1 and the plane z=y+3

    That is, the cylinder cuts out a surface from the plane.
    Last edited by Spec; September 18th 2008 at 01:13 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2008
    From
    Paris, France
    Posts
    1,174
    Drawing a sketch is very helpful (in this case, it is not very difficult to do).

    First we choose a coordinate system in the plane z=y+3, i.e. an origin O' and an orthonormal basis (\vec{u},\vec{v}), so we can write the equation of the intersection using it. For instance (as shows the sketch, it is not only a simple but also an appropriate choice), O'=(0,0,3), \vec{u}=(1,0,0) and \vec{v}=\frac{1}{\sqrt{2}}(0,1,1).

    Then, for any M=(x,y,z) in the intersection, z=y+3, hence M=O'+x\vec{u}+\sqrt{2}y\vec{v}, so the coordinates of M in the new system are X=x and Y=\sqrt{2}y. The equation x^2+y^2=1 becomes X^2+\frac{Y^2}{2}=1. In other words, the intersection is an ellipse with minor and major axes respectively equal to 1 and \sqrt{2}, centered at O'. The major axis is in direction \vec{v} and the minor one in direction \vec{u}.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Aug 2008
    Posts
    903
    That's very nice Laurent. I used your equation in the form of:

    f[\text{t$\_$}]\text{:=}\{0,0,3\}+\text{Cos}[t] \{1,0,0\}+\sqrt{2} \text{Sin}[t] \frac{1}{\sqrt{2}}\{0,1,1\};

    Then did a parametric plot of f(t) (in red) over the range [0,2\pi]

    and then plotted it with the cylinder and plane so defined:
    Attached Thumbnails Attached Thumbnails [SOLVED] Surface-cylinder-plane.jpg  
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Surface Area
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 22nd 2009, 01:10 AM
  2. Replies: 6
    Last Post: August 1st 2009, 03:29 AM
  3. [SOLVED] Another surface area
    Posted in the Calculus Forum
    Replies: 0
    Last Post: March 10th 2009, 09:32 PM
  4. [SOLVED] Area of a Surface of Revolution
    Posted in the Calculus Forum
    Replies: 2
    Last Post: December 1st 2008, 02:06 PM
  5. [SOLVED] Surface integrals and flux
    Posted in the Calculus Forum
    Replies: 5
    Last Post: September 30th 2008, 02:03 AM

Search Tags


/mathhelpforum @mathhelpforum