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Math Help - Parametric Equations for 2 Parallel Planes Joined by Funnel?

  1. #1
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    Parametric Equations for 2 Parallel Planes Joined by Funnel?

    Trying to render a 3d plot of two planes joined by a double-ended funnel, and would like to describe it using parametric equations because of the plotting software (gnuplot). I think the solution is simple, but parametric equations are new to me so I'm a bit lost.




    For example, I can render a Catenoid using equations of the form
    x = c*cos(v/c)*cos(u)
    y = c*cosh(v/c)*sin(u)
    z = v
    but the surfaces above and below the funnel aren't sufficiently flat.

    Parametric Equations for 2 Parallel Planes Joined by Funnel?-catenoidexample.gif



    I prefer the look of something more like y=1/x, as in a Gabriel's Horn
    x = c*cos(v)/u
    y = c*sin(v)/u
    z = -u
    but that only has one plane and funnel, and what I need is for it to be mirrored below, visually like a tornado reflected in a lake.

    Parametric Equations for 2 Parallel Planes Joined by Funnel?-gabrielshornexample.gif




    Another one-sided example I've found is
    x = u*cos(v)
    y = u*sin(v)
    z = log(u)
    but it's also not as nice and planar as the Gabriel's Horn solution.

    Parametric Equations for 2 Parallel Planes Joined by Funnel?-funnelexample.gif


    Thanks for any help!
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  2. #2
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    Re: Parametric Equations for 2 Parallel Planes Joined by Funnel?

    Perhaps y= \frac 1 {|x|} or y= \frac 1 {x^2}?
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  3. #3
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    Re: Parametric Equations for 2 Parallel Planes Joined by Funnel?

    Yes, thank you, that is the basic shape I would like to be using. My difficulty is in describing that shape mathematically when the funnel has two ends instead of one, and in 3 dimensions instead of 2, preferably using parametric equations. A finished product would yield something like the catenoid image in my original post, but would level off nicely at each end of the funnel as in the one-sided, 2-D equations you offered.

    Parametric Equations for 2 Parallel Planes Joined by Funnel?-1overabsx.gif

    Parametric Equations for 2 Parallel Planes Joined by Funnel?-1overxsquared.gif
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  4. #4
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    Re: Parametric Equations for 2 Parallel Planes Joined by Funnel?

    how about replacing Cosh(v/c) in the catenoid with something like:

    (Cosh(v/c)+ a v^m) where m is large.

    Then you can choose "a" so that this extra term has little effect on most of the curve but its differential 20 v ^19 will add significantly to the flatness at the ends of the funnel.

    I tried:
    c=1
    a=0.0001
    m=20

    plotted for -2.5<z<2.5

    and it looked quite good.

    Note in matlab:
    u=-pi:2*pi/25i;
    v=-2.5:.1:2.5;
    k=20
    c=1
    for i=1:length(u)
    for j=1:length(v)

    X(i,j)=(0.0001*v(j)^k+c*cosh(v(j)/c))*cos(u(i));
    Y(i,j)=(0.0001*v(j)^k+c*cosh(v(j)/c))*sin(u(i));

    Z(i,j)=v(j);
    end
    end
    mesh(X,Y,Z)
    axis (2*[-3 3 -3 3 -3 3])
    Last edited by Kiwi_Dave; July 9th 2012 at 02:04 AM.
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  5. #5
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    Re: Parametric Equations for 2 Parallel Planes Joined by Funnel?

    Hmmmm. That does look good. And tweaking the constants lets me shape it nicely. Thanks, Dave.
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