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

1. ## 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.

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.

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.

Thanks for any help!

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

Perhaps $y= \frac 1 {|x|}$ or $y= \frac 1 {x^2}$?

3. ## 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.

4. ## 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])

5. ## 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.