Perhaps or ?
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 formx = c*cos(v/c)*cos(u)y = c*cosh(v/c)*sin(u)z = vbut 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 Hornx = c*cos(v)/uy = c*sin(v)/uz = -ubut 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 isx = 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!
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.
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])