Hi,

Does anybody know an elegant way to create a tube with a parabolic
form? I can manually create a bunch of spheres on the curve and HULL
those pair-by-pair and union the hulls. Does anybody know a more
elegant way?

Roger.

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The plan was simple, like my brother-in-law Phil. But unlike
Phil, this plan just might work.

Can you clarify what you mean.
Parabolic form in what way?
parabolic along: profile, radius, outside, inside or maybe parabolic
thickness?

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instead of unioning pairs of hulled spheres you could use a union of pairs of
hulled "short" cylinders. Of course each cylinder must be properly rotated.

The best and fastest way to realise your design is a sweep:

use <Naca_sweep.scad>  // download from
https://www.thingiverse.com/thing:900137

a = 1;
x0 = -2;
x1 = 2;
N = 30;
P = parabola(a, x0, x1, N);
sweep(tube_dat(P));

function tube_dat(P) =
[
for(i=[0:len(P)-1])
let(w = atan(2aP[i][0]))  // f'(x) = 2ax
T(P[i][0], P[i][1], 0,
Rz(w, Ry(-90, circle(.3, N))))
];

// symmetric parabola: f(x)= ax²
function parabola(a=.005, x0=-100, x1=100, N=10) =
[for(i=[0:N-1]) let(x = x0+i/(N-1)(x1-x0)) [x, axx]];

function circle(r, N) = [for(i=[0:N-1]) let(a= 360/Ni)
r[sin(a), cos(a), 0]];

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When I've wanted something like that, I connect up a bunch of
cylinderoids and spheroids using a module called pipe_polygon.

Here's an implementation using my current pipe_polygon.  It's got
several more features than you're asking for, like the ability to
justify the polygon relative to the origin.

Notes:

• I don't remember why I insist on setting $fn rather than letting $fa
  and $fs work.  Maybe it was to default $fnz, or maybe it was just
  because I forgot about them.
• It looks like I planned to generate just a slice at a corner and
  then rotate into place, rather than generating a full spheroid.  I
  don't remember why I didn't follow through.  As a result the joints
  are not quite perfect.
• There's probably a problem at the ends if the last segment is
  shorter than the radius and round ends are turned off.

parabola(1,0,0, .3, -2, 2, 20);

module parabola (p1, p2, p3, r, x0, x1, fn )
{
s = (x1-x0)/fn;

points = [ for (x=[x0:s:x1]) [x,p1*x*x+p2*x+p3,0] ];
pipe_polygon(points, r=r, open=true, round_ends=false);

}

// OpenSCAD polygons have their radius/diameter
// measured to the vertices.  These functions
// give you adjusted r/d that will yield the
// specified r/d to the edges.
function adjustr(r, n) = r / cos(360/n/2);
function adjustd(d, n) = 2*adjustr(d/2, n);

function default(val, def) = (val != undef) ? val : def;

// @brief    Justify children as specified by argument.
// @param center  Equivalent to justify=[0,0,0].
// @param justify  [jx,jy,jz]:  1=toward-positive, 0=center, -1=toward-negative
// @param dims    Dimensions of child to justify
module justify(center, justify, dims) {
function o(flag, dim) = flag > 0 ? 0 :
flag < 0 ? -dim :
-dim/2;
j = default(justify, center ? [0,0,0] : [1,1,1]);
translate([o(j[0], dims[0]),
o(j[1], dims[1]),
o(j[2], dims[2])]) children();
}

// Given a series of points, connect them with a pipe.
// This module has more features than are used in this model.
// poly - the array of points.
// r, d - the radius or diameter of the pipe.  Defaults to d=1.  (NB this is measured to the edges.)
// fill - if true, fill the interior of the figure, yielding a
// filled polygon with rounded edges.  Requires open=false.  Defaults to false.
// open - if false, connect the start and end points to close
// the polygon.  Defaults to false.
// justify - justify the polygon as specified.  Defaults to [1,1,1].
// round_ends - if true, cap the ends.  Defaults to true.
// $fn - controls smoothness at corners.  Defaults to 32.
// $fnz - the $fn value to use for the cylinder and the z-curves at the
// corners.  This lets you make pipes with polygonal cross-sections.  Defaults to $fn.
// NEEDSWORK:  $fa, $fs, and $fn should control the smoothness at corners.
module pipe_polygon(poly, r, d, fill=false, open=false, justify=[1,1,1], round_ends=true) {
fn = $fn ? $fn : 32;
fnz = is_undef($fnz) ? fn : $fnz;
dims = [
max([for (p=poly) p[0]]),
max([for (p=poly) p[1]]),
0
];
_r = default(r, default(d,1)/2);
radjusted = adjustr(_r, fnz);
justify(justify=justify, dims=dims) {
$fn = fn;
$fnz = fnz;
for (i = [(round_ends ? 0 : 1) : len(poly)-(round_ends ? 1 : 2)]) {
p = poly[i];
translate([p[0], p[1], 0]) corner(r=radjusted);
}
poly2 = open ? poly : concat(poly, [poly[0]]);
for (i=[0:len(poly2)-2]) {
p1 = poly2[i];
p2 = poly2[i+1];
dy = p2[1] - p1[1];
dx = p2[0] - p1[0];
translate([p1[0], p1[1], 0]) {
rotate([0,90,atan2(dy, dx)])
rotate([0,0,360/fnz/2])
cylinder(r=radjusted, h=norm([dx, dy]), $fn=$fnz);
}
}
if (fill) linear_extrude(_r2, center=true) polygon(poly);
}
// corner() defaults to a slice of a spheroid.  It's done
// this way so that it can have different smoothing in x-y
// and z, so that it can mate with a polygonal pipe.
module corner(r, a) {
rotate_extrude(angle=a) {
intersection() {
rotate([0,0,360/$fnz/2])
circle(r=r, $fn=$fnz);
translate([0,-r2])
square([r2, r4]);
}
}
}
}

Here's another option for sweep using BOSL2 (though be warned that BOSL2 is
under development):

https://github.com/revarbat/BOSL2/wiki/skin.scad#path_sweep

include<BOSL2/std.scad>
include<BOSL2/skin.scad>

curve = [for(x=[-2:.1:2]) [x,x*x,0]];
path_sweep(circle(r=.1, $fn=16), curve);

The result looks like this:

http://forum.openscad.org/file/t2477/parabola.png

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