openscad/NopSCADlib
1
0
Fork 0
mirror of https://github.com/nophead/NopSCADlib.git synced 2025-08-10 17:33:57 +02:00
Code Issues Releases Wiki Activity

Added catenary curves.

Browse Source
This commit is contained in:
Chris Palmer
2020-09-19 12:11:54 +01:00
parent b11c5914b3
commit 1255e71271
4 changed files with 156 additions and 17 deletions
Download Patch File Download Diff File Expand all files Collapse all files

53
utils/catenary.scad Normal file
View File

@@ -0,0 +1,53 @@
//
// NopSCADlib Copyright Chris Palmer 2020
// nop.head@gmail.com
// hydraraptor.blogspot.com
//
// This file is part of NopSCADlib.
//
// NopSCADlib is free software: you can redistribute it and/or modify it under the terms of the
// GNU General Public License as published by the Free Software Foundation, either version 3 of
// the License, or (at your option) any later version.
//
// NopSCADlib is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY;
// without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
// See the GNU General Public License for more details.
//
// You should have received a copy of the GNU General Public License along with NopSCADlib.
// If not, see <https://www.gnu.org/licenses/>.
//
//
//! Catenary curve to model hanging wires, etc.
//!
//! Although the equation of the curve is simply ```y = a cosh(x / a)``` there is no explicit formula to calculate the constant ```a``` or the range of ```x``` given the
//! length of the cable and the end point coordinates. See <https://en.wikipedia.org/wiki/Catenary#Determining_parameters>. The Newton-Raphson method is used to find
//! ```a``` numerically, see <https://en.wikipedia.org/wiki/Newton%27s_method>.
//!
//! The coordinates of the lowest point on the curve can be retrieved by calling ```catenary_points()``` with ```steps``` equal to zero.
//
include <core/core.scad>
use <maths.scad>
function catenary(t, a) = let(u = argsinh(t)) a * [u, cosh(u)]; //! Parametric catenary function linear along the length of the curve.
function catenary_s(d, a) = 2 * a * sinh(d / a); //! Length of a symmetric catenary with width ```2d```.
function catenary_ds_by_da(d, a) = 2 * sinh(d / a) - 2 * d / a * cosh(d / a); //! First derivative of the length with respect to ```a```.
function catenary_find_a(d, l, a = 1) = //! Find the catenary constant ```a```, given half the horizontal span and the length.
assert(l > 2 * d, "Not long enough to span the gap") assert(d)
abs(catenary_s(d, a) - l) < 0.0001 ? a
: catenary_find_a(d, l, max(a - (catenary_s(d, a) - l) / catenary_ds_by_da(d, a), 0.001));
function catenary_points(l, x, y, steps = 100) = //! Returns a list of 2D points on the curve that goes from the origin to ```(x,y)``` and has length ```l```.
let(
d = x / 2,
a = catenary_find_a(d, sqrt(sqr(l) - sqr(y)), d / 2), // Find a to get the correct length
offset = argsinh(y / catenary_s(d, a)),
t0 = sinh(-d / a + offset),
t1 = sinh( d / a + offset),
h = a * cosh(-d / a + offset) - a,
lowest = offset > d / a ? [0, 0] : offset < -d / a ? [x, y] : [d - offset * a, -h],
//dummy = echo(l = l, d = d, a = a, t0=t0, t1=t1, sinh(d / a), s = catenary_s(d, a), offset = offset * a, h = h),
p0 = catenary(t0, a)
)
steps ? [for(t = [t0 : (t1 - t0) / steps : t1]) catenary(t, a) - p0] : lowest;