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Added catenary curves.

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Chris Palmer 2020-09-19 12:11:54 +01:00
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@ -20,23 +20,23 @@ See [usage](docs/usage.md) for requirements, installation instructions and a usa
<th align="left"> Vitamins A-I </th><th align="left"> Vitamins J-Q </th><th align="left"> Vitamins R-Z </th><th align="left"> Printed </th><th align="left"> Utilities </th><th align="left"> Core Utilities </th></tr>
<tr><td> <a href = "#Axials">Axials</a> </td><td> <a href = "#Jack">Jack</a> </td><td> <a href = "#Rails">Rails</a> </td><td> <a href = "#Box">Box</a> </td><td> <a href = "#Annotation">Annotation</a> </td><td> <a href = "#BOM">BOM</a> </td></tr>
<tr><td> <a href = "#Ball_bearings">Ball_bearings</a> </td><td> <a href = "#KP_pillow_blocks">KP_pillow_blocks</a> </td><td> <a href = "#Ring_terminals">Ring_terminals</a> </td><td> <a href = "#Butt_box">Butt_box</a> </td><td> <a href = "#Bezier">Bezier</a> </td><td> <a href = "#Clip">Clip</a> </td></tr>
<tr><td> <a href = "#Batteries">Batteries</a> </td><td> <a href = "#LDRs">LDRs</a> </td><td> <a href = "#Rockers">Rockers</a> </td><td> <a href = "#Cable_grommets">Cable_grommets</a> </td><td> <a href = "#Dogbones">Dogbones</a> </td><td> <a href = "#Global">Global</a> </td></tr>
<tr><td> <a href = "#Belts">Belts</a> </td><td> <a href = "#LED_meters">LED_meters</a> </td><td> <a href = "#Rod">Rod</a> </td><td> <a href = "#Carriers">Carriers</a> </td><td> <a href = "#Fillet">Fillet</a> </td><td> <a href = "#Polyholes">Polyholes</a> </td></tr>
<tr><td> <a href = "#Blowers">Blowers</a> </td><td> <a href = "#LEDs">LEDs</a> </td><td> <a href = "#SCS_bearing_blocks">SCS_bearing_blocks</a> </td><td> <a href = "#Corner_block">Corner_block</a> </td><td> <a href = "#Gears">Gears</a> </td><td> <a href = "#Rounded_rectangle">Rounded_rectangle</a> </td></tr>
<tr><td> <a href = "#Bulldogs">Bulldogs</a> </td><td> <a href = "#Leadnuts">Leadnuts</a> </td><td> <a href = "#SK_brackets">SK_brackets</a> </td><td> <a href = "#Door_hinge">Door_hinge</a> </td><td> <a href = "#Hanging_hole">Hanging_hole</a> </td><td> <a href = "#Sphere">Sphere</a> </td></tr>
<tr><td> <a href = "#Buttons">Buttons</a> </td><td> <a href = "#Light_strips">Light_strips</a> </td><td> <a href = "#SMDs">SMDs</a> </td><td> <a href = "#Door_latch">Door_latch</a> </td><td> <a href = "#Horiholes">Horiholes</a> </td><td> <a href = "#Teardrops">Teardrops</a> </td></tr>
<tr><td> <a href = "#Cable_strips">Cable_strips</a> </td><td> <a href = "#Linear_bearings">Linear_bearings</a> </td><td> <a href = "#SSRs">SSRs</a> </td><td> <a href = "#Fan_guard">Fan_guard</a> </td><td> <a href = "#Layout">Layout</a> </td><td></td></tr>
<tr><td> <a href = "#Cameras">Cameras</a> </td><td> <a href = "#Magnets">Magnets</a> </td><td> <a href = "#Screws">Screws</a> </td><td> <a href = "#Fixing_block">Fixing_block</a> </td><td> <a href = "#Maths">Maths</a> </td><td></td></tr>
<tr><td> <a href = "#Circlips">Circlips</a> </td><td> <a href = "#Mains_sockets">Mains_sockets</a> </td><td> <a href = "#Sealing_strip">Sealing_strip</a> </td><td> <a href = "#Flat_hinge">Flat_hinge</a> </td><td> <a href = "#Offset">Offset</a> </td><td></td></tr>
<tr><td> <a href = "#Components">Components</a> </td><td> <a href = "#Microswitches">Microswitches</a> </td><td> <a href = "#Sheets">Sheets</a> </td><td> <a href = "#Foot">Foot</a> </td><td> <a href = "#Quadrant">Quadrant</a> </td><td></td></tr>
<tr><td> <a href = "#DIP">DIP</a> </td><td> <a href = "#Microview">Microview</a> </td><td> <a href = "#Spades">Spades</a> </td><td> <a href = "#Handle">Handle</a> </td><td> <a href = "#Round">Round</a> </td><td></td></tr>
<tr><td> <a href = "#D_connectors">D_connectors</a> </td><td> <a href = "#Modules">Modules</a> </td><td> <a href = "#Spools">Spools</a> </td><td> <a href = "#PCB_mount">PCB_mount</a> </td><td> <a href = "#Rounded_cylinder">Rounded_cylinder</a> </td><td></td></tr>
<tr><td> <a href = "#Displays">Displays</a> </td><td> <a href = "#Nuts">Nuts</a> </td><td> <a href = "#Springs">Springs</a> </td><td> <a href = "#PSU_shroud">PSU_shroud</a> </td><td> <a href = "#Rounded_polygon">Rounded_polygon</a> </td><td></td></tr>
<tr><td> <a href = "#Extrusion_brackets">Extrusion_brackets</a> </td><td> <a href = "#O_ring">O_ring</a> </td><td> <a href = "#Stepper_motors">Stepper_motors</a> </td><td> <a href = "#Printed_box">Printed_box</a> </td><td> <a href = "#Sector">Sector</a> </td><td></td></tr>
<tr><td> <a href = "#Extrusions">Extrusions</a> </td><td> <a href = "#Opengrab">Opengrab</a> </td><td> <a href = "#Swiss_clips">Swiss_clips</a> </td><td> <a href = "#Ribbon_clamp">Ribbon_clamp</a> </td><td> <a href = "#Sweep">Sweep</a> </td><td></td></tr>
<tr><td> <a href = "#Fans">Fans</a> </td><td> <a href = "#PCB">PCB</a> </td><td> <a href = "#Toggles">Toggles</a> </td><td> <a href = "#SSR_shroud">SSR_shroud</a> </td><td> <a href = "#Thread">Thread</a> </td><td></td></tr>
<tr><td> <a href = "#Fuseholder">Fuseholder</a> </td><td> <a href = "#PCBs">PCBs</a> </td><td> <a href = "#Transformers">Transformers</a> </td><td> <a href = "#Screw_knob">Screw_knob</a> </td><td> <a href = "#Tube">Tube</a> </td><td></td></tr>
<tr><td> <a href = "#Geared_steppers">Geared_steppers</a> </td><td> <a href = "#PSUs">PSUs</a> </td><td> <a href = "#Tubings">Tubings</a> </td><td> <a href = "#Socket_box">Socket_box</a> </td><td></td><td></td></tr>
<tr><td> <a href = "#Batteries">Batteries</a> </td><td> <a href = "#LDRs">LDRs</a> </td><td> <a href = "#Rockers">Rockers</a> </td><td> <a href = "#Cable_grommets">Cable_grommets</a> </td><td> <a href = "#Catenary">Catenary</a> </td><td> <a href = "#Global">Global</a> </td></tr>
<tr><td> <a href = "#Belts">Belts</a> </td><td> <a href = "#LED_meters">LED_meters</a> </td><td> <a href = "#Rod">Rod</a> </td><td> <a href = "#Carriers">Carriers</a> </td><td> <a href = "#Dogbones">Dogbones</a> </td><td> <a href = "#Polyholes">Polyholes</a> </td></tr>
<tr><td> <a href = "#Blowers">Blowers</a> </td><td> <a href = "#LEDs">LEDs</a> </td><td> <a href = "#SCS_bearing_blocks">SCS_bearing_blocks</a> </td><td> <a href = "#Corner_block">Corner_block</a> </td><td> <a href = "#Fillet">Fillet</a> </td><td> <a href = "#Rounded_rectangle">Rounded_rectangle</a> </td></tr>
<tr><td> <a href = "#Bulldogs">Bulldogs</a> </td><td> <a href = "#Leadnuts">Leadnuts</a> </td><td> <a href = "#SK_brackets">SK_brackets</a> </td><td> <a href = "#Door_hinge">Door_hinge</a> </td><td> <a href = "#Gears">Gears</a> </td><td> <a href = "#Sphere">Sphere</a> </td></tr>
<tr><td> <a href = "#Buttons">Buttons</a> </td><td> <a href = "#Light_strips">Light_strips</a> </td><td> <a href = "#SMDs">SMDs</a> </td><td> <a href = "#Door_latch">Door_latch</a> </td><td> <a href = "#Hanging_hole">Hanging_hole</a> </td><td> <a href = "#Teardrops">Teardrops</a> </td></tr>
<tr><td> <a href = "#Cable_strips">Cable_strips</a> </td><td> <a href = "#Linear_bearings">Linear_bearings</a> </td><td> <a href = "#SSRs">SSRs</a> </td><td> <a href = "#Fan_guard">Fan_guard</a> </td><td> <a href = "#Horiholes">Horiholes</a> </td><td></td></tr>
<tr><td> <a href = "#Cameras">Cameras</a> </td><td> <a href = "#Magnets">Magnets</a> </td><td> <a href = "#Screws">Screws</a> </td><td> <a href = "#Fixing_block">Fixing_block</a> </td><td> <a href = "#Layout">Layout</a> </td><td></td></tr>
<tr><td> <a href = "#Circlips">Circlips</a> </td><td> <a href = "#Mains_sockets">Mains_sockets</a> </td><td> <a href = "#Sealing_strip">Sealing_strip</a> </td><td> <a href = "#Flat_hinge">Flat_hinge</a> </td><td> <a href = "#Maths">Maths</a> </td><td></td></tr>
<tr><td> <a href = "#Components">Components</a> </td><td> <a href = "#Microswitches">Microswitches</a> </td><td> <a href = "#Sheets">Sheets</a> </td><td> <a href = "#Foot">Foot</a> </td><td> <a href = "#Offset">Offset</a> </td><td></td></tr>
<tr><td> <a href = "#DIP">DIP</a> </td><td> <a href = "#Microview">Microview</a> </td><td> <a href = "#Spades">Spades</a> </td><td> <a href = "#Handle">Handle</a> </td><td> <a href = "#Quadrant">Quadrant</a> </td><td></td></tr>
<tr><td> <a href = "#D_connectors">D_connectors</a> </td><td> <a href = "#Modules">Modules</a> </td><td> <a href = "#Spools">Spools</a> </td><td> <a href = "#PCB_mount">PCB_mount</a> </td><td> <a href = "#Round">Round</a> </td><td></td></tr>
<tr><td> <a href = "#Displays">Displays</a> </td><td> <a href = "#Nuts">Nuts</a> </td><td> <a href = "#Springs">Springs</a> </td><td> <a href = "#PSU_shroud">PSU_shroud</a> </td><td> <a href = "#Rounded_cylinder">Rounded_cylinder</a> </td><td></td></tr>
<tr><td> <a href = "#Extrusion_brackets">Extrusion_brackets</a> </td><td> <a href = "#O_ring">O_ring</a> </td><td> <a href = "#Stepper_motors">Stepper_motors</a> </td><td> <a href = "#Printed_box">Printed_box</a> </td><td> <a href = "#Rounded_polygon">Rounded_polygon</a> </td><td></td></tr>
<tr><td> <a href = "#Extrusions">Extrusions</a> </td><td> <a href = "#Opengrab">Opengrab</a> </td><td> <a href = "#Swiss_clips">Swiss_clips</a> </td><td> <a href = "#Ribbon_clamp">Ribbon_clamp</a> </td><td> <a href = "#Sector">Sector</a> </td><td></td></tr>
<tr><td> <a href = "#Fans">Fans</a> </td><td> <a href = "#PCB">PCB</a> </td><td> <a href = "#Toggles">Toggles</a> </td><td> <a href = "#SSR_shroud">SSR_shroud</a> </td><td> <a href = "#Sweep">Sweep</a> </td><td></td></tr>
<tr><td> <a href = "#Fuseholder">Fuseholder</a> </td><td> <a href = "#PCBs">PCBs</a> </td><td> <a href = "#Transformers">Transformers</a> </td><td> <a href = "#Screw_knob">Screw_knob</a> </td><td> <a href = "#Thread">Thread</a> </td><td></td></tr>
<tr><td> <a href = "#Geared_steppers">Geared_steppers</a> </td><td> <a href = "#PSUs">PSUs</a> </td><td> <a href = "#Tubings">Tubings</a> </td><td> <a href = "#Socket_box">Socket_box</a> </td><td> <a href = "#Tube">Tube</a> </td><td></td></tr>
<tr><td> <a href = "#Green_terminals">Green_terminals</a> </td><td> <a href = "#Panel_meters">Panel_meters</a> </td><td> <a href = "#Variacs">Variacs</a> </td><td> <a href = "#Strap_handle">Strap_handle</a> </td><td></td><td></td></tr>
<tr><td> <a href = "#Hot_ends">Hot_ends</a> </td><td> <a href = "#Pillars">Pillars</a> </td><td> <a href = "#Veroboard">Veroboard</a> </td><td></td><td></td><td></td></tr>
<tr><td> <a href = "#Hygrometer">Hygrometer</a> </td><td> <a href = "#Pin_headers">Pin_headers</a> </td><td> <a href = "#Washers">Washers</a> </td><td></td><td></td><td></td></tr>
@ -5245,6 +5245,36 @@ Bezier curves and function to get and adjust the length or minimum z point.
![bezier](tests/png/bezier.png)
<a href="#top">Top</a>
---
<a name="Catenary"></a>
## Catenary
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.
[utils/catenary.scad](utils/catenary.scad) Implementation.
[tests/catenary.scad](tests/catenary.scad) Code for this example.
### Functions
| Function | Description |
|:--- |:--- |
| ```catenary(t, a)``` | Parametric catenary function linear along the length of the curve. |
| ```catenary_ds_by_da(d, a)``` | First derivative of the length with respect to ```a```. |
| ```catenary_find_a(d, l, a = 1)``` | Find the catenary constant ```a```, given half the horizontal span and the length. |
| ```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```. |
| ```catenary_s(d, a)``` | Length of a symmetric catenary with width ```2d```. |
![catenary](tests/png/catenary.png)
<a href="#top">Top</a>
---

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//
// 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/>.
//
l = 250; // [1: 1000]
x = 200; // [1: 1000]
y = 50; //[-500 : 500]
include <../utils/core/core.scad>
use <../utils/catenary.scad>
use <../utils/sweep.scad>
use <../utils/annotation.scad>
module catenaries() {
//
// catenary curve path from control points
//
curve = [for(p = catenary_points(l, x, y)) [p.x, p.y, 0]];
//
// Draw the curve
//
r = 0.5;
sweep(curve, circle_points(r, $fn = 64));
//
// Minimum Z
//
min_z = catenary_points(l, x, y, 0);
color("blue") {
translate([min_z.x, min_z.y + r])
rotate([-90, 0, 0])
arrow();
translate([min_z.x, min_z.y - r])
rotate([90, 0, 0])
arrow();
}
}
if($preview)
rotate(is_undef($bom) ? 0 : [70, 0, 315])
catenaries();

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//
// 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;