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Improved numerical accuarcy of catenary calculations.

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Chris Palmer 2020-09-19 23:52:57 +01:00
parent 1255e71271
commit a7803b1efb
2 changed files with 6 additions and 7 deletions
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2
readme.md
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@ -5268,7 +5268,7 @@ The coordinates of the lowest point on the curve can be retrieved by calling ```
|:--- |:--- |
| ```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_find_a(d, l, a = 1, best_e = inf, best_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```. |

11
utils/catenary.scad
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@ -33,21 +33,20 @@ function catenary(t, a) = let(u = argsinh(t)) a * [u, cosh(u)]; //! Parametric c
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_find_a(d, l, a = 1, best_e = inf, best_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) let(error = abs(catenary_s(d, a) - l))
error >= best_e && error < 0.0001 ? /*echo(best=best_e, a = best_a)*/ best_a
: catenary_find_a(d, l, max(a - (catenary_s(d, a) - l) / catenary_ds_by_da(d, a), d / argsinh(1e99)), error, a);
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
a = catenary_find_a(d, sqrt(sqr(l) - sqr(y))), // 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;