From dbf930fd366c28aa9e581d9fe01e3654b882923a Mon Sep 17 00:00:00 2001 From: Chris Palmer Date: Sun, 21 Apr 2024 12:13:31 +0100 Subject: [PATCH] Added cubic splines and Catmull Rom splines. --- lib.scad | 1 + libtest.png | Bin 1024804 -> 1024805 bytes readme.md | 34 ++++++++++++-- tests/png/splines.png | Bin 0 -> 46697 bytes tests/splines.scad | 38 ++++++++++++++++ utils/splines.scad | 101 ++++++++++++++++++++++++++++++++++++++++++ 6 files changed, 170 insertions(+), 4 deletions(-) create mode 100644 tests/png/splines.png create mode 100644 tests/splines.scad create mode 100644 utils/splines.scad diff --git a/lib.scad b/lib.scad index f64ee80..e10e864 100644 --- a/lib.scad +++ b/lib.scad @@ -97,6 +97,7 @@ use use use use +use use use use diff --git a/libtest.png b/libtest.png index 4041888059a09ecfcd89582a2057ff391406e8c0..a85c3438a37328aebd9904110b424d73971e6f6f 100644 GIT binary patch delta 747167 zcmXt9Wmr^QxE(r$k&^Ciq`Mm=q+5`X?uG+MBOqM@0@4jKl!VeHD9z9*9YYOy=ezg0 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Printed_pulleys + Extrusion_brackets PSUs Terminals Pocket_handle Splines + Extrusions Panel_meters Toggles Press_fit Sweep + Fans Photo_interrupters Transformers Printed_box Thread + Fastons Pillars Ttracks Printed_pulleys Tube Fuseholder Pillow_blocks Tubings Ribbon_clamp Gear_motors Pin_headers Variacs SSR_shroud Geared_steppers Potentiometers Veroboard Screw_knob @@ -7420,6 +7420,32 @@ A sector of a circle between two angles. ![sector](tests/png/sector.png) +Top + +--- + +## Splines +Cubic splines that interpolate between a list of 2D points passing through all of them. +Translated from the Python version at . +Note the x values of the points must be strictly increasing. + +Catmull-Rom splines are well behaved but the ends points are control points and the curve only goes from the second point to the penultimate point. +Coded from . +No restrictions on points and they can be 3D. + +[utils/splines.scad](utils/splines.scad) Implementation. + +[tests/splines.scad](tests/splines.scad) Code for this example. + +### Functions +| Function | Description | +|:--- |:--- | +| `catmull_rom_spline(points, n, alpha = 0.5)` | Interpolate n new points between the specified points with a Catmull-Rom spline, alpha = 0.5 for centripetal, 0 for uniform and 1 for chordal. | +| `cubic_spline(points, N = 100)` | Interpolate the list of points given to produce N points on a cubic spline that passes through points given. | + +![splines](tests/png/splines.png) + + Top --- diff --git a/tests/png/splines.png b/tests/png/splines.png new file mode 100644 index 0000000000000000000000000000000000000000..43ec855a39e9725412b0c9b8d0b2e71cffae8dda GIT binary patch literal 46697 zcmaI7byQT}7dAdL(nu;X#D{LBI|V^fM!G~mx;uuFlu|mSyQLe1QCg7h?vff97=Bm% zuHXCq^$u$mYwkVkp0oFPo*ifJ8>ykDh>!CW2LuA)zf_Xf0)f!}nQxFzBVcjJDVO z;gY9cBBT@Zbf?|k@)Oag_;J~j|1Q)Xiz*Ql)td6ORaHeD7!U1N9nViUc+OPWQjn1* zF00701_9%kLWxAvu996Jyy#IGMx^J@?-$1c+D)BgKc!iXR#Ow>Vamt@AF@wc6NG^g zz(*Oco{pH9ShK&MQwg}C?aYV!82Ak_91i|FkmCP+JH=yr-ZN>`)~XZ4 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<../utils/core/core.scad> + +use <../utils/splines.scad> +use <../utils/sweep.scad> + +points = [[0, 1.5], [2, 2], [3, 1], [4, -2], [5, 1], [6, 2], [7, 3]]; + +module splines() { + cm_spline = catmull_rom_spline(points, 100 / len(points), 0.5); + color("green") show_path(cm_spline, 0.01); + + cu_spline = cubic_spline(points, 100); + color("blue") show_path(cu_spline, 0.01); + + for(p = points) + translate(p) color("red") + cylinder($fn = 64, r = 0.03, h = 0.02, center = true); +} + +rotate([70, 0, 315]) splines(); diff --git a/utils/splines.scad b/utils/splines.scad new file mode 100644 index 0000000..221ddae --- /dev/null +++ b/utils/splines.scad @@ -0,0 +1,101 @@ +// +// NopSCADlib Copyright Chris Palmer 2024 +// 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 . +// + +// +//! Cubic splines that interpolate between a list of 2D points passing through all of them. +//! Translated from the Python version at . +//! Note the x values of the points must be strictly increasing. +//! +//! Catmull-Rom splines are well behaved but the ends points are control points and the curve only goes from the second point to the penultimate point. +//! Coded from . +//! No restrictions on points and they can be 3D. +// +include <../utils/core/core.scad> + +use +use + +function cubic_spline(points, N = 100) = let( //! Interpolate the list of points given to produce N points on a cubic spline that passes through points given. + N = N - 1, + n = len(points), + ass1 = assert(n >= 3, "must be at least 3 points")0, + + dx = [for(i = [0 : n - 2]) points[i + 1].x - points[i].x], // x deltas + ass2 = assert(min(dx) > 0, "X must strictly increase")0, + + // + // A and C are diagonals above and below the main diagonal B, which is all 2's + // + A = [for(i = [0 : n - 3]) dx[i] / (dx[i] + dx[i + 1]), 0], + C = [0, for(i = [0 : n - 3]) dx[i + 1] / (dx[i] + dx[i + 1]), 0], + // + // D are the target values on the right hand side of the equation + // + D = [0, for(i = [1 : n - 2]) 6 * ((points[i + 1].y - points[i].y) / dx[i] - (points[i].y - points[i - 1].y) / dx[i - 1]) / (dx[i] + dx[i - 1]), 0], + // + // Solve the tridiagonal equation using the Thomas algorithm + // + c = [for(i = 1, c = 0; i < n; c = C[i] / (2 - c * A[i - 1]), i = i + 1) c, 0], + d = [for(i = 1, d = 0; i < n; d = (D[i] - d * A[i - 1]) / (2 - c[i - 1] * A[i - 1]), i = i + 1) d, 0], + M = [for(i = n - 2, x = 0; i >= 0; x = d[i] - c[i] * x, i = i - 1) x, 0], + // + // Calculate the coefficients of each cubic curve + // + coefficients = [for(i = [0 : n - 2], dx2 = sqr(dx[i]), j = n - 1 - i) + [(M[j - 1] - M[j]) * dx2 / 6, + M[j] * dx2 / 2, + points[i + 1].y - points[i].y - (M[j - 1] + 2 * M[j]) * dx2 / 6, + points[i].y] + ], + // + // Use the coefficients to interpolate between the points + // + x0 = points[0].x, + x1 = points[n - 1].x, + spline = [for(i = 0, j = 0, z = 0, x = x0; i <= N + 1; + x = x0 + (x1 - x0) * i / N, + j = i < N - 1 && x > points[j + 1].x ? j + 1 : j, + z = (x - points[j].x) / dx[j], + i = i + 1, + C = coefficients[j] + ) if(i) [x, (((C[0] * z) + C[1]) * z + C[2]) * z + C[3]] + ] + ) spline; + +function tj(ti, pi, pj, alpha = 0.5) = ti + pow(norm(pi - pj), alpha); + +function catmull_rom_segment(P0, P1, P2, P3, n, alpha = 0.5, last = false) = let( + t0 = 0, + t1 = tj(t0, P0, P1, alpha), + t2 = tj(t1, P1, P2, alpha), + t3 = tj(t2, P2, P3, alpha), + end = last ? n : n - 1, + points = [for(i = [0 : end], t = t1 + (t2 - t1) * i / n) let( + A1 = (t1 - t) / (t1 - t0) * P0 + (t - t0) / (t1 - t0) * P1, + A2 = (t2 - t) / (t2 - t1) * P1 + (t - t1) / (t2 - t1) * P2, + A3 = (t3 - t) / (t3 - t2) * P2 + (t - t2) / (t3 - t2) * P3, + B1 = (t2 - t) / (t2 - t0) * A1 + (t - t0) / (t2 - t0) * A2, + B2 = (t3 - t) / (t3 - t1) * A2 + (t - t1) / (t3 - t1) * A3 + ) (t2 - t) / (t2 - t1) * B1 + (t - t1) / (t2 - t1) * B2], + ) points; + + +function catmull_rom_spline(points, n, alpha = 0.5) = let( //! Interpolate n new points between the specified points with a Catmull-Rom spline, alpha = 0.5 for centripetal, 0 for uniform and 1 for chordal. + segs = len(points) - 3 + ) [for(i = [0 : segs - 1]) each catmull_rom_segment(points[i], points[i + 1], points[i + 2], points[i + 3], n, alpha, last = i == segs - 1)];