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Updated bezier code to support N-point beziers.

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Revar Desmera 2018-02-16 14:49:32 -08:00
parent 453ebcb83f
commit 2d9fa565b3
3 changed files with 62 additions and 48 deletions
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88
beziers.scad
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@ -33,40 +33,46 @@ OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
include <math.scad> include <math.scad>
include <paths.scad> include <paths.scad>
include <transforms.scad>
// Formulae to calculate points on a cubic bezier curve. // Formulae to calculate points on an N-point bezier curve.
function bez_B0(curve,u) = curve[0]*pow((1-u),3); function bez_point(curve,u)=
function bez_B1(curve,u) = curve[1]*(3*u*pow((1-u),2)); (len(curve) <= 1) ?
function bez_B2(curve,u) = curve[2]*(3*pow(u,2)*(1-u)); curve[0] :
function bez_B3(curve,u) = curve[3]*pow(u,3); bez_point(
function bez_point(curve,u) = bez_B0(curve,u) + bez_B1(curve,u) + bez_B2(curve,u) + bez_B3(curve,u); [for(i=[0:len(curve)-2]) curve[i]*(1-u)+curve[i+1]*u],
u
);
// Takes an array of bezier points and converts it into a 3D polyline. // Takes an array of bezier points and converts it into a 3D polyline.
function bezier_polyline(bezier, splinesteps=16) = concat( function bezier_polyline(bezier, splinesteps=16, N=3) = concat(
[ [
for ( for (
b = [0 : 3 : len(bezier)-4], b = [0 : N : len(bezier)-N-1],
l = [0 : splinesteps-1] l = [0 : splinesteps-1]
) let ( ) let (
crv = [bezier[b+0], bezier[b+1], bezier[b+2], bezier[b+3]], crv = [for (i=[0 : N]) bezier[b+i]],
u = l / splinesteps u = l / splinesteps
) bez_point(crv, u) ) bez_point(crv, u)
], ],
[bez_point([bezier[len(bezier)-4], bezier[len(bezier)-3], bezier[len(bezier)-2], bezier[len(bezier)-1]], 1.0)] [bez_point([for (i=[-(N+1) : -1]) bezier[len(bezier)+i]], 1.0)]
); );
// Takes a closed 2D bezier path, and creates a 2D polygon from it. // Takes a closed 2D bezier path, and creates a 2D polygon from it.
module bezier_polygon(bezier, splinesteps=16) { // bezier = array of 2D bezier path points
polypoints=bezier_polyline(bezier, splinesteps); // splinesteps = number of straight lines to split each bezier segment into
// N = number of points in each bezier segment. Cubic = 3
module bezier_polygon(bezier, splinesteps=16, N=3) {
polypoints=bezier_polyline(bezier, splinesteps, N);
polygon(points=slice(polypoints, 0, -1)); polygon(points=slice(polypoints, 0, -1));
} }
// Generate bezier curve to fillet 2 line segments between 3 points. // Generate bezier curve to fillet 2 line segments between 3 points.
// Returns two path points with surrounding cubic bezier control points. // Returns two path points with surrounding cubic (N=3) bezier control points.
function fillet3pts(p0, p1, p2, r) = let( function fillet3pts(p0, p1, p2, r) = let(
v0 = normalize(p0-p1), v0 = normalize(p0-p1),
v1 = normalize(p2-p1), v1 = normalize(p2-p1),
@ -82,7 +88,7 @@ function fillet3pts(p0, p1, p2, r) = let(
) [tp0, tp0, cp0, cp1, tp1, tp1]; ) [tp0, tp0, cp0, cp1, tp1, tp1];
// Takes a 3D polyline path and fillets it into a 3d cubic bezier path. // Takes a 3D polyline path and fillets it into a 3d cubic (N=3) bezier path.
function fillet_path(pts, fillet) = concat( function fillet_path(pts, fillet) = concat(
[pts[0], pts[0]], [pts[0], pts[0]],
(len(pts) < 3)? [] : [ (len(pts) < 3)? [] : [
@ -98,6 +104,7 @@ function fillet_path(pts, fillet) = concat(
// Takes a closed 2D bezier and rotates it around the X axis, forming a solid. // Takes a closed 2D bezier and rotates it around the X axis, forming a solid.
// bezier = array of points for the bezier path to rotate. // bezier = array of points for the bezier path to rotate.
// splinesteps = number of segments to divide each bezier segment into. // splinesteps = number of segments to divide each bezier segment into.
// N = number of points in each bezier segment. Cubic = 3
// Example: // Example:
// path = [ // path = [
// [ 0, 10], [ 50, 0], [ 50, 40], // [ 0, 10], [ 50, 0], [ 50, 40],
@ -107,35 +114,36 @@ function fillet_path(pts, fillet) = concat(
// [ 0, 10] // [ 0, 10]
// ]; // ];
// revolve_bezier(path, splinesteps=32, $fn=180); // revolve_bezier(path, splinesteps=32, $fn=180);
module revolve_bezier(bezier, splinesteps=16) { module revolve_bezier(bezier, splinesteps=16, N=3) {
yrot(90) rotate_extrude(convexity=10) { yrot(90) rotate_extrude(convexity=10) {
xrot(180) zrot(-90) bezier_polygon(bezier, splinesteps); xrot(180) zrot(-90) bezier_polygon(bezier, splinesteps, N);
} }
} }
// Takes a bezier path and closes it to the X axis. // Takes a 2D bezier path and closes it to the X axis.
function bezier_close_to_axis(bezier) = function bezier_close_to_axis(bezier, N=3) =
let(bezend = len(bezier)-1) let(
concat( bezend = len(bezier)-1
[ [bezier[0][0], 0], [bezier[0][0], 0], bezier[0] ], ) concat(
[for (i=[0:N-1]) lerp([bezier[0][0], 0], bezier[0], i/N)],
bezier, bezier,
[ bezier[bezend], [bezier[bezend][0], 0], [bezier[bezend][0], 0] ] [for (i=[1:N]) lerp(bezier[bezend], [bezier[bezend][0], 0], i/N)],
[for (i=[1:N]) lerp([bezier[bezend][0], 0], [bezier[0][0], 0], i/N)]
); );
// Takes a bezier curve and closes it with a matching path that is // Takes a bezier curve and closes it with a matching path that is
// lowered by a given amount towards the X axis. // lowered by a given amount towards the X axis.
function bezier_offset(inset, bezier) = function bezier_offset(inset, bezier, N=3) =
let(backbez = reverse([ for (pt = bezier) [pt[0], pt[1]-inset] ])) let(
concat( backbez = reverse([ for (pt = bezier) [pt[0], pt[1]-inset] ]),
bezend = len(bezier)-1
) concat(
bezier, bezier,
[bezier[len(bezier)-1]], [for (i=[1:N-1]) lerp(bezier[bezend], backbez[0], i/N)],
[backbez[0]],
backbez, backbez,
[backbez[len(backbez)-1]], [for (i=[1:N]) lerp(backbez[bezend], bezier[0], i/N)]
[bezier[0]],
[bezier[0]]
); );
@ -145,8 +153,8 @@ function bezier_offset(inset, bezier) =
// Example: // Example:
// path = [ [0, 10], [33, 10], [66, 40], [100, 40] ]; // path = [ [0, 10], [33, 10], [66, 40], [100, 40] ];
// revolve_bezier_solid_to_axis(path, splinesteps=32, $fn=72); // revolve_bezier_solid_to_axis(path, splinesteps=32, $fn=72);
module revolve_bezier_solid_to_axis(bezier, splinesteps=16) { module revolve_bezier_solid_to_axis(bezier, splinesteps=16, N=3) {
revolve_bezier(bezier=bezier_close_to_axis(bezier), splinesteps=splinesteps); revolve_bezier(bezier=bezier_close_to_axis(bezier), splinesteps=splinesteps, N=N);
} }
@ -157,8 +165,8 @@ module revolve_bezier_solid_to_axis(bezier, splinesteps=16) {
// Example: // Example:
// path = [ [0, 10], [33, 10], [66, 40], [100, 40] ]; // path = [ [0, 10], [33, 10], [66, 40], [100, 40] ];
// revolve_bezier_offset_shell(path, offset=1, splinesteps=32, $fn=72); // revolve_bezier_offset_shell(path, offset=1, splinesteps=32, $fn=72);
module revolve_bezier_offset_shell(bezier, offset=1, splinesteps=16) { module revolve_bezier_offset_shell(bezier, offset=1, splinesteps=16, N=3) {
revolve_bezier(bezier=bezier_offset(offset, bezier), splinesteps=splinesteps); revolve_bezier(bezier=bezier_offset(offset, bezier), splinesteps=splinesteps, N=N);
} }
@ -169,8 +177,8 @@ module revolve_bezier_offset_shell(bezier, offset=1, splinesteps=16) {
// path = [ [0, 0, 0], [33, 33, 33], [66, -33, -33], [100, 0, 0] ]; // path = [ [0, 0, 0], [33, 33, 33], [66, -33, -33], [100, 0, 0] ];
// extrude_2d_shapes_along_bezier(path, splinesteps=32) // extrude_2d_shapes_along_bezier(path, splinesteps=32)
// circle(r=10, center=true); // circle(r=10, center=true);
module extrude_2d_shapes_along_bezier(bezier, splinesteps=16) { module extrude_2d_shapes_along_bezier(bezier, splinesteps=16, N=3) {
pointslist = slice(bezier_polyline(bezier, splinesteps), 0, -1); pointslist = slice(bezier_polyline(bezier, splinesteps, N), 0, -1);
ptcount = len(pointslist); ptcount = len(pointslist);
for (i = [0 : ptcount-2]) { for (i = [0 : ptcount-2]) {
pt1 = pointslist[i]; pt1 = pointslist[i];
@ -218,13 +226,15 @@ module extrude_2d_shapes_along_bezier(bezier, splinesteps=16) {
// path = Array of points of a bezier path, to extrude along. // path = Array of points of a bezier path, to extrude along.
// pathsteps = number of steps to divide each path segment into. // pathsteps = number of steps to divide each path segment into.
// bezsteps = number of steps to divide each bezier segment into. // bezsteps = number of steps to divide each bezier segment into.
// bezN = number of points in each extruded bezier segment. Cubic = 3
// pathN = number of points in each path bezier segment. Cubic = 3
// Example: // Example:
// bez = [ [-15, 0], [25, -15], [-5, 10], [0, 10], [5, 10], [10, 5], [15, 0], [10, -5], [5, -10], [0, -10], [-5, -10], [-10, -5], [-15, 0] ]; // bez = [ [-15, 0], [25, -15], [-5, 10], [0, 10], [5, 10], [10, 5], [15, 0], [10, -5], [5, -10], [0, -10], [-5, -10], [-10, -5], [-15, 0] ];
// path = [ [0, 0, 0], [33, 33, 33], [66, -33, -33], [100, 0, 0] ]; // path = [ [0, 0, 0], [33, 33, 33], [66, -33, -33], [100, 0, 0] ];
// extrude_bezier_along_bezier(bez, path, pathsteps=64, bezsteps=32); // extrude_bezier_along_bezier(bez, path, pathsteps=64, bezsteps=32);
module extrude_bezier_along_bezier(bezier, path, pathsteps=16, bezsteps=16) { module extrude_bezier_along_bezier(bezier, path, pathsteps=16, bezsteps=16, bezN=3, pathN=3) {
bez_points = simplify2d_path(bezier_polyline(bezier, bezsteps)); bez_points = simplify2d_path(bezier_polyline(bezier, bezsteps, bezN));
path_points = simplify3d_path(path3d(bezier_polyline(path, pathsteps))); path_points = simplify3d_path(path3d(bezier_polyline(path, pathsteps, pathN)));
extrude_2dpath_along_3dpath(bez_points, path_points); extrude_2dpath_along_3dpath(bez_points, path_points);
} }

4
math.scad
View file

@ -51,6 +51,10 @@ function quantup(x,y) = ceil(x/y)*y;
function segs(r) = $fn>0?($fn>3?$fn:3):(ceil(max(min(360.0/$fa,abs(r)*2*Cpi()/$fs),5))); function segs(r) = $fn>0?($fn>3?$fn:3):(ceil(max(min(360.0/$fa,abs(r)*2*Cpi()/$fs),5)));
// Interpolate between two values or vectors. 0.0 <= u <= 1.0
function lerp(a,b,u) = (b-a)*u + a;
// Calculate hypotenuse length of 2D triangle. // Calculate hypotenuse length of 2D triangle.
function hypot(x,y) = sqrt(x*x+y*y); function hypot(x,y) = sqrt(x*x+y*y);

2
paths.scad
View file

@ -189,7 +189,7 @@ function points_along_path3d(
// Example: // Example:
// shape = [ [-15, 0], [0, 0], [-5, 10], [0, 10], [5, 10], [10, 5], [15, 0], [10, -5], [5, -10], [0, -10], [-5, -10], [-10, -5], [-15, 0] ]; // shape = [ [-15, 0], [0, 0], [-5, 10], [0, 10], [5, 10], [10, 5], [15, 0], [10, -5], [5, -10], [0, -10], [-5, -10], [-10, -5], [-15, 0] ];
// path = [ [0, 0, 0], [100, 33, 33], [200, -33, -33], [300, 0, 0] ]; // path = [ [0, 0, 0], [100, 33, 33], [200, -33, -33], [300, 0, 0] ];
// extrude_2dpath_along_3dpath(shape, path, tilt=false); // extrude_2dpath_along_3dpath(shape, path);
module extrude_2dpath_along_3dpath(polyline, path, convexity=10) { module extrude_2dpath_along_3dpath(polyline, path, convexity=10) {
pline_count = len(polyline); pline_count = len(polyline);
path_count = len(path); path_count = len(path);