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1471 lines
59 KiB
OpenSCAD
1471 lines
59 KiB
OpenSCAD
//////////////////////////////////////////////////////////////////////
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// LibFile: paths.scad
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// Support for polygons and paths.
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// To use, add the following lines to the beginning of your file:
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// ```
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// include <BOSL2/std.scad>
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// ```
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//////////////////////////////////////////////////////////////////////
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include <triangulation.scad>
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// Section: Functions
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// Function: is_path()
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// Usage:
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// is_path(list, [dim], [fast])
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// Description:
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// Returns true if `list` is a path. A path is a list of two or more numeric vectors (AKA points).
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// All vectors must of the same size, and may only contain numbers that are not inf or nan.
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// By default the vectors in a path must be 2d or 3d. Set the `dim` parameter to specify a list
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// of allowed dimensions, or set it to `undef` to allow any dimension.
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// Examples:
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// is_path([[3,4],[5,6]]); // Returns true
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// is_path([[3,4]]); // Returns false
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// is_path([[3,4],[4,5]],2); // Returns true
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// is_path([[3,4,3],[5,4,5]],2); // Returns false
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// is_path([[3,4,3],[5,4,5]],2); // Returns false
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// is_path([[3,4,5],undef,[4,5,6]]); // Returns false
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// is_path([[3,5],[undef,undef],[4,5]]); // Returns false
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// is_path([[3,4],[5,6],[5,3]]); // Returns true
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// is_path([3,4,5,6,7,8]); // Returns false
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// is_path([[3,4],[5,6]], dim=[2,3]);// Returns true
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// is_path([[3,4],[5,6]], dim=[1,3]);// Returns false
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// is_path([[3,4],"hello"], fast=true); // Returns true
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// is_path([[3,4],[3,4,5]]); // Returns false
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// is_path([[1,2,3,4],[2,3,4,5]]); // Returns false
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// is_path([[1,2,3,4],[2,3,4,5]],undef);// Returns true
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// Arguments:
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// list = list to check
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// dim = list of allowed dimensions of the vectors in the path. Default: [2,3]
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// fast = set to true for fast check that only looks at first entry. Default: false
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function is_path(list, dim=[2,3], fast=false) =
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fast
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? is_list(list) && is_vector(list[0])
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: is_matrix(list)
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&& len(list)>1
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&& len(list[0])>0
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&& (is_undef(dim) || in_list(len(list[0]), force_list(dim)));
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// Function: is_closed_path()
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// Usage:
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// is_closed_path(path, [eps]);
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// Description:
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// Returns true if the first and last points in the given path are coincident.
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function is_closed_path(path, eps=EPSILON) = approx(path[0], path[len(path)-1], eps=eps);
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// Function: close_path()
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// Usage:
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// close_path(path);
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// Description:
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// If a path's last point does not coincide with its first point, closes the path so it does.
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function close_path(path, eps=EPSILON) = is_closed_path(path,eps=eps)? path : concat(path,[path[0]]);
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// Function: cleanup_path()
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// Usage:
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// cleanup_path(path);
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// Description:
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// If a path's last point coincides with its first point, deletes the last point in the path.
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function cleanup_path(path, eps=EPSILON) = is_closed_path(path,eps=eps)? select(path,0,-2) : path;
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// Function: path_subselect()
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// Usage:
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// path_subselect(path,s1,u1,s2,u2,[closed]):
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// Description:
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// Returns a portion of a path, from between the `u1` part of segment `s1`, to the `u2` part of
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// segment `s2`. Both `u1` and `u2` are values between 0.0 and 1.0, inclusive, where 0 is the start
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// of the segment, and 1 is the end. Both `s1` and `s2` are integers, where 0 is the first segment.
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// Arguments:
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// path = The path to get a section of.
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// s1 = The number of the starting segment.
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// u1 = The proportion along the starting segment, between 0.0 and 1.0, inclusive.
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// s2 = The number of the ending segment.
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// u2 = The proportion along the ending segment, between 0.0 and 1.0, inclusive.
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// closed = If true, treat path as a closed polygon.
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function path_subselect(path, s1, u1, s2, u2, closed=false) =
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let(
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lp = len(path),
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l = lp-(closed?0:1),
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u1 = s1<0? 0 : s1>l? 1 : u1,
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u2 = s2<0? 0 : s2>l? 1 : u2,
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s1 = constrain(s1,0,l),
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s2 = constrain(s2,0,l),
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pathout = concat(
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(s1<l && u1<1)? [lerp(path[s1],path[(s1+1)%lp],u1)] : [],
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[for (i=[s1+1:1:s2]) path[i]],
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(s2<l && u2>0)? [lerp(path[s2],path[(s2+1)%lp],u2)] : []
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)
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) pathout;
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// Function: simplify_path()
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// Description:
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// Takes a path and removes unnecessary subsequent collinear points.
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// Usage:
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// simplify_path(path, [eps])
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// Arguments:
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// path = A list of path points of any dimension.
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// eps = Largest positional variance allowed. Default: `EPSILON` (1-e9)
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function simplify_path(path, eps=EPSILON) =
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assert( is_path(path), "Invalid path." )
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assert( is_undef(eps) || (is_finite(eps) && (eps>=0) ), "Invalid tolerance." )
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len(path)<=2 ? path
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: let(
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indices = [ 0,
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for (i=[1:1:len(path)-2])
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if (!collinear(path[i-1],path[i],path[i+1], eps=eps)) i,
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len(path)-1
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]
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)
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[for (i = indices) path[i] ];
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// Function: simplify_path_indexed()
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// Description:
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// Takes a list of points, and a list of indices into `points`,
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// and removes from the list all indices of subsequent indexed points that are unecessarily collinear.
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// Returns the list of the remained indices.
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// Usage:
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// simplify_path_indexed(points,indices, eps)
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// Arguments:
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// points = A list of points.
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// indices = A list of indices into `points` that forms a path.
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// eps = Largest angle variance allowed. Default: EPSILON (1-e9) degrees.
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function simplify_path_indexed(points, indices, eps=EPSILON) =
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len(indices)<=2? indices
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: let(
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indices = concat( indices[0],
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[for (i=[1:1:len(indices)-2])
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let(
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i1 = indices[i-1],
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i2 = indices[i],
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i3 = indices[i+1]
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)
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if (!collinear(points[i1],points[i2],points[i3], eps=eps)) indices[i]],
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indices[len(indices)-1] )
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)
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indices;
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// Function: path_length()
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// Usage:
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// path_length(path,[closed])
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// Description:
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// Returns the length of the path.
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// Arguments:
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// path = The list of points of the path to measure.
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// closed = true if the path is closed. Default: false
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// Example:
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// path = [[0,0], [5,35], [60,-25], [80,0]];
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// echo(path_length(path));
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function path_length(path,closed=false) =
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len(path)<2? 0 :
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sum([for (i = [0:1:len(path)-2]) norm(path[i+1]-path[i])])+(closed?norm(path[len(path)-1]-path[0]):0);
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// Function: path_segment_lengths()
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// Usage:
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// path_segment_lengths(path,[closed])
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// Description:
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// Returns list of the length of each segment in a path
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// Arguments:
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// path = path to measure
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// closed = true if the path is closed. Default: false
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function path_segment_lengths(path, closed=false) =
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[
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for (i=[0:1:len(path)-2]) norm(path[i+1]-path[i]),
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if (closed) norm(path[0]-path[len(path)-1])
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];
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// Function: path_pos_from_start()
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// Usage:
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// pos = path_pos_from_start(path,length,[closed]);
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// Description:
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// Finds the segment and relative position along that segment that is `length` distance from the
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// front of the given `path`. Returned as [SEGNUM, U] where SEGNUM is the segment number, and U is
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// the relative distance along that segment, a number from 0 to 1. If the path is shorter than the
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// asked for length, this returns `undef`.
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// Arguments:
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// path = The path to find the position on.
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// length = The length from the start of the path to find the segment and position of.
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// Example(2D):
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// path = circle(d=50,$fn=18);
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// pos = path_pos_from_start(path,20,closed=false);
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// stroke(path,width=1,endcaps=false);
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// pt = lerp(path[pos[0]], path[(pos[0]+1)%len(path)], pos[1]);
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// color("red") translate(pt) circle(d=2,$fn=12);
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function path_pos_from_start(path,length,closed=false,_d=0,_i=0) =
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let (lp = len(path))
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_i >= lp - (closed?0:1)? undef :
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let (l = norm(path[(_i+1)%lp]-path[_i]))
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_d+l <= length? path_pos_from_start(path,length,closed,_d+l,_i+1) :
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[_i, (length-_d)/l];
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// Function: path_pos_from_end()
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// Usage:
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// pos = path_pos_from_end(path,length,[closed]);
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// Description:
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// Finds the segment and relative position along that segment that is `length` distance from the
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// end of the given `path`. Returned as [SEGNUM, U] where SEGNUM is the segment number, and U is
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// the relative distance along that segment, a number from 0 to 1. If the path is shorter than the
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// asked for length, this returns `undef`.
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// Arguments:
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// path = The path to find the position on.
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// length = The length from the end of the path to find the segment and position of.
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// Example(2D):
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// path = circle(d=50,$fn=18);
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// pos = path_pos_from_end(path,20,closed=false);
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// stroke(path,width=1,endcaps=false);
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// pt = lerp(path[pos[0]], path[(pos[0]+1)%len(path)], pos[1]);
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// color("red") translate(pt) circle(d=2,$fn=12);
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function path_pos_from_end(path,length,closed=false,_d=0,_i=undef) =
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let (
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lp = len(path),
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_i = _i!=undef? _i : lp - (closed?1:2)
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)
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_i < 0? undef :
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let (l = norm(path[(_i+1)%lp]-path[_i]))
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_d+l <= length? path_pos_from_end(path,length,closed,_d+l,_i-1) :
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[_i, 1-(length-_d)/l];
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// Function: path_trim_start()
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// Usage:
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// path_trim_start(path,trim);
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// Description:
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// Returns the `path`, with the start shortened by the length `trim`.
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// Arguments:
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// path = The path to trim.
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// trim = The length to trim from the start.
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// Example(2D):
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// path = circle(d=50,$fn=18);
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// path2 = path_trim_start(path,5);
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// path3 = path_trim_start(path,20);
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// color("blue") stroke(path3,width=5,endcaps=false);
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// color("cyan") stroke(path2,width=3,endcaps=false);
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// color("red") stroke(path,width=1,endcaps=false);
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function path_trim_start(path,trim,_d=0,_i=0) =
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_i >= len(path)-1? [] :
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let (l = norm(path[_i+1]-path[_i]))
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_d+l <= trim? path_trim_start(path,trim,_d+l,_i+1) :
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let (v = unit(path[_i+1]-path[_i]))
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concat(
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[path[_i+1]-v*(l-(trim-_d))],
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[for (i=[_i+1:1:len(path)-1]) path[i]]
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);
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// Function: path_trim_end()
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// Usage:
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// path_trim_end(path,trim);
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// Description:
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// Returns the `path`, with the end shortened by the length `trim`.
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// Arguments:
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// path = The path to trim.
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// trim = The length to trim from the end.
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// Example(2D):
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// path = circle(d=50,$fn=18);
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// path2 = path_trim_end(path,5);
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// path3 = path_trim_end(path,20);
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// color("blue") stroke(path3,width=5,endcaps=false);
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// color("cyan") stroke(path2,width=3,endcaps=false);
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// color("red") stroke(path,width=1,endcaps=false);
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function path_trim_end(path,trim,_d=0,_i=undef) =
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let (_i = _i!=undef? _i : len(path)-1)
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_i <= 0? [] :
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let (l = norm(path[_i]-path[_i-1]))
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_d+l <= trim? path_trim_end(path,trim,_d+l,_i-1) :
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let (v = unit(path[_i]-path[_i-1]))
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concat(
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[for (i=[0:1:_i-1]) path[i]],
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[path[_i-1]+v*(l-(trim-_d))]
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);
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// Function: path_closest_point()
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// Usage:
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// path_closest_point(path, pt);
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// Description:
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// Finds the closest path segment, and point on that segment to the given point.
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// Returns `[SEGNUM, POINT]`
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// Arguments:
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// path = The path to find the closest point on.
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// pt = the point to find the closest point to.
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// Example(2D):
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// path = circle(d=100,$fn=6);
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// pt = [20,10];
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// closest = path_closest_point(path, pt);
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// stroke(path, closed=true);
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// color("blue") translate(pt) circle(d=3, $fn=12);
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// color("red") translate(closest[1]) circle(d=3, $fn=12);
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function path_closest_point(path, pt) =
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let(
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pts = [for (seg=idx(path)) segment_closest_point(select(path,seg,seg+1),pt)],
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dists = [for (p=pts) norm(p-pt)],
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min_seg = min_index(dists)
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) [min_seg, pts[min_seg]];
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// Function: path_tangents()
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// Usage:
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// tangs = path_tangents(path, <closed>, <uniform>);
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// Description:
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// Compute the tangent vector to the input path. The derivative approximation is described in deriv().
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// The returns vectors will be normalized to length 1. If any derivatives are zero then
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// the function fails with an error. If you set `uniform` to false then the sampling is
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// assumed to be non-uniform and the derivative is computed with adjustments to produce corrected
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// values.
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// Arguments:
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// path = path to find the tagent vectors for
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// closed = set to true of the path is closed. Default: false
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// uniform = set to false to correct for non-uniform sampling. Default: true
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// Example: A shape with non-uniform sampling gives distorted derivatives that may be undesirable
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// rect = square([10,3]);
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// tangents = path_tangents(rect,closed=true);
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// stroke(rect,closed=true, width=0.1);
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// color("purple")
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// for(i=[0:len(tangents)-1])
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// stroke([rect[i]-tangents[i], rect[i]+tangents[i]],width=.1, endcap2="arrow2");
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// Example: A shape with non-uniform sampling gives distorted derivatives that may be undesirable
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// rect = square([10,3]);
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// tangents = path_tangents(rect,closed=true,uniform=false);
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// stroke(rect,closed=true, width=0.1);
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// color("purple")
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// for(i=[0:len(tangents)-1])
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// stroke([rect[i]-tangents[i], rect[i]+tangents[i]],width=.1, endcap2="arrow2");
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function path_tangents(path, closed=false, uniform=true) =
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assert(is_path(path))
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!uniform ? [for(t=deriv(path,closed=closed, h=path_segment_lengths(path,closed))) unit(t)]
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: [for(t=deriv(path,closed=closed)) unit(t)];
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// Function: path_normals()
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// Usage:
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// norms = path_normals(path, <tangents>, <closed>);
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// Description:
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// Compute the normal vector to the input path. This vector is perpendicular to the
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// path tangent and lies in the plane of the curve. When there are collinear points,
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// the curve does not define a unique plane and the normal is not uniquely defined.
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function path_normals(path, tangents, closed=false) =
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assert(is_path(path))
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assert(is_bool(closed))
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let( tangents = default(tangents, path_tangents(path,closed)) )
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assert(is_path(tangents))
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[
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for(i=idx(path)) let(
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pts = i==0? (closed? select(path,-1,1) : select(path,0,2)) :
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i==len(path)-1? (closed? select(path,i-1,i+1) : select(path,i-2,i)) :
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select(path,i-1,i+1)
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) unit(cross(
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cross(pts[1]-pts[0], pts[2]-pts[0]),
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tangents[i]
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))
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];
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// Function: path_curvature()
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// Usage:
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// curvs = path_curvature(path, <closed>);
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// Description:
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// Numerically estimate the curvature of the path (in any dimension).
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function path_curvature(path, closed=false) =
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let(
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d1 = deriv(path, closed=closed),
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d2 = deriv2(path, closed=closed)
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) [
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for(i=idx(path))
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sqrt(
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sqr(norm(d1[i])*norm(d2[i])) -
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sqr(d1[i]*d2[i])
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) / pow(norm(d1[i]),3)
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];
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// Function: path_torsion()
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// Usage:
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// tortions = path_torsion(path, <closed>);
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// Description:
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// Numerically estimate the torsion of a 3d path.
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function path_torsion(path, closed=false) =
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let(
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d1 = deriv(path,closed=closed),
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d2 = deriv2(path,closed=closed),
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d3 = deriv3(path,closed=closed)
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) [
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for (i=idx(path)) let(
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crossterm = cross(d1[i],d2[i])
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) crossterm * d3[i] / sqr(norm(crossterm))
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];
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// Function: path_chamfer_and_rounding()
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// Usage:
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// path2 = path_chamfer_and_rounding(path, [closed], [chamfer], [rounding]);
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// Description:
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// Rounds or chamfers corners in the given path.
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// Arguments:
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// path = The path to chamfer and/or round.
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// closed = If true, treat path like a closed polygon. Default: true
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// chamfer = The length of the chamfer faces at the corners. If given as a list of numbers, gives individual chamfers for each corner, from first to last. Default: 0 (no chamfer)
|
|
// rounding = The rounding radius for the corners. If given as a list of numbers, gives individual radii for each corner, from first to last. Default: 0 (no rounding)
|
|
// Example(2D): Chamfering a Path
|
|
// path = star(5, step=2, d=100);
|
|
// path2 = path_chamfer_and_rounding(path, closed=true, chamfer=5);
|
|
// stroke(path2, closed=true);
|
|
// Example(2D): Per-Corner Chamfering
|
|
// path = star(5, step=2, d=100);
|
|
// chamfs = [for (i=[0:1:4]) each 3*[i,i]];
|
|
// path2 = path_chamfer_and_rounding(path, closed=true, chamfer=chamfs);
|
|
// stroke(path2, closed=true);
|
|
// Example(2D): Rounding a Path
|
|
// path = star(5, step=2, d=100);
|
|
// path2 = path_chamfer_and_rounding(path, closed=true, rounding=5);
|
|
// stroke(path2, closed=true);
|
|
// Example(2D): Per-Corner Chamfering
|
|
// path = star(5, step=2, d=100);
|
|
// rs = [for (i=[0:1:4]) each 2*[i,i]];
|
|
// path2 = path_chamfer_and_rounding(path, closed=true, rounding=rs);
|
|
// stroke(path2, closed=true);
|
|
// Example(2D): Mixing Chamfers and Roundings
|
|
// path = star(5, step=2, d=100);
|
|
// chamfs = [for (i=[0:4]) each [5,0]];
|
|
// rs = [for (i=[0:4]) each [0,10]];
|
|
// path2 = path_chamfer_and_rounding(path, closed=true, chamfer=chamfs, rounding=rs);
|
|
// stroke(path2, closed=true);
|
|
function path_chamfer_and_rounding(path, closed, chamfer, rounding) =
|
|
let (
|
|
path = deduplicate(path,closed=true),
|
|
lp = len(path),
|
|
chamfer = is_undef(chamfer)? repeat(0,lp) :
|
|
is_vector(chamfer)? list_pad(chamfer,lp,0) :
|
|
is_num(chamfer)? repeat(chamfer,lp) :
|
|
assert(false, "Bad chamfer value."),
|
|
rounding = is_undef(rounding)? repeat(0,lp) :
|
|
is_vector(rounding)? list_pad(rounding,lp,0) :
|
|
is_num(rounding)? repeat(rounding,lp) :
|
|
assert(false, "Bad rounding value."),
|
|
corner_paths = [
|
|
for (i=(closed? [0:1:lp-1] : [1:1:lp-2])) let(
|
|
p1 = select(path,i-1),
|
|
p2 = select(path,i),
|
|
p3 = select(path,i+1)
|
|
)
|
|
chamfer[i] > 0? _corner_chamfer_path(p1, p2, p3, side=chamfer[i]) :
|
|
rounding[i] > 0? _corner_roundover_path(p1, p2, p3, r=rounding[i]) :
|
|
[p2]
|
|
],
|
|
out = [
|
|
if (!closed) path[0],
|
|
for (i=(closed? [0:1:lp-1] : [1:1:lp-2])) let(
|
|
p1 = select(path,i-1),
|
|
p2 = select(path,i),
|
|
crn1 = select(corner_paths,i-1),
|
|
crn2 = corner_paths[i],
|
|
l1 = norm(select(crn1,-1)-p1),
|
|
l2 = norm(crn2[0]-p2),
|
|
needed = l1 + l2,
|
|
seglen = norm(p2-p1),
|
|
check = assert(seglen >= needed, str("Path segment ",i," is too short to fulfill rounding/chamfering for the adjacent corners."))
|
|
) each crn2,
|
|
if (!closed) select(path,-1)
|
|
]
|
|
) deduplicate(out);
|
|
|
|
|
|
function _corner_chamfer_path(p1, p2, p3, dist1, dist2, side, angle) =
|
|
let(
|
|
v1 = unit(p1 - p2),
|
|
v2 = unit(p3 - p2),
|
|
n = vector_axis(v1,v2),
|
|
ang = vector_angle(v1,v2),
|
|
path = (is_num(dist1) && is_undef(dist2) && is_undef(side))? (
|
|
// dist1 & optional angle
|
|
assert(dist1 > 0)
|
|
let(angle = default(angle,(180-ang)/2))
|
|
assert(is_num(angle))
|
|
assert(angle > 0 && angle < 180)
|
|
let(
|
|
pta = p2 + dist1*v1,
|
|
a3 = 180 - angle - ang
|
|
) assert(a3>0, "Angle too extreme.")
|
|
let(
|
|
side = sin(angle) * dist1/sin(a3),
|
|
ptb = p2 + side*v2
|
|
) [pta, ptb]
|
|
) : (is_undef(dist1) && is_num(dist2) && is_undef(side))? (
|
|
// dist2 & optional angle
|
|
assert(dist2 > 0)
|
|
let(angle = default(angle,(180-ang)/2))
|
|
assert(is_num(angle))
|
|
assert(angle > 0 && angle < 180)
|
|
let(
|
|
ptb = p2 + dist2*v2,
|
|
a3 = 180 - angle - ang
|
|
) assert(a3>0, "Angle too extreme.")
|
|
let(
|
|
side = sin(angle) * dist2/sin(a3),
|
|
pta = p2 + side*v1
|
|
) [pta, ptb]
|
|
) : (is_undef(dist1) && is_undef(dist2) && is_num(side))? (
|
|
// side & optional angle
|
|
assert(side > 0)
|
|
let(angle = default(angle,(180-ang)/2))
|
|
assert(is_num(angle))
|
|
assert(angle > 0 && angle < 180)
|
|
let(
|
|
a3 = 180 - angle - ang
|
|
) assert(a3>0, "Angle too extreme.")
|
|
let(
|
|
dist1 = sin(a3) * side/sin(ang),
|
|
dist2 = sin(angle) * side/sin(ang),
|
|
pta = p2 + dist1*v1,
|
|
ptb = p2 + dist2*v2
|
|
) [pta, ptb]
|
|
) : (is_num(dist1) && is_num(dist2) && is_undef(side) && is_undef(side))? (
|
|
// dist1 & dist2
|
|
assert(dist1 > 0)
|
|
assert(dist2 > 0)
|
|
let(
|
|
pta = p2 + dist1*v1,
|
|
ptb = p2 + dist2*v2
|
|
) [pta, ptb]
|
|
) : (
|
|
assert(false,"Bad arguments.")
|
|
)
|
|
) path;
|
|
|
|
|
|
function _corner_roundover_path(p1, p2, p3, r, d) =
|
|
let(
|
|
r = get_radius(r=r,d=d,dflt=undef),
|
|
res = circle_2tangents(p1, p2, p3, r=r, tangents=true),
|
|
cp = res[0],
|
|
n = res[1],
|
|
tp1 = res[2],
|
|
ang = res[4]+res[5],
|
|
steps = floor(segs(r)*ang/360+0.5),
|
|
step = ang / steps,
|
|
path = [for (i=[0:1:steps]) move(cp, p=rot(a=-i*step, v=n, p=tp1-cp))]
|
|
) path;
|
|
|
|
|
|
|
|
// Function: path3d_spiral()
|
|
// Description:
|
|
// Returns a 3D spiral path.
|
|
// Usage:
|
|
// path3d_spiral(turns, h, n, r|d, [cp], [scale]);
|
|
// Arguments:
|
|
// h = Height of spiral.
|
|
// turns = Number of turns in spiral.
|
|
// n = Number of spiral sides.
|
|
// r = Radius of spiral.
|
|
// d = Radius of spiral.
|
|
// cp = Centerpoint of spiral. Default: `[0,0]`
|
|
// scale = [X,Y] scaling factors for each axis. Default: `[1,1]`
|
|
// Example(3D):
|
|
// trace_path(path3d_spiral(turns=2.5, h=100, n=24, r=50), N=1, showpts=true);
|
|
function path3d_spiral(turns=3, h=100, n=12, r, d, cp=[0,0], scale=[1,1]) = let(
|
|
rr=get_radius(r=r, d=d, dflt=100),
|
|
cnt=floor(turns*n),
|
|
dz=h/cnt
|
|
) [
|
|
for (i=[0:1:cnt]) [
|
|
rr * cos(i*360/n) * scale.x + cp.x,
|
|
rr * sin(i*360/n) * scale.y + cp.y,
|
|
i*dz
|
|
]
|
|
];
|
|
|
|
|
|
// Function: path_self_intersections()
|
|
// Usage:
|
|
// isects = path_self_intersections(path, [eps]);
|
|
// Description:
|
|
// Locates all self intersections of the given path. Returns a list of intersections, where
|
|
// each intersection is a list like [POINT, SEGNUM1, PROPORTION1, SEGNUM2, PROPORTION2] where
|
|
// POINT is the coordinates of the intersection point, SEGNUMs are the integer indices of the
|
|
// intersecting segments along the path, and the PROPORTIONS are the 0.0 to 1.0 proportions
|
|
// of how far along those segments they intersect at. A proportion of 0.0 indicates the start
|
|
// of the segment, and a proportion of 1.0 indicates the end of the segment.
|
|
// Arguments:
|
|
// path = The path to find self intersections of.
|
|
// closed = If true, treat path like a closed polygon. Default: true
|
|
// eps = The epsilon error value to determine whether two points coincide. Default: `EPSILON` (1e-9)
|
|
// Example(2D):
|
|
// path = [
|
|
// [-100,100], [0,-50], [100,100], [100,-100], [0,50], [-100,-100]
|
|
// ];
|
|
// isects = path_self_intersections(path, closed=true);
|
|
// // isects == [[[-33.3333, 0], 0, 0.666667, 4, 0.333333], [[33.3333, 0], 1, 0.333333, 3, 0.666667]]
|
|
// stroke(path, closed=true, width=1);
|
|
// for (isect=isects) translate(isect[0]) color("blue") sphere(d=10);
|
|
function path_self_intersections(path, closed=true, eps=EPSILON) =
|
|
let(
|
|
path = cleanup_path(path, eps=eps),
|
|
plen = len(path)
|
|
) [
|
|
for (i = [0:1:plen-(closed?2:3)], j=[i+1:1:plen-(closed?1:2)]) let(
|
|
a1 = path[i],
|
|
a2 = path[(i+1)%plen],
|
|
b1 = path[j],
|
|
b2 = path[(j+1)%plen],
|
|
isect =
|
|
(max(a1.x, a2.x) < min(b1.x, b2.x))? undef :
|
|
(min(a1.x, a2.x) > max(b1.x, b2.x))? undef :
|
|
(max(a1.y, a2.y) < min(b1.y, b2.y))? undef :
|
|
(min(a1.y, a2.y) > max(b1.y, b2.y))? undef :
|
|
let(
|
|
c = a1-a2,
|
|
d = b1-b2,
|
|
denom = (c.x*d.y)-(c.y*d.x)
|
|
) abs(denom)<eps? undef : let(
|
|
e = a1-b1,
|
|
t = ((e.x*d.y)-(e.y*d.x)) / denom,
|
|
u = ((e.x*c.y)-(e.y*c.x)) / denom
|
|
) [a1+t*(a2-a1), t, u]
|
|
) if (
|
|
isect != undef &&
|
|
isect[1]>eps && isect[1]<=1+eps &&
|
|
isect[2]>eps && isect[2]<=1+eps
|
|
) [isect[0], i, isect[1], j, isect[2]]
|
|
];
|
|
|
|
|
|
// Function: split_path_at_self_crossings()
|
|
// Usage:
|
|
// paths = split_path_at_self_crossings(path, [closed], [eps]);
|
|
// Description:
|
|
// Splits a path into sub-paths wherever the original path crosses itself.
|
|
// Splits may occur mid-segment, so new vertices will be created at the intersection points.
|
|
// Arguments:
|
|
// path = The path to split up.
|
|
// closed = If true, treat path as a closed polygon. Default: true
|
|
// eps = Acceptable variance. Default: `EPSILON` (1e-9)
|
|
// Example(2D):
|
|
// path = [ [-100,100], [0,-50], [100,100], [100,-100], [0,50], [-100,-100] ];
|
|
// paths = split_path_at_self_crossings(path);
|
|
// rainbow(paths) stroke($item, closed=false, width=2);
|
|
function split_path_at_self_crossings(path, closed=true, eps=EPSILON) =
|
|
let(
|
|
path = cleanup_path(path, eps=eps),
|
|
isects = deduplicate(
|
|
eps=eps,
|
|
concat(
|
|
[[0, 0]],
|
|
sort([
|
|
for (
|
|
a = path_self_intersections(path, closed=closed, eps=eps),
|
|
ss = [ [a[1],a[2]], [a[3],a[4]] ]
|
|
) if (ss[0] != undef) ss
|
|
]),
|
|
[[len(path)-(closed?1:2), 1]]
|
|
)
|
|
)
|
|
) [
|
|
for (p = pair(isects))
|
|
let(
|
|
s1 = p[0][0],
|
|
u1 = p[0][1],
|
|
s2 = p[1][0],
|
|
u2 = p[1][1],
|
|
section = path_subselect(path, s1, u1, s2, u2, closed=closed),
|
|
outpath = deduplicate(eps=eps, section)
|
|
)
|
|
outpath
|
|
];
|
|
|
|
|
|
function _tag_self_crossing_subpaths(path, closed=true, eps=EPSILON) =
|
|
let(
|
|
subpaths = split_path_at_self_crossings(
|
|
path, closed=closed, eps=eps
|
|
)
|
|
) [
|
|
for (subpath = subpaths) let(
|
|
seg = select(subpath,0,1),
|
|
mp = mean(seg),
|
|
n = line_normal(seg) / 2048,
|
|
p1 = mp + n,
|
|
p2 = mp - n,
|
|
p1in = point_in_polygon(p1, path) >= 0,
|
|
p2in = point_in_polygon(p2, path) >= 0,
|
|
tag = (p1in && p2in)? "I" : "O"
|
|
) [tag, subpath]
|
|
];
|
|
|
|
|
|
// Function: decompose_path()
|
|
// Usage:
|
|
// splitpaths = decompose_path(path, [closed], [eps]);
|
|
// Description:
|
|
// Given a possibly self-crossing path, decompose it into non-crossing paths that are on the perimeter
|
|
// of the areas bounded by that path.
|
|
// Arguments:
|
|
// path = The path to split up.
|
|
// closed = If true, treat path like a closed polygon. Default: true
|
|
// eps = The epsilon error value to determine whether two points coincide. Default: `EPSILON` (1e-9)
|
|
// Example(2D):
|
|
// path = [
|
|
// [-100,100], [0,-50], [100,100], [100,-100], [0,50], [-100,-100]
|
|
// ];
|
|
// splitpaths = decompose_path(path, closed=true);
|
|
// rainbow(splitpaths) stroke($item, closed=true, width=3);
|
|
function decompose_path(path, closed=true, eps=EPSILON) =
|
|
let(
|
|
path = cleanup_path(path, eps=eps),
|
|
tagged = _tag_self_crossing_subpaths(path, closed=closed, eps=eps),
|
|
kept = [for (sub = tagged) if(sub[0] == "O") sub[1]],
|
|
completed = [for (frag=kept) if(is_closed_path(frag)) frag],
|
|
incomplete = [for (frag=kept) if(!is_closed_path(frag)) frag],
|
|
defrag = _path_fast_defragment(incomplete, eps=eps),
|
|
completed2 = assemble_path_fragments(defrag, eps=eps)
|
|
) concat(completed2,completed);
|
|
|
|
|
|
function _path_fast_defragment(fragments, eps=EPSILON, _done=[]) =
|
|
len(fragments)==0? _done :
|
|
let(
|
|
path = fragments[0],
|
|
endpt = select(path,-1),
|
|
extenders = [
|
|
for (i = [1:1:len(fragments)-1]) let(
|
|
test1 = approx(endpt,fragments[i][0],eps=eps),
|
|
test2 = approx(endpt,select(fragments[i],-1),eps=eps)
|
|
) if (test1 || test2) (test1? i : -1)
|
|
]
|
|
) len(extenders) == 1 && extenders[0] >= 0? _path_fast_defragment(
|
|
fragments=[
|
|
concat(select(path,0,-2),fragments[extenders[0]]),
|
|
for (i = [1:1:len(fragments)-1])
|
|
if (i != extenders[0]) fragments[i]
|
|
],
|
|
eps=eps,
|
|
_done=_done
|
|
) : _path_fast_defragment(
|
|
fragments=[for (i = [1:1:len(fragments)-1]) fragments[i]],
|
|
eps=eps,
|
|
_done=concat(_done,[deduplicate(path,closed=true,eps=eps)])
|
|
);
|
|
|
|
|
|
function _extreme_angle_fragment(seg, fragments, rightmost=true, eps=EPSILON) =
|
|
!fragments? [undef, []] :
|
|
let(
|
|
delta = seg[1] - seg[0],
|
|
segang = atan2(delta.y,delta.x),
|
|
frags = [
|
|
for (i = idx(fragments)) let(
|
|
fragment = fragments[i],
|
|
fwdmatch = approx(seg[1], fragment[0], eps=eps),
|
|
bakmatch = approx(seg[1], select(fragment,-1), eps=eps)
|
|
) [
|
|
fwdmatch,
|
|
bakmatch,
|
|
bakmatch? reverse(fragment) : fragment
|
|
]
|
|
],
|
|
angs = [
|
|
for (frag = frags)
|
|
(frag[0] || frag[1])? let(
|
|
delta2 = frag[2][1] - frag[2][0],
|
|
segang2 = atan2(delta2.y, delta2.x)
|
|
) modang(segang2 - segang) : (
|
|
rightmost? 999 : -999
|
|
)
|
|
],
|
|
fi = rightmost? min_index(angs) : max_index(angs)
|
|
) abs(angs[fi]) > 360? [undef, fragments] : let(
|
|
remainder = [for (i=idx(fragments)) if (i!=fi) fragments[i]],
|
|
frag = frags[fi],
|
|
foundfrag = frag[2]
|
|
) [foundfrag, remainder];
|
|
|
|
|
|
// Function: assemble_a_path_from_fragments()
|
|
// Usage:
|
|
// assemble_a_path_from_fragments(subpaths);
|
|
// Description:
|
|
// Given a list of paths, assembles them together into one complete closed polygon path, and
|
|
// remainder fragments. Returns [PATH, FRAGMENTS] where FRAGMENTS is the list of remaining
|
|
// unused path fragments.
|
|
// Arguments:
|
|
// fragments = List of paths to be assembled into complete polygons.
|
|
// rightmost = If true, assemble paths using rightmost turns. Leftmost if false.
|
|
// startfrag = The fragment to start with. Default: 0
|
|
// eps = The epsilon error value to determine whether two points coincide. Default: `EPSILON` (1e-9)
|
|
function assemble_a_path_from_fragments(fragments, rightmost=true, startfrag=0, eps=EPSILON) =
|
|
len(fragments)==0? _finished :
|
|
let(
|
|
path = fragments[startfrag],
|
|
newfrags = [for (i=idx(fragments)) if (i!=startfrag) fragments[i]]
|
|
) is_closed_path(path, eps=eps)? (
|
|
// starting fragment is already closed
|
|
[path, newfrags]
|
|
) : let(
|
|
// Find rightmost/leftmost continuation fragment
|
|
seg = select(path,-2,-1),
|
|
extrema = _extreme_angle_fragment(seg=seg, fragments=newfrags, rightmost=rightmost, eps=eps),
|
|
foundfrag = extrema[0],
|
|
remainder = extrema[1]
|
|
) is_undef(foundfrag)? (
|
|
// No remaining fragments connect! INCOMPLETE PATH!
|
|
// Treat it as complete.
|
|
[path, remainder]
|
|
) : is_closed_path(foundfrag, eps=eps)? (
|
|
// Found fragment is already closed
|
|
[foundfrag, concat([path], remainder)]
|
|
) : let(
|
|
fragend = select(foundfrag,-1),
|
|
hits = [for (i = idx(path,end=-2)) if(approx(path[i],fragend,eps=eps)) i]
|
|
) hits? (
|
|
let(
|
|
// Found fragment intersects with initial path
|
|
hitidx = select(hits,-1),
|
|
newpath = slice(path,0,hitidx+1),
|
|
newfrags = concat(len(newpath)>1? [newpath] : [], remainder),
|
|
outpath = concat(slice(path,hitidx,-2), foundfrag)
|
|
)
|
|
[outpath, newfrags]
|
|
) : let(
|
|
// Path still incomplete. Continue building it.
|
|
newpath = concat(path, slice(foundfrag, 1, -1)),
|
|
newfrags = concat([newpath], remainder)
|
|
)
|
|
assemble_a_path_from_fragments(
|
|
fragments=newfrags,
|
|
rightmost=rightmost,
|
|
eps=eps
|
|
);
|
|
|
|
|
|
// Function: assemble_path_fragments()
|
|
// Usage:
|
|
// assemble_path_fragments(subpaths);
|
|
// Description:
|
|
// Given a list of paths, assembles them together into complete closed polygon paths if it can.
|
|
// Arguments:
|
|
// fragments = List of paths to be assembled into complete polygons.
|
|
// eps = The epsilon error value to determine whether two points coincide. Default: `EPSILON` (1e-9)
|
|
function assemble_path_fragments(fragments, eps=EPSILON, _finished=[]) =
|
|
len(fragments)==0? _finished :
|
|
let(
|
|
minxidx = min_index([
|
|
for (frag=fragments) min(subindex(frag,0))
|
|
]),
|
|
result_l = assemble_a_path_from_fragments(
|
|
fragments=fragments,
|
|
startfrag=minxidx,
|
|
rightmost=false,
|
|
eps=eps
|
|
),
|
|
result_r = assemble_a_path_from_fragments(
|
|
fragments=fragments,
|
|
startfrag=minxidx,
|
|
rightmost=true,
|
|
eps=eps
|
|
),
|
|
l_area = abs(polygon_area(result_l[0])),
|
|
r_area = abs(polygon_area(result_r[0])),
|
|
result = l_area < r_area? result_l : result_r,
|
|
newpath = cleanup_path(result[0]),
|
|
remainder = result[1],
|
|
finished = concat(_finished, [newpath])
|
|
) assemble_path_fragments(
|
|
fragments=remainder,
|
|
eps=eps,
|
|
_finished=finished
|
|
);
|
|
|
|
|
|
|
|
// Section: 2D Modules
|
|
|
|
|
|
// Module: modulated_circle()
|
|
// Usage:
|
|
// modulated_circle(r|d, sines);
|
|
// Description:
|
|
// Creates a 2D polygon circle, modulated by one or more superimposed sine waves.
|
|
// Arguments:
|
|
// r = Radius of the base circle. Default: 40
|
|
// d = Diameter of the base circle.
|
|
// sines = array of [amplitude, frequency] pairs or [amplitude, frequency, phase] triples, where the frequency is the number of times the cycle repeats around the circle.
|
|
// Example(2D):
|
|
// modulated_circle(r=40, sines=[[3, 11], [1, 31]], $fn=6);
|
|
module modulated_circle(r, sines=[[1,1]], d)
|
|
{
|
|
r = get_radius(r=r, d=d, dflt=40);
|
|
assert(is_list(sines)
|
|
&& all([for(s=sines) is_vector(s,2) || is_vector(s,3)]),
|
|
"sines must be given as a list of pairs or triples");
|
|
sines_ = [for(s=sines) [s[0], s[1], len(s)==2 ? 0 : s[2]]];
|
|
freqs = len(sines_)>0? [for (i=sines_) i[1]] : [5];
|
|
points = [
|
|
for (a = [0 : (360/segs(r)/max(freqs)) : 360])
|
|
let(nr=r+sum_of_sines(a,sines_)) [nr*cos(a), nr*sin(a)]
|
|
];
|
|
polygon(points);
|
|
}
|
|
|
|
|
|
// Section: 3D Modules
|
|
|
|
|
|
// Module: extrude_from_to()
|
|
// Description:
|
|
// Extrudes a 2D shape between the points pt1 and pt2. Takes as children a set of 2D shapes to extrude.
|
|
// Arguments:
|
|
// pt1 = starting point of extrusion.
|
|
// pt2 = ending point of extrusion.
|
|
// convexity = max number of times a line could intersect a wall of the 2D shape being extruded.
|
|
// twist = number of degrees to twist the 2D shape over the entire extrusion length.
|
|
// scale = scale multiplier for end of extrusion compared the start.
|
|
// slices = Number of slices along the extrusion to break the extrusion into. Useful for refining `twist` extrusions.
|
|
// Example(FlatSpin):
|
|
// extrude_from_to([0,0,0], [10,20,30], convexity=4, twist=360, scale=3.0, slices=40) {
|
|
// xcopies(3) circle(3, $fn=32);
|
|
// }
|
|
module extrude_from_to(pt1, pt2, convexity, twist, scale, slices) {
|
|
rtp = xyz_to_spherical(pt2-pt1);
|
|
translate(pt1) {
|
|
rotate([0, rtp[2], rtp[1]]) {
|
|
if (rtp[0] > 0) {
|
|
linear_extrude(height=rtp[0], convexity=convexity, center=false, slices=slices, twist=twist, scale=scale) {
|
|
children();
|
|
}
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
|
|
|
|
// Module: spiral_sweep()
|
|
// Description:
|
|
// Takes a closed 2D polygon path, centered on the XY plane, and sweeps/extrudes it along a 3D spiral path
|
|
// of a given radius, height and twist.
|
|
// Arguments:
|
|
// path = Array of points of a polygon path, to be extruded.
|
|
// h = height of the spiral to extrude along.
|
|
// r = Radius of the spiral to extrude along. Default: 50
|
|
// d = Diameter of the spiral to extrude along.
|
|
// twist = number of degrees of rotation to spiral up along height.
|
|
// anchor = Translate so anchor point is at origin (0,0,0). See [anchor](attachments.scad#anchor). Default: `CENTER`
|
|
// spin = Rotate this many degrees around the Z axis after anchor. See [spin](attachments.scad#spin). Default: `0`
|
|
// orient = Vector to rotate top towards, after spin. See [orient](attachments.scad#orient). Default: `UP`
|
|
// center = If given, overrides `anchor`. A true value sets `anchor=CENTER`, false sets `anchor=BOTTOM`.
|
|
// Example:
|
|
// poly = [[-10,0], [-3,-5], [3,-5], [10,0], [0,-30]];
|
|
// spiral_sweep(poly, h=200, r=50, twist=1080, $fn=36);
|
|
module spiral_sweep(poly, h, r, twist=360, center, d, anchor, spin=0, orient=UP) {
|
|
r = get_radius(r=r, d=d, dflt=50);
|
|
poly = path3d(poly);
|
|
pline_count = len(poly);
|
|
steps = ceil(segs(r)*(twist/360));
|
|
anchor = get_anchor(anchor,center,BOT,BOT);
|
|
|
|
poly_points = [
|
|
for (
|
|
p = [0:1:steps]
|
|
) let (
|
|
a = twist * (p/steps),
|
|
dx = r*cos(a),
|
|
dy = r*sin(a),
|
|
dz = h * (p/steps),
|
|
pts = apply_list(
|
|
poly, [
|
|
affine3d_xrot(90),
|
|
affine3d_zrot(a),
|
|
affine3d_translate([dx, dy, dz-h/2])
|
|
]
|
|
)
|
|
) for (pt = pts) pt
|
|
];
|
|
|
|
poly_faces = concat(
|
|
[[for (b = [0:1:pline_count-1]) b]],
|
|
[
|
|
for (
|
|
p = [0:1:steps-1],
|
|
b = [0:1:pline_count-1],
|
|
i = [0:1]
|
|
) let (
|
|
b2 = (b == pline_count-1)? 0 : b+1,
|
|
p0 = p * pline_count + b,
|
|
p1 = p * pline_count + b2,
|
|
p2 = (p+1) * pline_count + b2,
|
|
p3 = (p+1) * pline_count + b,
|
|
pt = (i==0)? [p0, p2, p1] : [p0, p3, p2]
|
|
) pt
|
|
],
|
|
[[for (b = [pline_count-1:-1:0]) b+(steps)*pline_count]]
|
|
);
|
|
|
|
tri_faces = triangulate_faces(poly_points, poly_faces);
|
|
attachable(anchor,spin,orient, r=r, l=h) {
|
|
polyhedron(points=poly_points, faces=tri_faces, convexity=10);
|
|
children();
|
|
}
|
|
}
|
|
|
|
|
|
|
|
// Module: path_extrude()
|
|
// Description:
|
|
// Extrudes 2D children along a 3D path. This may be slow.
|
|
// Arguments:
|
|
// path = array of points for the bezier path to extrude along.
|
|
// convexity = maximum number of walls a ran can pass through.
|
|
// clipsize = increase if artifacts are left. Default: 1000
|
|
// Example(FlatSpin):
|
|
// path = [ [0, 0, 0], [33, 33, 33], [66, 33, 40], [100, 0, 0], [150,0,0] ];
|
|
// path_extrude(path) circle(r=10, $fn=6);
|
|
module path_extrude(path, convexity=10, clipsize=100) {
|
|
function polyquats(path, q=Q_Ident(), v=[0,0,1], i=0) = let(
|
|
v2 = path[i+1] - path[i],
|
|
ang = vector_angle(v,v2),
|
|
axis = ang>0.001? unit(cross(v,v2)) : [0,0,1],
|
|
newq = Q_Mul(Quat(axis, ang), q),
|
|
dist = norm(v2)
|
|
) i < (len(path)-2)?
|
|
concat([[dist, newq, ang]], polyquats(path, newq, v2, i+1)) :
|
|
[[dist, newq, ang]];
|
|
|
|
epsilon = 0.0001; // Make segments ever so slightly too long so they overlap.
|
|
ptcount = len(path);
|
|
pquats = polyquats(path);
|
|
for (i = [0:1:ptcount-2]) {
|
|
pt1 = path[i];
|
|
pt2 = path[i+1];
|
|
dist = pquats[i][0];
|
|
q = pquats[i][1];
|
|
difference() {
|
|
translate(pt1) {
|
|
Qrot(q) {
|
|
down(clipsize/2/2) {
|
|
if ((dist+clipsize/2) > 0) {
|
|
linear_extrude(height=dist+clipsize/2, convexity=convexity) {
|
|
children();
|
|
}
|
|
}
|
|
}
|
|
}
|
|
}
|
|
translate(pt1) {
|
|
hq = (i > 0)? Q_Slerp(q, pquats[i-1][1], 0.5) : q;
|
|
Qrot(hq) down(clipsize/2+epsilon) cube(clipsize, center=true);
|
|
}
|
|
translate(pt2) {
|
|
hq = (i < ptcount-2)? Q_Slerp(q, pquats[i+1][1], 0.5) : q;
|
|
Qrot(hq) up(clipsize/2+epsilon) cube(clipsize, center=true);
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
|
|
// Module: path_spread()
|
|
//
|
|
// Description:
|
|
// Uniformly spreads out copies of children along a path. Copies are located based on path length. If you specify `n` but not spacing then `n` copies will be placed
|
|
// with one at path[0] of `closed` is true, or spanning the entire path from start to end if `closed` is false.
|
|
// If you specify `spacing` but not `n` then copies will spread out starting from one at path[0] for `closed=true` or at the path center for open paths.
|
|
// If you specify `sp` then the copies will start at `sp`.
|
|
//
|
|
// Usage:
|
|
// path_spread(path), [n], [spacing], [sp], [rotate_children], [closed]) ...
|
|
//
|
|
// Arguments:
|
|
// path = the path where children are placed
|
|
// n = number of copies
|
|
// spacing = space between copies
|
|
// sp = if given, copies will start distance sp from the path start and spread beyond that point
|
|
//
|
|
// Side Effects:
|
|
// `$pos` is set to the center of each copy
|
|
// `$idx` is set to the index number of each copy. In the case of closed paths the first copy is at `path[0]` unless you give `sp`.
|
|
// `$dir` is set to the direction vector of the path at the point where the copy is placed.
|
|
// `$normal` is set to the direction of the normal vector to the path direction that is coplanar with the path at this point
|
|
//
|
|
// Example(2D):
|
|
// spiral = [for(theta=[0:360*8]) theta * [cos(theta), sin(theta)]]/100;
|
|
// stroke(spiral,width=.25);
|
|
// color("red") path_spread(spiral, n=100) circle(r=1);
|
|
// Example(2D):
|
|
// circle = regular_ngon(n=64, or=10);
|
|
// stroke(circle,width=1,closed=true);
|
|
// color("green") path_spread(circle, n=7, closed=true) circle(r=1+$idx/3);
|
|
// Example(2D):
|
|
// heptagon = regular_ngon(n=7, or=10);
|
|
// stroke(heptagon, width=1, closed=true);
|
|
// color("purple") path_spread(heptagon, n=9, closed=true) rect([0.5,3],anchor=FRONT);
|
|
// Example(2D): Direction at the corners is the average of the two adjacent edges
|
|
// heptagon = regular_ngon(n=7, or=10);
|
|
// stroke(heptagon, width=1, closed=true);
|
|
// color("purple") path_spread(heptagon, n=7, closed=true) rect([0.5,3],anchor=FRONT);
|
|
// Example(2D): Don't rotate the children
|
|
// heptagon = regular_ngon(n=7, or=10);
|
|
// stroke(heptagon, width=1, closed=true);
|
|
// color("red") path_spread(heptagon, n=9, closed=true, rotate_children=false) rect([0.5,3],anchor=FRONT);
|
|
// Example(2D): Open path, specify `n`
|
|
// sinwav = [for(theta=[0:360]) 5*[theta/180, sin(theta)]];
|
|
// stroke(sinwav,width=.1);
|
|
// color("red") path_spread(sinwav, n=5) rect([.2,1.5],anchor=FRONT);
|
|
// Example(2D): Open path, specify `n` and `spacing`
|
|
// sinwav = [for(theta=[0:360]) 5*[theta/180, sin(theta)]];
|
|
// stroke(sinwav,width=.1);
|
|
// color("red") path_spread(sinwav, n=5, spacing=1) rect([.2,1.5],anchor=FRONT);
|
|
// Example(2D): Closed path, specify `n` and `spacing`, copies centered around circle[0]
|
|
// circle = regular_ngon(n=64,or=10);
|
|
// stroke(circle,width=.1,closed=true);
|
|
// color("red") path_spread(circle, n=10, spacing=1, closed=true) rect([.2,1.5],anchor=FRONT);
|
|
// Example(2D): Open path, specify `spacing`
|
|
// sinwav = [for(theta=[0:360]) 5*[theta/180, sin(theta)]];
|
|
// stroke(sinwav,width=.1);
|
|
// color("red") path_spread(sinwav, spacing=5) rect([.2,1.5],anchor=FRONT);
|
|
// Example(2D): Open path, specify `sp`
|
|
// sinwav = [for(theta=[0:360]) 5*[theta/180, sin(theta)]];
|
|
// stroke(sinwav,width=.1);
|
|
// color("red") path_spread(sinwav, n=5, sp=18) rect([.2,1.5],anchor=FRONT);
|
|
// Example(2D):
|
|
// wedge = arc(angle=[0,100], r=10, $fn=64);
|
|
// difference(){
|
|
// polygon(concat([[0,0]],wedge));
|
|
// path_spread(wedge,n=5,spacing=3) fwd(.1) rect([1,4],anchor=FRONT);
|
|
// }
|
|
// Example(Spin): 3d example, with children rotated into the plane of the path
|
|
// tilted_circle = lift_plane(regular_ngon(n=64, or=12), [0,0,0], [5,0,5], [0,2,3]);
|
|
// path_sweep(regular_ngon(n=16,or=.1),tilted_circle);
|
|
// path_spread(tilted_circle, n=15,closed=true) {
|
|
// color("blue") cyl(h=3,r=.2, anchor=BOTTOM); // z-aligned cylinder
|
|
// color("red") xcyl(h=10,r=.2, anchor=FRONT+LEFT); // x-aligned cylinder
|
|
// }
|
|
// Example(Spin): 3d example, with rotate_children set to false
|
|
// tilted_circle = lift_plane(regular_ngon(n=64, or=12), [0,0,0], [5,0,5], [0,2,3]);
|
|
// path_sweep(regular_ngon(n=16,or=.1),tilted_circle);
|
|
// path_spread(tilted_circle, n=25,rotate_children=false,closed=true) {
|
|
// color("blue") cyl(h=3,r=.2, anchor=BOTTOM); // z-aligned cylinder
|
|
// color("red") xcyl(h=10,r=.2, anchor=FRONT+LEFT); // x-aligned cylinder
|
|
// }
|
|
module path_spread(path, n, spacing, sp=undef, rotate_children=true, closed=false)
|
|
{
|
|
length = path_length(path,closed);
|
|
distances = is_def(sp)? (
|
|
is_def(n) && is_def(spacing)? list_range(s=sp, step=spacing, n=n) :
|
|
is_def(n)? list_range(s=sp, e=length, n=n) :
|
|
list_range(s=sp, step=spacing, e=length)
|
|
) : is_def(n) && is_undef(spacing)? (
|
|
closed?
|
|
let(range=list_range(s=0,e=length, n=n+1)) slice(range,0,-2) :
|
|
list_range(s=0, e=length, n=n)
|
|
) : (
|
|
let(
|
|
n = is_def(n)? n : floor(length/spacing)+(closed?0:1),
|
|
ptlist = list_range(s=0,step=spacing,n=n),
|
|
listcenter = mean(ptlist)
|
|
) closed?
|
|
sort([for(entry=ptlist) posmod(entry-listcenter,length)]) :
|
|
[for(entry=ptlist) entry + length/2-listcenter ]
|
|
);
|
|
distOK = min(distances)>=0 && max(distances)<=length;
|
|
assert(distOK,"Cannot fit all of the copies");
|
|
cutlist = path_cut(path, distances, closed, direction=true);
|
|
planar = len(path[0])==2;
|
|
if (true) for(i=[0:1:len(cutlist)-1]) {
|
|
$pos = cutlist[i][0];
|
|
$idx = i;
|
|
$dir = rotate_children ? (planar?[1,0]:[1,0,0]) : cutlist[i][2];
|
|
$normal = rotate_children? (planar?[0,1]:[0,0,1]) : cutlist[i][3];
|
|
translate($pos) {
|
|
if (rotate_children) {
|
|
if(planar) {
|
|
rot(from=[0,1],to=cutlist[i][3]) children();
|
|
} else {
|
|
multmatrix(affine2d_to_3d(transpose([cutlist[i][2],cross(cutlist[i][3],cutlist[i][2]), cutlist[i][3]])))
|
|
children();
|
|
}
|
|
} else {
|
|
children();
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
|
|
// Function: path_cut()
|
|
//
|
|
// Usage:
|
|
// path_cut(path, dists, [closed], [direction])
|
|
//
|
|
// Description:
|
|
// Cuts a path at a list of distances from the first point in the path. Returns a list of the cut
|
|
// points and indices of the next point in the path after that point. So for example, a return
|
|
// value entry of [[2,3], 5] means that the cut point was [2,3] and the next point on the path after
|
|
// this point is path[5]. If the path is too short then path_cut returns undef. If you set
|
|
// `direction` to true then `path_cut` will also return the tangent vector to the path and a normal
|
|
// vector to the path. It tries to find a normal vector that is coplanar to the path near the cut
|
|
// point. If this fails it will return a normal vector parallel to the xy plane. The output with
|
|
// direction vectors will be `[point, next_index, tangent, normal]`.
|
|
//
|
|
// Arguments:
|
|
// path = path to cut
|
|
// dists = distances where the path should be cut (a list) or a scalar single distance
|
|
// closed = set to true if the curve is closed. Default: false
|
|
// direction = set to true to return direction vectors. Default: false
|
|
//
|
|
// Example(NORENDER):
|
|
// square=[[0,0],[1,0],[1,1],[0,1]];
|
|
// path_cut(square, [.5,1.5,2.5]); // Returns [[[0.5, 0], 1], [[1, 0.5], 2], [[0.5, 1], 3]]
|
|
// path_cut(square, [0,1,2,3]); // Returns [[[0, 0], 1], [[1, 0], 2], [[1, 1], 3], [[0, 1], 4]]
|
|
// path_cut(square, [0,0.8,1.6,2.4,3.2], closed=true); // Returns [[[0, 0], 1], [[0.8, 0], 1], [[1, 0.6], 2], [[0.6, 1], 3], [[0, 0.8], 4]]
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// path_cut(square, [0,0.8,1.6,2.4,3.2]); // Returns [[[0, 0], 1], [[0.8, 0], 1], [[1, 0.6], 2], [[0.6, 1], 3], undef]
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function path_cut(path, dists, closed=false, direction=false) =
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let(long_enough = len(path) >= (closed ? 3 : 2))
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assert(long_enough,len(path)<2 ? "Two points needed to define a path" : "Closed path must include three points")
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!is_list(dists)? path_cut(path, [dists],closed, direction)[0] :
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let(cuts = _path_cut(path,dists,closed))
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!direction ? cuts : let(
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dir = _path_cuts_dir(path, cuts, closed),
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normals = _path_cuts_normals(path, cuts, dir, closed)
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) zip(cuts, array_group(dir,1), array_group(normals,1));
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// Main recursive path cut function
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function _path_cut(path, dists, closed=false, pind=0, dtotal=0, dind=0, result=[]) =
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dind == len(dists) ? result :
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let(
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lastpt = len(result)>0? select(result,-1)[0] : [],
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dpartial = len(result)==0? 0 : norm(lastpt-path[pind]),
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nextpoint = dpartial > dists[dind]-dtotal?
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[lerp(lastpt,path[pind], (dists[dind]-dtotal)/dpartial),pind] :
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_path_cut_single(path, dists[dind]-dtotal-dpartial, closed, pind)
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) is_undef(nextpoint)?
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concat(result, repeat(undef,len(dists)-dind)) :
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_path_cut(path, dists, closed, nextpoint[1], dists[dind],dind+1, concat(result, [nextpoint]));
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// Search for a single cut point in the path
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function _path_cut_single(path, dist, closed=false, ind=0, eps=1e-7) =
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ind>=len(path)? undef :
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ind==len(path)-1 && !closed? (dist<eps? [path[ind],ind+1] : undef) :
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let(d = norm(path[ind]-select(path,ind+1))) d > dist ?
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[lerp(path[ind],select(path,ind+1),dist/d), ind+1] :
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_path_cut_single(path, dist-d,closed, ind+1, eps);
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// Find normal directions to the path, coplanar to local part of the path
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// Or return a vector parallel to the x-y plane if the above fails
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function _path_cuts_normals(path, cuts, dirs, closed=false) =
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[for(i=[0:len(cuts)-1])
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len(path[0])==2? [-dirs[i].y, dirs[i].x] : (
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let(
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plane = len(path)<3 ? undef :
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let(start = max(min(cuts[i][1],len(path)-1),2)) _path_plane(path, start, start-2)
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)
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plane==undef?
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unit([-dirs[i].y, dirs[i].x,0]) :
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unit(cross(dirs[i],cross(plane[0],plane[1])))
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)
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];
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// Scan from the specified point (ind) to find a noncoplanar triple to use
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// to define the plane of the path.
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function _path_plane(path, ind, i,closed) =
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i<(closed?-1:0) ? undef :
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!collinear(path[ind],path[ind-1], select(path,i))?
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[select(path,i)-path[ind-1],path[ind]-path[ind-1]] :
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_path_plane(path, ind, i-1);
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// Find the direction of the path at the cut points
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function _path_cuts_dir(path, cuts, closed=false, eps=1e-2) =
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[for(ind=[0:len(cuts)-1])
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let(
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zeros = path[0]*0,
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nextind = cuts[ind][1],
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nextpath = unit(select(path, nextind+1)-select(path, nextind),zeros),
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thispath = unit(select(path, nextind) - path[nextind-1],zeros),
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lastpath = unit(path[nextind-1] - select(path, nextind-2),zeros),
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nextdir =
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nextind==len(path) && !closed? lastpath :
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(nextind<=len(path)-2 || closed) && approx(cuts[ind][0], path[nextind],eps)?
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unit(nextpath+thispath) :
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(nextind>1 || closed) && approx(cuts[ind][0],path[nextind-1],eps)?
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unit(thispath+lastpath) :
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thispath
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) nextdir
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];
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// Input `data` is a list that sums to an integer.
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// Returns rounded version of input data so that every
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// entry is rounded to an integer and the sum is the same as
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// that of the input. Works by rounding an entry in the list
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// and passing the rounding error forward to the next entry.
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// This will generally distribute the error in a uniform manner.
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function _sum_preserving_round(data, index=0) =
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index == len(data)-1 ? list_set(data, len(data)-1, round(data[len(data)-1])) :
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let(
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newval = round(data[index]),
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error = newval - data[index]
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) _sum_preserving_round(
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list_set(data, [index,index+1], [newval, data[index+1]-error]),
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index+1
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);
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// Function: subdivide_path()
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// Usage:
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// newpath = subdivide_path(path, [N|refine], method);
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// Description:
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// Takes a path as input (closed or open) and subdivides the path to produce a more
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// finely sampled path. The new points can be distributed proportional to length
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// (`method="length"`) or they can be divided up evenly among all the path segments
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// (`method="segment"`). If the extra points don't fit evenly on the path then the
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// algorithm attempts to distribute them uniformly. The `exact` option requires that
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// the final length is exactly as requested. If you set it to `false` then the
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// algorithm will favor uniformity and the output path may have a different number of
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// points due to rounding error.
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// .
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// With the `"segment"` method you can also specify a vector of lengths. This vector,
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// `N` specfies the desired point count on each segment: with vector input, `subdivide_path`
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// attempts to place `N[i]-1` points on segment `i`. The reason for the -1 is to avoid
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// double counting the endpoints, which are shared by pairs of segments, so that for
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// a closed polygon the total number of points will be sum(N). Note that with an open
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// path there is an extra point at the end, so the number of points will be sum(N)+1.
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// Arguments:
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// path = path to subdivide
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// N = scalar total number of points desired or with `method="segment"` can be a vector requesting `N[i]-1` points on segment i.
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// refine = number of points to add each segment.
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// closed = set to false if the path is open. Default: True
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// exact = if true return exactly the requested number of points, possibly sacrificing uniformity. If false, return uniform point sample that may not match the number of points requested. Default: True
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// method = One of `"length"` or `"segment"`. If `"length"`, adds vertices evenly along the total path length. If `"segment"`, adds points evenly among the segments. Default: `"length"`
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// Example(2D):
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// mypath = subdivide_path(square([2,2],center=true), 12);
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// move_copies(mypath)circle(r=.1,$fn=32);
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// Example(2D):
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// mypath = subdivide_path(square([8,2],center=true), 12);
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// move_copies(mypath)circle(r=.2,$fn=32);
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// Example(2D):
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// mypath = subdivide_path(square([8,2],center=true), 12, method="segment");
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// move_copies(mypath)circle(r=.2,$fn=32);
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// Example(2D):
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// mypath = subdivide_path(square([2,2],center=true), 17, closed=false);
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// move_copies(mypath)circle(r=.1,$fn=32);
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// Example(2D): Specifying different numbers of points on each segment
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// mypath = subdivide_path(hexagon(side=2), [2,3,4,5,6,7], method="segment");
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// move_copies(mypath)circle(r=.1,$fn=32);
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// Example(2D): Requested point total is 14 but 15 points output due to extra end point
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// mypath = subdivide_path(pentagon(side=2), [3,4,3,4], method="segment", closed=false);
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// move_copies(mypath)circle(r=.1,$fn=32);
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// Example(2D): Since 17 is not divisible by 5, a completely uniform distribution is not possible.
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// mypath = subdivide_path(pentagon(side=2), 17);
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// move_copies(mypath)circle(r=.1,$fn=32);
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// Example(2D): With `exact=false` a uniform distribution, but only 15 points
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// mypath = subdivide_path(pentagon(side=2), 17, exact=false);
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// move_copies(mypath)circle(r=.1,$fn=32);
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// Example(2D): With `exact=false` you can also get extra points, here 20 instead of requested 18
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// mypath = subdivide_path(pentagon(side=2), 18, exact=false);
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// move_copies(mypath)circle(r=.1,$fn=32);
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// Example(FlatSpin): Three-dimensional paths also work
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// mypath = subdivide_path([[0,0,0],[2,0,1],[2,3,2]], 12);
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// move_copies(mypath)sphere(r=.1,$fn=32);
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function subdivide_path(path, N, refine, closed=true, exact=true, method="length") =
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assert(is_path(path))
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assert(method=="length" || method=="segment")
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assert(num_defined([N,refine]),"Must give exactly one of N and refine")
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let(
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N = !is_undef(N)? N :
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!is_undef(refine)? len(path) * refine :
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undef
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)
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assert((is_num(N) && N>0) || is_vector(N),"Parameter N to subdivide_path must be postive number or vector")
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let(
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count = len(path) - (closed?0:1),
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add_guess = method=="segment"? (
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is_list(N)? (
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assert(len(N)==count,"Vector parameter N to subdivide_path has the wrong length")
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add_scalar(N,-1)
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) : repeat((N-len(path)) / count, count)
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) : // method=="length"
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assert(is_num(N),"Parameter N to subdivide path must be a number when method=\"length\"")
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let(
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path_lens = concat(
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[ for (i = [0:1:len(path)-2]) norm(path[i+1]-path[i]) ],
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closed? [norm(path[len(path)-1]-path[0])] : []
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),
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add_density = (N - len(path)) / sum(path_lens)
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)
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path_lens * add_density,
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add = exact? _sum_preserving_round(add_guess) :
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[for (val=add_guess) round(val)]
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) concat(
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[
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for (i=[0:1:count]) each [
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for(j=[0:1:add[i]])
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lerp(path[i],select(path,i+1), j/(add[i]+1))
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]
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],
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closed? [] : [select(path,-1)]
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);
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// Function: path_length_fractions()
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// Usage:
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// fracs = path_length_fractions(path, <closed>);
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// Description:
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// Returns the distance fraction of each point in the path along the path, so the first
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// point is zero and the final point is 1. If the path is closed the length of the output
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// will have one extra point because of the final connecting segment that connects the last
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// point of the path to the first point.
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function path_length_fractions(path, closed=false) =
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assert(is_path(path))
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assert(is_bool(closed))
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let(
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lengths = [
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0,
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for (i=[0:1:len(path)-(closed?1:2)])
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norm(select(path,i+1)-path[i])
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],
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partial_len = cumsum(lengths),
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total_len = select(partial_len,-1)
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) partial_len / total_len;
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// Function: resample_path()
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// Usage:
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// newpath = resample_path(path, N|spacing, <closed>);
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|
// Description:
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// Compute a uniform resampling of the input path. If you specify `N` then the output path will have N
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// points spaced uniformly (by linear interpolation along the input path segments). The only points of the
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// input path that are guaranteed to appear in the output path are the starting and ending points.
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// If you specify `spacing` then the length you give will be rounded to the nearest spacing that gives
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// a uniform sampling of the path and the resulting uniformly sampled path is returned.
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// Note that because this function operates on a discrete input path the quality of the output depends on
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// the sampling of the input. If you want very accurate output, use a lot of points for the input.
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// Arguments:
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// path = path to resample
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// N = Number of points in output
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// spacing = Approximate spacing desired
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// closed = set to true if path is closed. Default: false
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function resample_path(path, N, spacing, closed=false) =
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assert(is_path(path))
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assert(num_defined([N,spacing])==1,"Must define exactly one of N and spacing")
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assert(is_bool(closed))
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let(
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length = path_length(path,closed),
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N = is_def(N) ? N : round(length/spacing) + (closed?0:1),
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spacing = length/(closed?N:N-1), // Note: worried about round-off error, so don't include
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distlist = list_range(closed?N:N-1,step=spacing), // last point when closed=false
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cuts = path_cut(path, distlist, closed=closed)
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)
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concat(subindex(cuts,0),closed?[]:[select(path,-1)]); // Then add last point here
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// vim: expandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap
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