////////////////////////////////////////////////////////////////////// // LibFile: paths.scad // Polylines, polygons and paths. // To use, add the following lines to the beginning of your file: // ``` // include // ``` ////////////////////////////////////////////////////////////////////// include // Section: Functions // Function: is_path() // Usage: // is_path(list, [dim], [fast]) // Description: // Returns true if `list` is a path. A path is a list of two or more numeric vectors (AKA points). // All vectors must of the same size, and may only contain numbers that are not inf or nan. // By default the vectors in a path must be 2d or 3d. Set the `dim` parameter to specify a list // of allowed dimensions, or set it to `undef` to allow any dimension. // Examples: // is_path([[3,4],[5,6]]); // Returns true // is_path([[3,4]]); // Returns false // is_path([[3,4],[4,5]],2); // Returns true // is_path([[3,4,3],[5,4,5]],2); // Returns false // is_path([[3,4,3],[5,4,5]],2); // Returns false // is_path([[3,4,5],undef,[4,5,6]]); // Returns false // is_path([[3,5],[undef,undef],[4,5]]); // Returns false // is_path([[3,4],[5,6],[5,3]]); // Returns true // is_path([3,4,5,6,7,8]); // Returns false // is_path([[3,4],[5,6]], dim=[2,3]);// Returns true // is_path([[3,4],[5,6]], dim=[1,3]);// Returns false // is_path([[3,4],"hello"], fast=true); // Returns true // is_path([[3,4],[3,4,5]]); // Returns false // is_path([[1,2,3,4],[2,3,4,5]]); // Returns false // is_path([[1,2,3,4],[2,3,4,5]],undef);// Returns true // Arguments: // list = list to check // dim = list of allowed dimensions of the vectors in the path. Default: [2,3] // fast = set to true for fast check that only looks at first entry. Default: false function is_path(list, dim=[2,3], fast=false) = fast ? is_list(list) && is_vector(list[0]) : is_matrix(list) && len(list)>1 && len(list[0])>0 && (is_undef(dim) || in_list(len(list[0]), force_list(dim))); // Function: is_closed_path() // Usage: // is_closed_path(path, [eps]); // Description: // Returns true if the first and last points in the given path are coincident. function is_closed_path(path, eps=EPSILON) = approx(path[0], path[len(path)-1], eps=eps); // Function: close_path() // Usage: // close_path(path); // Description: // If a path's last point does not coincide with its first point, closes the path so it does. function close_path(path, eps=EPSILON) = is_closed_path(path,eps=eps)? path : concat(path,[path[0]]); // Function: cleanup_path() // Usage: // cleanup_path(path); // Description: // If a path's last point coincides with its first point, deletes the last point in the path. function cleanup_path(path, eps=EPSILON) = is_closed_path(path,eps=eps)? select(path,0,-2) : path; // Function: path_subselect() // Usage: // path_subselect(path,s1,u1,s2,u2,[closed]): // Description: // Returns a portion of a path, from between the `u1` part of segment `s1`, to the `u2` part of // segment `s2`. Both `u1` and `u2` are values between 0.0 and 1.0, inclusive, where 0 is the start // of the segment, and 1 is the end. Both `s1` and `s2` are integers, where 0 is the first segment. // Arguments: // path = The path to get a section of. // s1 = The number of the starting segment. // u1 = The proportion along the starting segment, between 0.0 and 1.0, inclusive. // s2 = The number of the ending segment. // u2 = The proportion along the ending segment, between 0.0 and 1.0, inclusive. // closed = If true, treat path as a closed polygon. function path_subselect(path, s1, u1, s2, u2, closed=false) = let( lp = len(path), l = lp-(closed?0:1), u1 = s1<0? 0 : s1>l? 1 : u1, u2 = s2<0? 0 : s2>l? 1 : u2, s1 = constrain(s1,0,l), s2 = constrain(s2,0,l), pathout = concat( (s10)? [lerp(path[s2],path[(s2+1)%lp],u2)] : [] ) ) pathout; // Function: simplify_path() // Description: // Takes a path and removes unnecessary subsequent collinear points. // Usage: // simplify_path(path, [eps]) // Arguments: // path = A list of path points of any dimension. // eps = Largest positional variance allowed. Default: `EPSILON` (1-e9) function simplify_path(path, eps=EPSILON) = assert( is_path(path), "Invalid path." ) assert( is_undef(eps) || (is_finite(eps) && (eps>=0) ), "Invalid tolerance." ) len(path)<=2 ? path : let( indices = [ 0, for (i=[1:1:len(path)-2]) if (!collinear(path[i-1],path[i],path[i+1], eps=eps)) i, len(path)-1 ] ) [for (i = indices) path[i] ]; // Function: simplify_path_indexed() // Description: // Takes a list of points, and a list of indices into `points`, // and removes from the list all indices of subsequent indexed points that are unecessarily collinear. // Returns the list of the remained indices. // Usage: // simplify_path_indexed(points,indices, eps) // Arguments: // points = A list of points. // indices = A list of indices into `points` that forms a path. // eps = Largest angle variance allowed. Default: EPSILON (1-e9) degrees. function simplify_path_indexed(points, indices, eps=EPSILON) = len(indices)<=2? indices : let( indices = concat( indices[0], [for (i=[1:1:len(indices)-2]) let( i1 = indices[i-1], i2 = indices[i], i3 = indices[i+1] ) if (!collinear(points[i1],points[i2],points[i3], eps=eps)) indices[i]], indices[len(indices)-1] ) ) indices; // Function: path_length() // Usage: // path_length(path,[closed]) // Description: // Returns the length of the path. // Arguments: // path = The list of points of the path to measure. // closed = true if the path is closed. Default: false // Example: // path = [[0,0], [5,35], [60,-25], [80,0]]; // echo(path_length(path)); function path_length(path,closed=false) = len(path)<2? 0 : sum([for (i = [0:1:len(path)-2]) norm(path[i+1]-path[i])])+(closed?norm(path[len(path)-1]-path[0]):0); // Function: path_segment_lengths() // Usage: // path_segment_lengths(path,[closed]) // Description: // Returns list of the length of each segment in a path // Arguments: // path = path to measure // closed = true if the path is closed. Default: false function path_segment_lengths(path, closed=false) = [ for (i=[0:1:len(path)-2]) norm(path[i+1]-path[i]), if (closed) norm(path[0]-path[len(path)-1]) ]; // Function: path_pos_from_start() // Usage: // pos = path_pos_from_start(path,length,[closed]); // Description: // Finds the segment and relative position along that segment that is `length` distance from the // front of the given `path`. Returned as [SEGNUM, U] where SEGNUM is the segment number, and U is // the relative distance along that segment, a number from 0 to 1. If the path is shorter than the // asked for length, this returns `undef`. // Arguments: // path = The path to find the position on. // length = The length from the start of the path to find the segment and position of. // Example(2D): // path = circle(d=50,$fn=18); // pos = path_pos_from_start(path,20,closed=false); // stroke(path,width=1,endcaps=false); // pt = lerp(path[pos[0]], path[(pos[0]+1)%len(path)], pos[1]); // color("red") translate(pt) circle(d=2,$fn=12); function path_pos_from_start(path,length,closed=false,_d=0,_i=0) = let (lp = len(path)) _i >= lp - (closed?0:1)? undef : let (l = norm(path[(_i+1)%lp]-path[_i])) _d+l <= length? path_pos_from_start(path,length,closed,_d+l,_i+1) : [_i, (length-_d)/l]; // Function: path_pos_from_end() // Usage: // pos = path_pos_from_end(path,length,[closed]); // Description: // Finds the segment and relative position along that segment that is `length` distance from the // end of the given `path`. Returned as [SEGNUM, U] where SEGNUM is the segment number, and U is // the relative distance along that segment, a number from 0 to 1. If the path is shorter than the // asked for length, this returns `undef`. // Arguments: // path = The path to find the position on. // length = The length from the end of the path to find the segment and position of. // Example(2D): // path = circle(d=50,$fn=18); // pos = path_pos_from_end(path,20,closed=false); // stroke(path,width=1,endcaps=false); // pt = lerp(path[pos[0]], path[(pos[0]+1)%len(path)], pos[1]); // color("red") translate(pt) circle(d=2,$fn=12); function path_pos_from_end(path,length,closed=false,_d=0,_i=undef) = let ( lp = len(path), _i = _i!=undef? _i : lp - (closed?1:2) ) _i < 0? undef : let (l = norm(path[(_i+1)%lp]-path[_i])) _d+l <= length? path_pos_from_end(path,length,closed,_d+l,_i-1) : [_i, 1-(length-_d)/l]; // Function: path_trim_start() // Usage: // path_trim_start(path,trim); // Description: // Returns the `path`, with the start shortened by the length `trim`. // Arguments: // path = The path to trim. // trim = The length to trim from the start. // Example(2D): // path = circle(d=50,$fn=18); // path2 = path_trim_start(path,5); // path3 = path_trim_start(path,20); // color("blue") stroke(path3,width=5,endcaps=false); // color("cyan") stroke(path2,width=3,endcaps=false); // color("red") stroke(path,width=1,endcaps=false); function path_trim_start(path,trim,_d=0,_i=0) = _i >= len(path)-1? [] : let (l = norm(path[_i+1]-path[_i])) _d+l <= trim? path_trim_start(path,trim,_d+l,_i+1) : let (v = unit(path[_i+1]-path[_i])) concat( [path[_i+1]-v*(l-(trim-_d))], [for (i=[_i+1:1:len(path)-1]) path[i]] ); // Function: path_trim_end() // Usage: // path_trim_end(path,trim); // Description: // Returns the `path`, with the end shortened by the length `trim`. // Arguments: // path = The path to trim. // trim = The length to trim from the end. // Example(2D): // path = circle(d=50,$fn=18); // path2 = path_trim_end(path,5); // path3 = path_trim_end(path,20); // color("blue") stroke(path3,width=5,endcaps=false); // color("cyan") stroke(path2,width=3,endcaps=false); // color("red") stroke(path,width=1,endcaps=false); function path_trim_end(path,trim,_d=0,_i=undef) = let (_i = _i!=undef? _i : len(path)-1) _i <= 0? [] : let (l = norm(path[_i]-path[_i-1])) _d+l <= trim? path_trim_end(path,trim,_d+l,_i-1) : let (v = unit(path[_i]-path[_i-1])) concat( [for (i=[0:1:_i-1]) path[i]], [path[_i-1]+v*(l-(trim-_d))] ); // Function: path_closest_point() // Usage: // path_closest_point(path, pt); // Description: // Finds the closest path segment, and point on that segment to the given point. // Returns `[SEGNUM, POINT]` // Arguments: // path = The path to find the closest point on. // pt = the point to find the closest point to. // Example(2D): // path = circle(d=100,$fn=6); // pt = [20,10]; // closest = path_closest_point(path, pt); // stroke(path, closed=true); // color("blue") translate(pt) circle(d=3, $fn=12); // color("red") translate(closest[1]) circle(d=3, $fn=12); function path_closest_point(path, pt) = let( pts = [for (seg=idx(path)) segment_closest_point(select(path,seg,seg+1),pt)], dists = [for (p=pts) norm(p-pt)], min_seg = min_index(dists) ) [min_seg, pts[min_seg]]; // Function: path_tangents() // Usage: path_tangents(path, [closed], [uniform]) // Description: // Compute the tangent vector to the input path. The derivative approximation is described in deriv(). // The returns vectors will be normalized to length 1. If any derivatives are zero then // the function fails with an error. If you set `uniform` to false then the sampling is // assumed to be non-uniform and the derivative is computed with adjustments to produce corrected // values. // Arguments: // path = path to find the tagent vectors for // closed = set to true of the path is closed. Default: false // uniform = set to false to correct for non-uniform sampling. Default: true // Example: A shape with non-uniform sampling gives distorted derivatives that may be undesirable // rect = square([10,3]); // tangents = path_tangents(rect,closed=true); // stroke(rect,closed=true, width=0.1); // color("purple") // for(i=[0:len(tangents)-1]) // stroke([rect[i]-tangents[i], rect[i]+tangents[i]],width=.1, endcap2="arrow2"); // Example: A shape with non-uniform sampling gives distorted derivatives that may be undesirable // rect = square([10,3]); // tangents = path_tangents(rect,closed=true,uniform=false); // stroke(rect,closed=true, width=0.1); // color("purple") // for(i=[0:len(tangents)-1]) // stroke([rect[i]-tangents[i], rect[i]+tangents[i]],width=.1, endcap2="arrow2"); function path_tangents(path, closed=false, uniform=true) = assert(is_path(path)) !uniform ? [for(t=deriv(path,closed=closed, h=path_segment_lengths(path,closed))) unit(t)] : [for(t=deriv(path,closed=closed)) unit(t)]; // Function: path_normals() // Usage: path_normals(path, [tangents], [closed]) // Description: // Compute the normal vector to the input path. This vector is perpendicular to the // path tangent and lies in the plane of the curve. When there are collinear points, // the curve does not define a unique plane and the normal is not uniquely defined. function path_normals(path, tangents, closed=false) = assert(is_path(path)) assert(is_bool(closed)) let( tangents = default(tangents, path_tangents(path,closed)) ) assert(is_path(tangents)) [ for(i=idx(path)) let( pts = i==0? (closed? select(path,-1,1) : select(path,0,2)) : i==len(path)-1? (closed? select(path,i-1,i+1) : select(path,i-2,i)) : select(path,i-1,i+1) ) unit(cross( cross(pts[1]-pts[0], pts[2]-pts[0]), tangents[i] )) ]; // Function: path_curvature() // Usage: path_curvature(path, [closed]) // Description: // Numerically estimate the curvature of the path (in any dimension). function path_curvature(path, closed=false) = let( d1 = deriv(path, closed=closed), d2 = deriv2(path, closed=closed) ) [ for(i=idx(path)) sqrt( sqr(norm(d1[i])*norm(d2[i])) - sqr(d1[i]*d2[i]) ) / pow(norm(d1[i]),3) ]; // Function: path_torsion() // Usage: path_torsion(path, [closed]) // Description: // Numerically estimate the torsion of a 3d path. function path_torsion(path, closed=false) = let( d1 = deriv(path,closed=closed), d2 = deriv2(path,closed=closed), d3 = deriv3(path,closed=closed) ) [ for (i=idx(path)) let( crossterm = cross(d1[i],d2[i]) ) crossterm * d3[i] / sqr(norm(crossterm)) ]; // 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_polyline(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: points_along_path3d() // Usage: // points_along_path3d(polyline, path); // Description: // Calculates the vertices needed to create a `polyhedron()` of the // extrusion of `polyline` along `path`. The closed 2D path shold be // centered on the XY plane. The 2D path is extruded perpendicularly // along the 3D path. Produces a list of 3D vertices. Vertex count // is `len(polyline)*len(path)`. Gives all the reoriented vertices // for `polyline` at the first point in `path`, then for the second, // and so on. // Arguments: // polyline = A closed list of 2D path points. // path = A list of 3D path points. function points_along_path3d( polyline, // The 2D polyline to drag along the 3D path. path, // The 3D polyline path to follow. q=Q_Ident(), // Used in recursion n=0 // Used in recursion ) = let( end = len(path)-1, v1 = (n == 0)? [0, 0, 1] : unit(path[n]-path[n-1]), v2 = (n == end)? unit(path[n]-path[n-1]) : unit(path[n+1]-path[n]), crs = cross(v1, v2), axis = norm(crs) <= 0.001? [0, 0, 1] : crs, ang = vector_angle(v1, v2), hang = ang * (n==0? 1.0 : 0.5), hrot = Quat(axis, hang), arot = Quat(axis, ang), roth = Q_Mul(hrot, q), rotm = Q_Mul(arot, q) ) concat( [for (i = [0:1:len(polyline)-1]) Qrot(roth,p=point3d(polyline[i])) + path[n]], (n == end)? [] : points_along_path3d(polyline, path, rotm, n+1) ); // 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 && 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: // polylines = split_path_at_self_crossings(path, [closed], [eps]); // Description: // Splits a path into polyline sections wherever the 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] ]; // polylines = split_path_at_self_crossings(path); // rainbow(polylines) 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 incomplete paths, assembles them together into one complete closed path, and // remainder fragments. Returns [PATH, FRAGMENTS] where FRAGMENTS is the list of remaining // polyline path fragments. // Arguments: // fragments = List of polylines 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 incomplete paths, assembles them together into complete closed paths if it can. // Arguments: // fragments = List of polylines 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, 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=[10], d) { r = get_radius(r=r, d=d, dflt=40); 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=undef, twist=undef, scale=undef, slices=undef) { rtp = xyz_to_spherical(pt2-pt1); translate(pt1) { rotate([0, rtp[2], rtp[1]]) { linear_extrude(height=rtp[0], convexity=convexity, center=false, slices=slices, twist=twist, scale=scale) { children(); } } } } // Module: spiral_sweep() // Description: // Takes a closed 2D polyline path, centered on the XY plane, and // extrudes it along a 3D spiral path of a given radius, height and twist. // Arguments: // polyline = Array of points of a polyline 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(polyline, h, r, twist=360, center, d, anchor, spin=0, orient=UP) { r = get_radius(r=r, d=d, dflt=50); polyline = path3d(polyline); pline_count = len(polyline); 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( polyline, [ 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 polyline 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, $fn=6); module path_extrude(path, convexity=10, clipsize=100) { function polyquats(path, q=Q_Ident(), v=[0,0,1], i=0) = let( v2 = path[i+1] - path[i], ang = vector_angle(v,v2), axis = ang>0.001? unit(cross(v,v2)) : [0,0,1], newq = Q_Mul(Quat(axis, ang), q), dist = norm(v2) ) i < (len(path)-2)? concat([[dist, newq, ang]], polyquats(path, newq, v2, i+1)) : [[dist, newq, ang]]; epsilon = 0.0001; // Make segments ever so slightly too long so they overlap. ptcount = len(path); pquats = polyquats(path); for (i = [0:1:ptcount-2]) { pt1 = path[i]; pt2 = path[i+1]; dist = pquats[i][0]; q = pquats[i][1]; difference() { translate(pt1) { Qrot(q) { down(clipsize/2/2) { linear_extrude(height=dist+clipsize/2, convexity=convexity) { children(); } } } } translate(pt1) { hq = (i > 0)? Q_Slerp(q, pquats[i-1][1], 0.5) : q; Qrot(hq) down(clipsize/2+epsilon) cube(clipsize, center=true); } translate(pt2) { hq = (i < ptcount-2)? Q_Slerp(q, pquats[i+1][1], 0.5) : q; Qrot(hq) up(clipsize/2+epsilon) cube(clipsize, center=true); } } } } // Module: path_spread() // // Description: // Uniformly spreads out copies of children along a path. Copies are located based on path length. If you specify `n` but not spacing then `n` copies will be placed // with one at path[0] of `closed` is true, or spanning the entire path from start to end if `closed` is false. // If you specify `spacing` but not `n` then copies will spread out starting from one at path[0] for `closed=true` or at the path center for open paths. // If you specify `sp` then the copies will start at `sp`. // // Usage: // path_spread(path), [n], [spacing], [sp], [rotate_children], [closed]) ... // // Arguments: // path = the path where children are placed // n = number of copies // spacing = space between copies // sp = if given, copies will start distance sp from the path start and spread beyond that point // // Side Effects: // `$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]] // 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] function path_cut(path, dists, closed=false, direction=false) = let(long_enough = len(path) >= (closed ? 3 : 2)) assert(long_enough,len(path)<2 ? "Two points needed to define a path" : "Closed path must include three points") !is_list(dists)? path_cut(path, [dists],closed, direction)[0] : let(cuts = _path_cut(path,dists,closed)) !direction ? cuts : let( dir = _path_cuts_dir(path, cuts, closed), normals = _path_cuts_normals(path, cuts, dir, closed) ) zip(cuts, array_group(dir,1), array_group(normals,1)); // Main recursive path cut function function _path_cut(path, dists, closed=false, pind=0, dtotal=0, dind=0, result=[]) = dind == len(dists) ? result : let( lastpt = len(result)>0? select(result,-1)[0] : [], dpartial = len(result)==0? 0 : norm(lastpt-path[pind]), nextpoint = dpartial > dists[dind]-dtotal? [lerp(lastpt,path[pind], (dists[dind]-dtotal)/dpartial),pind] : _path_cut_single(path, dists[dind]-dtotal-dpartial, closed, pind) ) is_undef(nextpoint)? concat(result, repeat(undef,len(dists)-dind)) : _path_cut(path, dists, closed, nextpoint[1], dists[dind],dind+1, concat(result, [nextpoint])); // Search for a single cut point in the path function _path_cut_single(path, dist, closed=false, ind=0, eps=1e-7) = ind>=len(path)? undef : ind==len(path)-1 && !closed? (dist dist ? [lerp(path[ind],select(path,ind+1),dist/d), ind+1] : _path_cut_single(path, dist-d,closed, ind+1, eps); // Find normal directions to the path, coplanar to local part of the path // Or return a vector parallel to the x-y plane if the above fails function _path_cuts_normals(path, cuts, dirs, closed=false) = [for(i=[0:len(cuts)-1]) len(path[0])==2? [-dirs[i].y, dirs[i].x] : ( let( plane = len(path)<3 ? undef : let(start = max(min(cuts[i][1],len(path)-1),2)) _path_plane(path, start, start-2) ) plane==undef? unit([-dirs[i].y, dirs[i].x,0]) : unit(cross(dirs[i],cross(plane[0],plane[1]))) ) ]; // Scan from the specified point (ind) to find a noncoplanar triple to use // to define the plane of the path. function _path_plane(path, ind, i,closed) = i<(closed?-1:0) ? undef : !collinear(path[ind],path[ind-1], select(path,i))? [select(path,i)-path[ind-1],path[ind]-path[ind-1]] : _path_plane(path, ind, i-1); // Find the direction of the path at the cut points function _path_cuts_dir(path, cuts, closed=false, eps=1e-2) = [for(ind=[0:len(cuts)-1]) let( zeros = path[0]*0, nextind = cuts[ind][1], nextpath = unit(select(path, nextind+1)-select(path, nextind),zeros), thispath = unit(select(path, nextind) - path[nextind-1],zeros), lastpath = unit(path[nextind-1] - select(path, nextind-2),zeros), nextdir = nextind==len(path) && !closed? lastpath : (nextind<=len(path)-2 || closed) && approx(cuts[ind][0], path[nextind],eps)? unit(nextpath+thispath) : (nextind>1 || closed) && approx(cuts[ind][0],path[nextind-1],eps)? unit(thispath+lastpath) : thispath ) nextdir ]; // Input `data` is a list that sums to an integer. // Returns rounded version of input data so that every // entry is rounded to an integer and the sum is the same as // that of the input. Works by rounding an entry in the list // and passing the rounding error forward to the next entry. // This will generally distribute the error in a uniform manner. function _sum_preserving_round(data, index=0) = index == len(data)-1 ? list_set(data, len(data)-1, round(data[len(data)-1])) : let( newval = round(data[index]), error = newval - data[index] ) _sum_preserving_round( list_set(data, [index,index+1], [newval, data[index+1]-error]), index+1 ); // Function: subdivide_path() // Usage: // newpath = subdivide_path(path, [N|refine], method); // Description: // Takes a path as input (closed or open) and subdivides the path to produce a more // finely sampled path. The new points can be distributed proportional to length // (`method="length"`) or they can be divided up evenly among all the path segments // (`method="segment"`). If the extra points don't fit evenly on the path then the // algorithm attempts to distribute them uniformly. The `exact` option requires that // the final length is exactly as requested. If you set it to `false` then the // algorithm will favor uniformity and the output path may have a different number of // points due to rounding error. // . // With the `"segment"` method you can also specify a vector of lengths. This vector, // `N` specfies the desired point count on each segment: with vector input, `subdivide_path` // attempts to place `N[i]-1` points on segment `i`. The reason for the -1 is to avoid // double counting the endpoints, which are shared by pairs of segments, so that for // a closed polygon the total number of points will be sum(N). Note that with an open // path there is an extra point at the end, so the number of points will be sum(N)+1. // Arguments: // path = path to subdivide // N = scalar total number of points desired or with `method="segment"` can be a vector requesting `N[i]-1` points on segment i. // refine = number of points to add each segment. // closed = set to false if the path is open. Default: True // 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 // 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"` // Example(2D): // mypath = subdivide_path(square([2,2],center=true), 12); // move_copies(mypath)circle(r=.1,$fn=32); // Example(2D): // mypath = subdivide_path(square([8,2],center=true), 12); // move_copies(mypath)circle(r=.2,$fn=32); // Example(2D): // mypath = subdivide_path(square([8,2],center=true), 12, method="segment"); // move_copies(mypath)circle(r=.2,$fn=32); // Example(2D): // mypath = subdivide_path(square([2,2],center=true), 17, closed=false); // move_copies(mypath)circle(r=.1,$fn=32); // Example(2D): Specifying different numbers of points on each segment // mypath = subdivide_path(hexagon(side=2), [2,3,4,5,6,7], method="segment"); // move_copies(mypath)circle(r=.1,$fn=32); // Example(2D): Requested point total is 14 but 15 points output due to extra end point // mypath = subdivide_path(pentagon(side=2), [3,4,3,4], method="segment", closed=false); // move_copies(mypath)circle(r=.1,$fn=32); // Example(2D): Since 17 is not divisible by 5, a completely uniform distribution is not possible. // mypath = subdivide_path(pentagon(side=2), 17); // move_copies(mypath)circle(r=.1,$fn=32); // Example(2D): With `exact=false` a uniform distribution, but only 15 points // mypath = subdivide_path(pentagon(side=2), 17, exact=false); // move_copies(mypath)circle(r=.1,$fn=32); // Example(2D): With `exact=false` you can also get extra points, here 20 instead of requested 18 // mypath = subdivide_path(pentagon(side=2), 18, exact=false); // move_copies(mypath)circle(r=.1,$fn=32); // Example(FlatSpin): Three-dimensional paths also work // mypath = subdivide_path([[0,0,0],[2,0,1],[2,3,2]], 12); // move_copies(mypath)sphere(r=.1,$fn=32); function subdivide_path(path, N, refine, closed=true, exact=true, method="length") = assert(is_path(path)) assert(method=="length" || method=="segment") assert(num_defined([N,refine]),"Must give exactly one of N and refine") let( N = !is_undef(N)? N : !is_undef(refine)? len(path) * refine : undef ) assert((is_num(N) && N>0) || is_vector(N),"Parameter N to subdivide_path must be postive number or vector") let( count = len(path) - (closed?0:1), add_guess = method=="segment"? ( is_list(N)? ( assert(len(N)==count,"Vector parameter N to subdivide_path has the wrong length") add_scalar(N,-1) ) : repeat((N-len(path)) / count, count) ) : // method=="length" assert(is_num(N),"Parameter N to subdivide path must be a number when method=\"length\"") let( path_lens = concat( [ for (i = [0:1:len(path)-2]) norm(path[i+1]-path[i]) ], closed? [norm(path[len(path)-1]-path[0])] : [] ), add_density = (N - len(path)) / sum(path_lens) ) path_lens * add_density, add = exact? _sum_preserving_round(add_guess) : [for (val=add_guess) round(val)] ) concat( [ for (i=[0:1:count]) each [ for(j=[0:1:add[i]]) lerp(path[i],select(path,i+1), j/(add[i]+1)) ] ], closed? [] : [select(path,-1)] ); // Function: path_length_fractions() // Usage: path_length_fractions(path, [closed]) // Description: // Returns the distance fraction of each point in the path along the path, so the first // point is zero and the final point is 1. If the path is closed the length of the output // will have one extra point because of the final connecting segment that connects the last // point of the path to the first point. function path_length_fractions(path, closed=false) = assert(is_path(path)) assert(is_bool(closed)) let( lengths = [ 0, for (i=[0:1:len(path)-(closed?1:2)]) norm(select(path,i+1)-path[i]) ], partial_len = cumsum(lengths), total_len = select(partial_len,-1) ) partial_len / total_len; // Function: resample_path() // Usage: resample_path(path, N|spacing, [closed]) // Description: // Compute a uniform resampling of the input path. If you specify `N` then the output path will have N // points spaced uniformly (by linear interpolation along the input path segments). The only points of the // input path that are guaranteed to appear in the output path are the starting and ending points. // If you specify `spacing` then the length you give will be rounded to the nearest spacing that gives // a uniform sampling of the path and the resulting uniformly sampled path is returned. // Note that because this function operates on a discrete input path the quality of the output depends on // the sampling of the input. If you want very accurate output, use a lot of points for the input. // Arguments: // path = path to resample // N = Number of points in output // spacing = Approximate spacing desired // closed = set to true if path is closed. Default: false function resample_path(path, N, spacing, closed=false) = assert(is_path(path)) assert(num_defined([N,spacing])==1,"Must define exactly one of N and spacing") assert(is_bool(closed)) let( length = path_length(path,closed), N = is_def(N) ? N : round(length/spacing) + (closed?0:1), spacing = length/(closed?N:N-1), // Note: worried about round-off error, so don't include distlist = list_range(closed?N:N-1,step=spacing), // last point when closed=false cuts = path_cut(path, distlist, closed=closed) ) concat(subindex(cuts,0),closed?[]:[select(path,-1)]); // Then add last point here // vim: expandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap