path_sweep tangent example

README.md

@@ -21,7 +21,7 @@ The BOSL2 library is an enormous library that provides many different kinds of c
 * **Building Blocks.** OpenSCAD provides cubes, cones and spheres.  The BOSL2 library extends this to provide different kinds of prisms, tubes, and other abstract geometrical building blocks. In many cases the BOSL2 objects include options to round their edges.  Basic objects have extensions like the ability to specify the **inner** radius of a circle to create holes with a guaranteed minimum size.  
 * **Texturing.**  Many kinds of objects can be created with [textures](https://github.com/BelfrySCAD/BOSL2/wiki/skin.scad#section-texturing) applied.  This can create knurling, but it can do much more than that.  A texture can be any repeating pattern, and applying a texture can actually replace the base object with something different based on repeating copies of the texture element.  A texture can also be an image; using texturing you can emboss an arbitrary image onto your model. 
 * **Parts library.**  The parts library includes many useful specific functional parts including [gears](https://github.com/BelfrySCAD/BOSL2/wiki/gears.scad), generic [threading](https://github.com/BelfrySCAD/BOSL2/wiki/threading.scad#section-generic-threading), and specific threading to match plastic [bottles](https://github.com/BelfrySCAD/BOSL2/wiki/bottlecaps.scad), [pipe fittings](https://github.com/BelfrySCAD/BOSL2/wiki/threading.scad#module-npt_threaded_rod), or standard [screws.](https://github.com/BelfrySCAD/BOSL2/wiki/screws.scad)  Also included are [clips](https://github.com/BelfrySCAD/BOSL2/wiki/joiners.scad#section-tension-clips), [hinges](https://github.com/BelfrySCAD/BOSL2/wiki/hinges.scad), and [dovetail joints.](https://github.com/BelfrySCAD/BOSL2/wiki/joiners.scad#section-dovetails)
-* **Shorthands.** The shorthands make your code a little shorter, and more importantly, they can make it significantly easier to read.  Compare `up(z)` to `translate([0,0,z])`.  The shorthands include operations for creating [copies of objects](https://github.com/BelfrySCAD/BOSL2/wiki/distributors.scad) and for applying [transformations](https://github.com/BelfrySCAD/BOSL2/wiki/transforms.scad) to objects, including [rot()](https://github.com/BelfrySCAD/BOSL2/wiki/transforms.scad#functionmodule-rot) which extends rotate in some useful ways that are not easy to do directly. 
+* **Shorthands.** The shorthands make your code a little shorter, and more importantly, they can make it significantly easier to read.  Compare `up(z)` to `translate([0,0,z])`.  The shorthands include operations for creating [copies of objects](https://github.com/BelfrySCAD/BOSL2/wiki/distributors.scad) and for applying [transformations](https://github.com/BelfrySCAD/BOSL2/wiki/transforms.scad) to objects, including [rot()](https://github.com/BelfrySCAD/BOSL2/wiki/transforms.scad#functionmodule-rot) which extends `rotate()` in some useful ways that are not easy to do directly. 
 * **Geometrical operations on data.**  In OpenSCAD, geometrical operations happen on geometry, and information can never be extracted from geometry.  The BOLS2 library provides operations on 2d point lists (called "paths" or "regions") to make [rounded paths](https://github.com/BelfrySCAD/BOSL2/wiki/rounding.scad#function-round_corners) from ones with corners or do operations like [intersection](https://github.com/BelfrySCAD/BOSL2/wiki/regions.scad#functionmodule-intersection) and [offset](https://github.com/BelfrySCAD/BOSL2/wiki/regions.scad#function-offset).  It can also do some limited operations on three dimensional data.
 * **Programming aids.** The library provides basic [mathematical operations](https://github.com/BelfrySCAD/BOSL2/wiki/math.scad) including solutions to [linear systems of equations](https://github.com/BelfrySCAD/BOSL2/wiki/linalg.scad#function-linear_solve) and [generic](https://github.com/BelfrySCAD/BOSL2/wiki/math.scad#function-root_find) and [polynomial](https://github.com/BelfrySCAD/BOSL2/wiki/math.scad#function-real_roots) numerical root finding.  It provides [geometrical operations](https://github.com/BelfrySCAD/BOSL2/wiki/geometry.scad) like [line intersection](https://github.com/BelfrySCAD/BOSL2/wiki/geometry.scad#function-line_intersection) or [circle intersection](https://github.com/BelfrySCAD/BOSL2/wiki/geometry.scad#function-circle_circle_intersection), [coordinate transformations](https://github.com/BelfrySCAD/BOSL2/wiki/coords.scad), [string manipulation,](https://github.com/BelfrySCAD/BOSL2/wiki/strings.scad) and [list processing](https://github.com/BelfrySCAD/BOSL2/wiki/lists.scad).   
 

skin.scad

@@ -2038,6 +2038,17 @@ module spiral_sweep(poly, h, r, turns=1, taper, r1, r2, d, d1, d2, internal=fals
 //   knot_path   = [ for (i=[0:k-1]) knot(360*i/k/gcd(p,q),R,r,p,q) ];
 //   normals = [ for (i=[0:k-1]) knot_normal(360*i/k/gcd(p,q),R,r,p,q) ];
 //   path_sweep(ushape,knot_path,normal=normals, method="manual", closed=true);
+// Example(Med,NoScales,VPR=[355.50,0.00,355.60],VPD=140.00,VPT=[0.00,0.00,0.00],): The left hand example uses the automatically computed derivatives.  The example on the right uses a user-supplied tangent to ensure that the ends are aligned with the coordinate system.  Even with a simple circular arc you may find that the ends are not aligned as expedted by the automatically computed approximate tangent vectors.
+//   $fa=1; $fs=0.5;
+//   theta_range=[180:2:450];
+//   path = [for (theta = theta_range)
+//              polar_to_xy(0.5*PHI^(2*theta/180), theta)];
+//   tangents = [for (theta = theta_range)
+//                  [-sin(theta), cos(theta), 0]];
+//   circ = circle(1, $fn=32);
+//   path_sweep(circ, path);
+//   right(8)
+//      path_sweep(circ, path, tangent=tangents);
 // Example(NoScales): You can request the transformations and manipulate them before passing them on to sweep.  Here we construct a tube that changes scale by first generating the transforms and then applying the scale factor and connecting the inside and outside.  Note that the wall thickness varies because it is produced by scaling.
 //   shape = star(n=5, r=10, ir=5);
 //   rpath = arc(25, points=[[29,6,-4], [3,4,6], [1,1,7]]);