added golden_spiral.scad

src/golden_spiral.scad

@@ -0,0 +1,98 @@
+/**
+* archimedean_spiral.scad
+*
+*  Gets all points and angles on the path of an archimedean_spiral. The distance between two  points is almost constant. 
+* 
+*  It returns a vector of [[x, y], angle]. 
+*
+*  In polar coordinates (r, <20>c) Archimedean spiral can be described by the equation r = b<>c where
+*  <20>c is measured in radians. For being consistent with OpenSCAD, the function here use degrees.
+*
+*  An init_angle less than 180 degrees is not recommended because the function uses an approximate 
+*  approach. If you really want an init_angle less than 180 degrees, a larger arm_distance 
+*  is required. To avoid a small error value at the calculated distance between two points, you 
+*  may try a smaller point_distance.
+* 
+* @copyright Justin Lin, 2017
+* @license https://opensource.org/licenses/lgpl-3.0.html
+*
+* @see https://openhome.cc/eGossip/OpenSCAD/lib-archimedean_spiral.html
+*
+**/ 
+
+function _radian_step(b, theta, l) =
+    let(r_square = pow(b * theta, 2))
+    acos((2 * r_square - pow(l, 2)) / (2 * r_square)) / 180 * 3.14159;
+
+function _find_radians(b, point_distance, radians, n, count = 1) =
+    let(pre_radians = radians[count - 1])
+    count == n ? radians : (
+        _find_radians(
+            b, 
+            point_distance, 
+            concat(
+                radians,
+                [pre_radians + _radian_step(b, pre_radians, point_distance)]
+            ), 
+            n,
+        count + 1) 
+    );
+
+/* 
+    In polar coordinates (r, <20>c) Archimedean spiral can be described by the equation r = b<>c where
+    <20>c is measured in radians. For being consistent with OpenSCAD, the function here use degrees.
+    An init_angle angle less than 180 degrees is not recommended because the function uses an 
+    approximate approach. If you really want an angle less than 180 degrees, a larger arm_distance 
+    is required. To avoid a small error value at the calculated distance between two points, you 
+    may try a smaller point_distance.
+*/
+
+function _fast_fibonacci_sub(nth) = 
+    let(
+        _f = _fast_fibonacci_2_elems(floor(nth / 2)),
+        a = _f[0],
+        b = _f[1],
+        c = a * (b * 2 - a),
+        d = a * a + b * b
+    ) 
+    nth % 2 == 0 ? [c, d] : [d, c + d];
+
+function _fast_fibonacci_2_elems(nth) =
+    nth == 0 ? [0, 1] : _fast_fibonacci_sub(nth);
+    
+function _fast_fibonacci(nth) =
+    _fast_fibonacci_2_elems(nth)[0];
+    
+function _remove_same_pts(pts1, pts2) = 
+    pts1[len(pts1) - 1] == pts2[0] ? 
+        concat(pts1, [for(i = [1:len(pts2) - 1]) pts2[i]]) : 
+        concat(pts1, pts2);    
+
+function _golden_spiral_from_ls_or_eql_to(from, to, point_distance ) = 
+    let(
+        f1 = _fast_fibonacci(from),
+        f2 = _fast_fibonacci(from + 1),
+        fn = floor(f1 * 6.28312 / point_distance), 
+        $fn = fn + 4 - (fn % 4),
+        arc_points = [
+            for(pt = circle_path(radius = f1, n = $fn / 4 + 1))
+                [pt[0], pt[1], 0] // to 3D points because of rotate_p
+        ],
+        offset = f2 - f1
+    ) _remove_same_pts(
+        arc_points, 
+        [
+            for(pt = _golden_spiral(from + 1, to, point_distance)) 
+                rotate_p(pt, [0, 0, 90]) + [0, -offset, 0]
+        ]
+    );
+
+function _golden_spiral(from, to, point_distance) = 
+    from <= to ? 
+        _golden_spiral_from_ls_or_eql_to(from, to, point_distance) : [];
+
+function golden_spiral(from, to, point_distance) =    
+    [
+        for(pt = _golden_spiral(from, to, point_distance)) 
+            [pt[0], pt[1]]  // to 2D points
+    ];