added archimedean_spiral

README.md

@@ -27,6 +27,7 @@ I've been using OpenSCAD for years and created some funny things. Some of them i
     - [circle_path](https://openhome.cc/eGossip/OpenSCAD/lib-circle_path.html)
     - [bezier](https://openhome.cc/eGossip/OpenSCAD/lib-bezier.html)
     - [cylinder_spiral](https://openhome.cc/eGossip/OpenSCAD/lib-cylinder_spiral.html)
+    - [archimedean_spiral](https://openhome.cc/eGossip/OpenSCAD/lib-archimedean_spiral.html)
 
 - Other
     - [box_extrude](https://openhome.cc/eGossip/OpenSCAD/lib-box_extrude.html)

docs/lib-archimedean_spiral.md

@@ -0,0 +1,65 @@
+# archimedean_spiral
+
+Get 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]`. 
+
+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 reduce the error value at the calculated distance between two points, you may try a smaller `point_distance`.
+
+## Parameters
+
+- `arm_distance` : If any ray from the origin intersects two successive turnings of the spiral, we'll have two points. The `arm_distance` is the distance between these two points.
+- `init_angle` : In polar coordinates `(r, θ)` Archimedean spiral can be described by the equation `r = bθ ` where `θ` is measured in radians. For being consistent with OpenSCAD, the function here use degrees. The `init_angle` is which angle the first point want to start.
+- `point_distance` : Distance between two points on the path.
+- `num_of_points` : How many points do you want?
+
+## Examples
+    
+	include <polyline2d.scad>;
+	
+	points_angles = archimedean_spiral(
+	    arm_distance = 10,
+	    init_angle = 180,
+	    point_distance = 5,
+	    num_of_points = 100 
+	); 
+	
+	points = [for(pa = points_angles) pa[0]];
+	
+	polyline2d(points, width = 1);
+
+
+![archimedean_spiral](images/lib-archimedean_spiral-1.JPG)
+	
+	points_angles = archimedean_spiral(
+	    arm_distance = 10,  
+	    init_angle = 180, 
+	    point_distance = 5,
+	    num_of_points = 100 
+	); 
+	
+	for(pa = points_angles) {
+	    translate(pa[0]) 
+	        circle(2);
+	}
+
+![archimedean_spiral](images/lib-archimedean_spiral-2.JPG)
+
+t = "3.141592653589793238462643383279502884197169399375105820974944592307816406286";
+
+	points = archimedean_spiral(
+	    arm_distance = 15,
+	    init_angle = 450, 
+	    point_distance = 12, 
+	    num_of_points = len(t) 
+	); 
+	
+	for(i = [0: len(points) - 1]) {
+	    translate(points[i][0])          
+	        rotate(points[i][1] + 90)  
+	            text(t[i], valign = "center", halign = "center");
+	}
+
+![archimedean_spiral](images/lib-archimedean_spiral-3.JPG)
+
+

src/archimedean_spiral.scad

@@ -0,0 +1,55 @@
+/**
+* archimedean_spiral.scad
+*
+*  Get 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, £c) Archimedean spiral can be described by the equation r = b£c where
+*  £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, £c) Archimedean spiral can be described by the equation r = b£c where
+    £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 archimedean_spiral(arm_distance, init_angle, point_distance, num_of_points) =
+    let(b = arm_distance / 6.28318, init_radian = init_angle *3.14159 / 180)
+    [
+        for(theta = _find_radians(b, point_distance, [init_radian], num_of_points)) 
+           let(r = b * theta, a = theta * 57.2958)
+           [[r * cos(a), r * sin(a)], a]
+    ];