added squircle to shapes2d.scad

shapes2d.scad

@@ -1929,7 +1929,7 @@ function _superformula(theta,m1,m2,n1,n2=1,n3=1,a=1,b=1) =
 // Usage: As Function
 //   path = reuleaux_polygon(n, r|d=, ...);
 // Description:
-//   When called as a module, reates a 2D Reuleaux Polygon; a constant width shape that is not circular.  Uses "intersect" type anchoring.  
+//   When called as a module, creates a 2D Reuleaux Polygon; a constant width shape that is not circular.  Uses "intersect" type anchoring.  
 //   When called as a function, returns a 2D path for a Reulaux Polygon.
 // Arguments:
 //   n = Number of "sides" to the Reuleaux Polygon.  Must be an odd positive number.  Default: 3
@@ -1988,6 +1988,133 @@ function reuleaux_polygon(n=3, r, d, anchor=CENTER, spin=0) =
 
 
 
+// Function&Module: squircle()
+// Synopsis: Creates a shape between a circle and a square, centered on the origin.
+// SynTags: Geom, Path
+// Topics: Shapes (2D), Paths (2D), Path Generators, Attachable
+// See Also: circle(), square(), supershape()
+// Usage: As Module
+//   squircle(size, [squareness], [style]) [ATTACHMENTS];
+// Usage: As Function
+//   path = squircle(size, [squareness], [style]);
+// Description:
+//   A [squircle](https://en.wikipedia.org/wiki/Squircle) is a shape intermediate between a square/rectangle and a circle/ellipse.Squircles are sometimes used to make dinner plates (more area for the same radius as a circle), keyboard buttons, and smartphone icons. Old CRT television screens also resembled elongated squircles.
+//   .
+//   There are multiple approaches to constructing a squircle. One approach is a special case of superellipse (shown in {{supershape()}} example 3), and uses exponents between 2 and infinity to adjust the shape. Another, the Fernández-Guasti squircle or FG squircle, arises from work in optics and uses a "squareness" parameter between 0 and 1 to adjust the shape. We use the same squareness parameter for both types, adjusting the internal FG parameter or superellipse exponent as needed to achieve the same squircle corner extents.
+//   .
+//   The FG style and superellipse style squircles are visually almost indistinguishable, with the superellipse having slightly rounder "corners" than FG for a given value of squareness. Either style requires just the two parameters `squareness` and `size`. The vertex distribution is adjusted to be more dense at the corners for smoothness at low values of `$fn`.
+//   .
+//   When called as a module, creates a 2D squircle with the desired squareness.    
+//   When called as a function, returns a 2D path for a squircle.
+// Arguments:
+//   size = Same as the `size` parameter in `square()`, can be a single number or a vector `[xsize,ysize]`.
+//   squareness = Value between 0 and 1. Controls the shape, setting the location of a squircle "corner" at the specified interpolated position between a circle and a square. When `squareness=0` the shape is a circle, and when `squareness=1` the shape is a square. For the "superellipse" style, the special case where the superellipse exponent is 4 (also known as *Lamé's quartic curve*) results in a squircle at the geometric mean between radial points on the circle and square, corresponding to squareness=0.456786. Default: 0.5
+//   style = method for generating a squircle, "fg" for Fernández-Guasti and "superellipse" for superellipse. Default: "fg"
+//   atype = anchor type, "box" for bounding box corners and sides, "perim" for the squircle corners
+//   $fn = Number of points. The special variables `$fs` and `$fa` are ignored. If set, `$fn` must be 12 or greater, and is rounded to the nearest multiple of 4. Points are generated so they are more dense around sharper curves. Default if not set: 48
+// Examples(2D):
+//   squircle(size=50, squareness=0.4);
+//   squircle([80,60], 0.7, $fn=64);
+// Example(2D): Ten increments of squareness parameter for a superellipse squircle
+//   for(sq=[0:0.1:1])
+//       stroke(squircle(100, sq, style="superellipse", $fn=128), closed=true, width=0.5);
+// Example(2D): Standard vector anchors are based on the bounding box
+//   squircle(50, 0.6) show_anchors();
+// Example(2D): Perimeter anchors, anchoring at bottom left and spinning 20°
+//   squircle([60,40], 0.5, anchor=(BOTTOM+LEFT), atype="perim", spin=20)
+//       show_anchors();
+
+module squircle(size, squareness=0.5, style="fg", atype="box", anchor=CENTER, spin=0) {
+    check = assert(squareness >= 0 && squareness <= 1);
+    anchorchk = assert(in_list(atype, ["box", "perim"]));
+    size = is_num(size) ? [size,size] : point2d(size);
+    assert(all_positive(size), "All components of size must be positive.");
+    path = squircle(size, squareness, style, atype="box");
+    if (atype == "box") {
+        attachable(anchor, spin, two_d=true, size=size, extent=false) {
+            polygon(path);
+            children();
+        }
+    } else { // atype=="perim"
+        attachable(anchor, spin, two_d=true, extent=true, path=path) {
+            polygon(path);
+            children();
+        }
+    }
+}
+
+
+function squircle(size, squareness=0.5, style="fg", atype="box", anchor=CENTER, spin=0) =
+    assert(squareness >= 0 && squareness <= 1)
+    assert(is_num(size) || is_vector(size,2))
+    assert(in_list(atype, ["box", "perim"]))
+    let(
+        size = is_num(size) ? [size,size] : point2d(size),
+        path = style == "fg" ? _squircle_fg(size, squareness)
+            : style == "superellipse" ? _squircle_se(size, squareness)
+            : assert(false, "Style must be \"fg\" or \"superellipse\"")
+    ) reorient(anchor, spin, two_d=true, size=atype=="box"?size:undef, path=atype=="box"?undef:path, p=path, extent=true);
+
+
+/* FG squircle functions */
+
+function _squircle_fg(size, squareness) = [
+    let(
+        sq = _linearize_squareness(squareness),
+        size = is_num(size) ? [size,size] : point2d(size),
+        aspect = size[1] / size[0],
+        r = 0.5 * size[0],
+        astep = $fn>=12 ? 90/round($fn/4) : 360/48
+    ) for(a=[360:-astep:0.01]) let(
+        theta = a + sq * sin(4*a) * 30/PI, // tighter angle steps at corners
+        p = squircle_radius_fg(sq, r, theta)
+    ) p*[cos(theta), aspect*sin(theta)]
+];
+
+function squircle_radius_fg(squareness, r, angle) = let(
+    s2a = abs(squareness*sin(2*angle))
+    ) s2a>0 ? r*sqrt(2)/s2a * sqrt(1 - sqrt(1 - s2a*s2a)) : r;
+
+function _linearize_squareness(s) =
+    // from Chamberlain Fong (2016). "Squircular Calculations". arXiv.
+    // https://arxiv.org/pdf/1604.02174v5
+    let(c = 2 - 2*sqrt(2), d = 1 - 0.5*c*s)
+    2 * sqrt((1+c)*s*s - c*s) / (d*d);
+
+
+/* Superellipse squircle functions */
+
+function _squircle_se(size, squareness) = [
+    let(
+        n = _squircle_se_exponent(squareness),
+        size = is_num(size) ? [size,size] : point2d(size),
+        ra = 0.5*size[0],
+        rb = 0.5*size[1],
+        astep = $fn>=12 ? 90/round($fn/4) : 360/48,
+        fgsq = _linearize_squareness(min(0.998,squareness)) // works well for distributing theta
+    ) for(a=[360:-astep:0.01]) let(
+        theta = a + fgsq*sin(4*a)*30/PI, // tighter angle steps at corners
+        x = cos(theta),
+        y = sin(theta),
+        r = (abs(x)^n + abs(y)^n)^(1/n) // superellipse
+        //r = _superformula(theta=theta, m1=4,m2=4,n1=n,n2=n,n3=n,a=1,b=1)
+    ) [ra*x, rb*y] / r
+];
+
+function squircle_radius_se(n, r, angle) = let(
+    x = cos(angle),
+    y = sin(angle)
+) (abs(x)^n + abs(y)^n)^(1/n) / r;
+
+function _squircle_se_exponent(squareness) = let(
+    // limit squareness; error if >0.99889, limit is smaller for r>1
+    s=min(0.998,squareness),
+    rho = 1 + s*(sqrt(2)-1),
+    x = rho / sqrt(2)
+) log(0.5) / log(x);
+
+
+
 // Section: Text
 
 // Module: text()