Align rack pressure angle for helical meshes

README.md

@@ -17,6 +17,7 @@ A library for the parametric creation of gear racks, spur-, ring-, bevel- and wo
 Creates a gear rack.
 
 This script adjusts the pressure angle in the transverse section to the helix angle: e.g. with a 20° helix angle, a pressure angle of 20° becomes a pressure angle of 21.17° in the transverse section.
+The rack module matches the gear module directly, without additional helix-based scaling.
 
 ### Format
 

gears.scad

@@ -117,9 +117,10 @@ module rack(modul, length, height, width, pressure_angle = 20, helix_angle = 0)
     // Dimension Calculations
     modul=modul*(1-clearance);
     c = modul / 6;                                              // Tip Clearance
-    mx = modul/cos(helix_angle);                          // Module Shift by Helix Angle in the X-Direction
-    a = 2*mx*tan(pressure_angle)+c*tan(pressure_angle);       // Flank Width
-    b = pi*mx/2-2*mx*tan(pressure_angle);                      // Tip Width
+    mx = modul;                                                 // Keep module consistent with gears; no helix scaling
+    alpha_spur = atan(tan(pressure_angle)/cos(helix_angle));   // Helix Angle in Transverse Section
+    a = 2*mx*tan(alpha_spur)+c*tan(alpha_spur);                 // Flank Width
+    b = pi*mx/2-2*mx*tan(alpha_spur);                           // Tip Width
     x = width*tan(helix_angle);                          // Topside Shift by Helix Angle in the X-Direction
     nz = ceil((length+abs(2*x))/(pi*mx));                       // Number of Teeth
 
@@ -148,6 +149,74 @@ module rack(modul, length, height, width, pressure_angle = 20, helix_angle = 0)
     };
 }
 
+/* Involute rack; uses involute flank geometry to better match involute gears
+    modul = Height of the Tooth Tip above the Rolling LIne
+    length = Length of the Rack
+    height = Height of the Rack to the Pitch Line
+    width = Width of a Tooth
+    pressure_angle = Pressure Angle, Standard = 20° according to DIN 867. Should not exceed 45°.
+    helix_angle = Helix Angle of the Rack Transverse Axis; 0° = Spur Teeth */
+module involute_rack(modul, length, height, width, pressure_angle = 20, helix_angle = 0) {
+
+    // Dimension Calculations
+    c = modul / 6;                                              // Tip Clearance
+    mx = modul;                                                 // Keep module consistent with gears; no helix scaling
+    alpha_spur = atan(tan(pressure_angle)/cos(helix_angle));   // Helix Angle in Transverse Section
+    pitch = pi*mx;                                              // Pitch length
+    x_shift = width*tan(helix_angle);                           // Topside Shift by Helix Angle in the X-Direction
+    x_slope = (width == 0) ? 0 : x_shift/width;                 // Avoid division by zero for degenerate widths
+    nz = ceil((length+abs(2*x_shift))/pitch);                   // Number of Teeth
+    virtual_teeth = (nz < 12) ? 12 : nz;                        // Stabilize involute curvature for short racks
+
+    // Virtual gear parameters for involute flank generation
+    d = mx * virtual_teeth;                                     // Pitch Circle Diameter
+    r = d / 2;                                                  // Pitch Circle Radius
+    rb = r * cos(alpha_spur);                                   // Base Circle Radius
+    da = (mx < 1) ? d + mx * 2.2 : d + mx * 2;                   // Tip Diameter
+    ra = da / 2;                                                // Tip Circle Radius
+    rho_ra = acos(rb/ra);                                       // Maximum Rolling Angle
+    rho_r = acos(rb/r);                                         // Rolling Angle at Pitch Circle
+    phi_r = grad(tan(rho_r)-radian(rho_r));                     // Angle to Point of Involute on Pitch Circle
+    step = rho_ra/16;                                           // Involute is divided into 16 pieces
+    tooth_width = (180*(1-clearance))/virtual_teeth + 2*phi_r;  // Tooth thickness in degrees
+    angle_shift = -phi_r - 90*(1-clearance)/virtual_teeth;      // Center tooth around zero degrees
+
+    y_root = -(mx + c);                                         // Root Line
+    y_base = rb - r;                                            // Base Line (involute start)
+
+    // 2D involute tooth profile for a single pitch
+    points_2d = concat(
+        [[0, y_root], [pitch, y_root], [pitch, y_base]],
+        [for (rho = [0:step:rho_ra]) let (
+            angle = tooth_width - ev(rb, rho)[1] + angle_shift,
+            radius = ev(rb, rho)[0]
+        ) [r*radian(angle) + pitch/2, radius - r]],
+        [for (rho = [rho_ra:-step:0]) let (
+            angle = ev(rb, rho)[1] + angle_shift,
+            radius = ev(rb, rho)[0]
+        ) [r*radian(angle) + pitch/2, radius - r]],
+        [[0, y_base]]
+    );
+
+    translate([-pitch*(nz-1)/2,0,0]){
+        intersection(){
+            copier([1,0,0], nz, pitch, 0){
+                multmatrix(m = [
+                    [1,0,x_slope,0],
+                    [0,1,0,0],
+                    [0,0,1,0],
+                    [0,0,0,1]
+                ])
+                    linear_extrude(height = width, convexity = 10)
+                        polygon(points_2d);
+            };
+            translate([abs(x_shift),-height-0.5,-0.5]){
+                cube([length,height+mx+1,width+1]);             // Cuboid which includes the Volume of the Rack
+            }
+        };
+    };
+}
+
 /* Mountable-rack; uses module "rack"
     modul = Height of the Tooth Tip above the Rolling LIne
     length = Length of the Rack
@@ -334,9 +403,9 @@ module herringbone_rack(modul, length, height, width, pressure_angle = 20, helix
  width = width/2;
  translate([0,0,width]){
         union(){
-            rack(modul, length, height, width, pressure_angle, helix_angle);      // bottom Half
+            involute_rack(modul, length, height, width, pressure_angle, helix_angle);      // bottom Half
             mirror([0,0,1]){
-                rack(modul, length, height, width, pressure_angle, helix_angle);  // top Half
+                involute_rack(modul, length, height, width, pressure_angle, helix_angle);  // top Half
             }
         }
     }

requirements-dev.txt

@@ -1 +1 @@
-pillow
+Pillow>=10.0.0

setup.sh

@@ -0,0 +1,20 @@
+#!/usr/bin/env bash
+set -euo pipefail
+
+VENV_DIR="${1:-.env}"
+
+if [ ! -f "requirements-dev.txt" ]; then
+  echo "requirements-dev.txt not found in $(pwd)" >&2
+  exit 1
+fi
+
+if [ ! -d "$VENV_DIR" ]; then
+  mkdir -p "$(dirname "$VENV_DIR")"
+  python3 -m venv "$VENV_DIR"
+fi
+
+# shellcheck disable=SC1091
+source "$VENV_DIR/bin/activate"
+
+python -m pip install --upgrade pip
+python -m pip install -r requirements-dev.txt

tests/test_herringbone_rack.py

@@ -0,0 +1,86 @@
+import shutil
+import subprocess
+import tempfile
+import unittest
+from pathlib import Path
+
+from PIL import Image
+
+
+HERRINGBONE_RACK_SCAD_SNIPPET = """\
+include <gears.scad>
+modul = 1;
+gear_teeth = 30;
+width = 20;
+rack_length = 60;
+rack_height = 5;
+helix_angle = 30;
+pressure_angle = 20;
+
+intersection(){
+    rotate([0,180,0])
+        translate([.5, -20, -10])
+            herringbone_rack(modul=modul, length=rack_length, height=rack_height, width=width, pressure_angle=pressure_angle, helix_angle=helix_angle);
+    translate([0, -4.75, -20/2])
+        rotate([0,0,4])
+            herringbone_gear (modul=modul, tooth_number=gear_teeth, width=width, bore=4, pressure_angle=pressure_angle, helix_angle=helix_angle, optimized=true);
+}
+"""
+
+
+class HerringboneRackMeshTest(unittest.TestCase):
+    """Regression test for https://github.com/chrisspen/gears/issues/16."""
+
+    def setUp(self) -> None:
+        if shutil.which("openscad-nightly") is None:
+            self.skipTest("openscad-nightly is required for this test")
+
+    def test_herringbone_rack_mesh_does_not_intersect(self) -> None:
+        image = self._render_intersection()
+        pixels = list(image.getdata())
+        dark_pixels = sum(1 for pixel in pixels if pixel < 220)
+
+        self.assertLess(
+            dark_pixels,
+            100,
+            "Expected minimal intersection between herringbone rack and gear",
+        )
+
+    @staticmethod
+    def _render_intersection() -> Image.Image:
+        tests_dir = Path(__file__).resolve().parent
+        repo_root = tests_dir.parent
+        scad_source = repo_root / "gears.scad"
+
+        with tempfile.TemporaryDirectory(dir=repo_root) as tmp_dir:
+            tmp_path = Path(tmp_dir)
+            scad_path = tmp_path / "herringbone_mesh.scad"
+            png_path = tmp_path / ".." / ".cache" / "herringbone_mesh.png"
+
+            scad_path.write_text(HERRINGBONE_RACK_SCAD_SNIPPET)
+            shutil.copy(scad_source, tmp_path / "gears.scad")
+
+            cmd = [
+                "openscad-nightly",
+                str(scad_path),
+                "--viewall",
+                "--projection",
+                "ortho",
+                "-o",
+                str(png_path),
+            ]
+            try:
+                subprocess.run(cmd, check=True, capture_output=True)
+            except subprocess.CalledProcessError as exc:  # pragma: no cover - diagnostic aid
+                raise AssertionError(
+                    f"openscad-nightly failed with code {exc.returncode}\n"
+                    f"stdout: {exc.stdout.decode()}\n"
+                    f"stderr: {exc.stderr.decode()}"
+                ) from exc
+
+            image = Image.open(png_path).convert("L")
+            return image.copy()
+
+
+if __name__ == "__main__":
+    unittest.main()