I got really intrigued by the Hirth joint and have been working a library entry that would generate them. I have it fully working, apparently, because it renders perfectly as far as I can see. However when I load one of the parts (a concave part) into PrusaSlicer, it reports thousands of errors that it supposedly fixed. The visual rendering is fine, but when you look at the slice it is completely screwed up.

First some explanation. When I looked up hirth joints I only saw flat ones mentioned. The one I have actually encountered though is a conical one. There is a concave part and a convex part. They fit together perfectly. I had to invent some terminology since I did not find terms for some things I found to be important. The outer center line is the plane that passes through the midpoint of the teeth at the outer radius. In a flat hirth joint all the teeth radiate from the center of this circle, call it the origin. If the origin of the teeth is above the outer center line, then joint has a conical shape and the two parts are concave and convex.

Each tooth at every point has the same profile, the top of the tooth profile is either 60 or 90 degrees. The groove angle is the angle that the groove forms from the origin to the outer radius. It will pass below (for a convex part) the outer radius. It’s critical for making hirth joints because it forms the tool path. The ridge angle is the angle formed by the corresponding ridge.

If the joint is conical, then there is an inner center line that passes through the center of the teeth at the inner radius. The tooth height is the critical number to calculate. It is dependent on the number of teeth, the profile angle and the inner and out radii. Something that really complicates the mental gymnastics is that the tooth height is smaller at the inner radius. My code adds a “base” of extra material at the bottom of the convex part and the top of the concave part. What’s tricky is positioning the base of the concave part relative to the inner center line. The concave parts base is relatively easy to place because it depends on the outer radius and center line.

I generate the raw teeth using the groove and ridge angles and spherical coordinates to make two 2d tooth profiles. Then I complete a tooth using skin(). A for loop generates the full set of N teeth. Then intersection or difference is used the trim the teeth at the inner and outer radii. I discovered something at that point, but solved it pretty simply. When diffing a cylinder from the N teeth, each tooth took up a small portion of the cylinder being diffed out. I had to raise the $fn on the cylinder or tube being used so that I got enough points in each tooth that diff or intersection was accurate. I kept raising it until the results stopped improving. I ended up with the 16*360 you see in the code.

After I have the raw teeth, I generate a cone that is diffed away to produce the chamfer at the top of the teeth. Then I add in the base. The base also adds the “chamfer” at the bottom of the groove. It’s half the size of the top chamfer to leave a little gap when the parts are joined.

When I comment out the everything but the raw teeth and import the STL into PrusaSlicer, it reports errors fixed, but appears to slice correctly when supports are added.

When I add the intersection to trim the raw teeth, the number of reported errors goes way up.

When I add the diff to remove the chamfer, the number goes way up again.

When I add in the base, the number goes up again and it starts reporting many open edges.

Throughout, OpenSCAD reports no errors. Using $fn at 16360, manifold renders in 4+ seconds. At 32360, it’s 53+ seconds. I did check CSG and it did not help.

Maybe the biggest hint is that the convex part uses almost identical code, but produces no errors. Very weird.

So, is this an OpenSCAD problem? Or something in my code?

It’s also possible that my math is off somewhere. My formal trig training was 60 years ago. However I doubt a math error is responsible for the slicing problem and STL errors.

-Bob

include <BOSL2/std.scad>
include <BOSL2/structs.scad>

// Number of Teeth
_n = 36;          // Number Of Teeth

// Inner Radius
_ir = 30;
// Outer Radius
_or = 50;
// Is the coupling conical?
_conic = 10;
// Tooth Profile Angle
_profile = 60; // [60, 90]
// Percentage of tooth height
_chamfer = 5; // Default 5%
_base = 1;

$fn = 180;
tiny = 1 / 1024;

hirth_concave();

_irRatio = _ir / _or;
_toothAngle = 360/_n;
_toothHeight = (sin(_toothAngle/2) * _or) / tan(_profile/2);
_chamferHeight = _chamfer * _toothHeight / 100;
_grooveAngle =  atan(((_toothHeight/2) + _conic) / _or);
_ridgeAngle = -atan(((_toothHeight/2) - _conic) / _or);
_innerCenterLine = (1 - _irRatio) * _conic;
_stackHeight  = _innerCenterLine + _toothHeight + 2*_base;
_tubeHeight = 2*_conic + 2*_base + 2*_toothHeight;

module hirth_concave() {
zrot(_toothAngle/2)
union() {
difference() {
intersection() {
union() {
for (i = [0 : 360/_n : 359])
zrot(i)
_profileF();
}
tube(ir = _ir, or = _or, anchor = CENTER, l = _tubeHeight, orient = UP, $fn = 16360);
}
_chamferF();
zcyl(r = _ir, h = _tubeHeight, $fn = 16360, anchor = CENTER);
}
_baseF();
}
} // hirth_concave

module _profileF() {
IR = _ir * .5;
OR = _or * 1.5;
tI = [spherical_to_xyz(IR,  0,            _grooveAngle + 90),
spherical_to_xyz(IR,  _toothAngle/2, _ridgeAngle + 90),
spherical_to_xyz(IR, -_toothAngle/2, _ridgeAngle + 90)];

 tO = [spherical_to_xyz(OR,   0,            _grooveAngle + 90),
       spherical_to_xyz(OR,  _toothAngle/2, _ridgeAngle + 90),
       spherical_to_xyz(OR, -_toothAngle/2, _ridgeAngle + 90)];
 up(_conic)
 skin([tI, tO], slices = 0);

}

module _chamferF() {
A = -_toothHeight/2 - .1;
B = -_toothHeight/2 + _chamferHeight;

 pts = [[0, _conic], 
        [_or+tiny, A], 
        [_or+tiny, B]];

 rotate_extrude(angle = 360, $fn = 16*360)
 polygon(pts);

}

module _baseF() {
A = _base + _irRatio*_toothHeight/2 + _innerCenterLine;
B = _irRatio * (_toothHeight/2 - _chamferHeight/2) + _innerCenterLine;
C = _toothHeight/2 - _chamferHeight/2;
pts = [
[_ir, A],
[_ir, B],
[_or, C],
[_or, A]
];

 rotate_extrude(angle = 360, $fn = 360*16)
 polygon(pts);

}

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I got really intrigued by the Hirth joint and have been working a library
entry that would generate them. I have it fully working, apparently,
because it renders perfectly as far as I can see. However when I load one
of the parts (a concave part) into PrusaSlicer, it reports thousands of
errors that it supposedly fixed. The visual rendering is fine, but when you
look at the slice it is completely screwed up.

First some explanation. When I looked up hirth joints I only saw flat
ones mentioned. The one I have actually encountered though is a conical
one. There is a concave part and a convex part. They fit together
perfectly. I had to invent some terminology since I did not find terms for
some things I found to be important. The outer center line is the plane
that passes through the midpoint of the teeth at the outer radius. In a
flat hirth joint all the teeth radiate from the center of this circle, call
it the origin. If the origin of the teeth is above the outer center line,
then joint has a conical shape and the two parts are concave and convex.

Each tooth at every point has the same profile, the top of the tooth
profile is either 60 or 90 degrees. The groove angle is the angle that the
groove forms from the origin to the outer radius. It will pass below (for a
convex part) the outer radius. It’s critical for making hirth joints
because it forms the tool path. The ridge angle is the angle formed by the
corresponding ridge.

If the joint is conical, then there is an inner center line that passes
through the center of the teeth at the inner radius. The tooth height is
the critical number to calculate. It is dependent on the number of teeth,
the profile angle and the inner and out radii. Something that really
complicates the mental gymnastics is that the tooth height is smaller at
the inner radius. My code adds a “base” of extra material at the bottom of
the convex part and the top of the concave part. What’s tricky is
positioning the base of the concave part relative to the inner center line.
The concave parts base is relatively easy to place because it depends on
the outer radius and center line.

I generate the raw teeth using the groove and ridge angles and spherical
coordinates to make two 2d tooth profiles. Then I complete a tooth using
skin(). A for loop generates the full set of N teeth. Then intersection or
difference is used the trim the teeth at the inner and outer radii. I
discovered something at that point, but solved it pretty simply. When
diffing a cylinder from the N teeth, each tooth took up a small portion of
the cylinder being diffed out. I had to raise the $fn on the cylinder or
tube being used so that I got enough points in each tooth that diff or
intersection was accurate. I kept raising it until the results stopped
improving. I ended up with the 16*360 you see in the code.

After I have the raw teeth, I generate a cone that is diffed away to
produce the chamfer at the top of the teeth. Then I add in the base. The
base also adds the “chamfer” at the bottom of the groove. It’s half the
size of the top chamfer to leave a little gap when the parts are joined.

When I comment out the everything but the raw teeth and import the STL
into PrusaSlicer, it reports errors fixed, but appears to slice correctly
when supports are added.

When I add the intersection to trim the raw teeth, the number of reported
errors goes way up.

When I add the diff to remove the chamfer, the number goes way up again.

When I add in the base, the number goes up again and it starts reporting
many open edges.

Throughout, OpenSCAD reports no errors. Using $fn at 16360, manifold
renders in 4+ seconds. At 32360, it’s 53+ seconds. I did check CSG and it
did not help.

Maybe the biggest hint is that the convex part uses almost identical
code, but produces no errors. Very weird.

So, is this an OpenSCAD problem? Or something in my code?

It’s also possible that my math is off somewhere. My formal trig training
was 60 years ago. However I doubt a math error is responsible for the
slicing problem and STL errors.

-Bob

include <BOSL2/std.scad>
include <BOSL2/structs.scad>

// Number of Teeth
_n = 36;          // Number Of Teeth

// Inner Radius
_ir = 30;
// Outer Radius
_or = 50;
// Is the coupling conical?
_conic = 10;
// Tooth Profile Angle
_profile = 60; // [60, 90]
// Percentage of tooth height
_chamfer = 5; // Default 5%
_base = 1;

$fn = 180;
tiny = 1 / 1024;

hirth_concave();

_irRatio = _ir / _or;
_toothAngle = 360/_n;
_toothHeight = (sin(_toothAngle/2) * _or) / tan(_profile/2);
_chamferHeight = _chamfer * _toothHeight / 100;
_grooveAngle =  atan(((_toothHeight/2) + _conic) / _or);
_ridgeAngle = -atan(((_toothHeight/2) - _conic) / _or);
_innerCenterLine = (1 - _irRatio) * _conic;
_stackHeight  = _innerCenterLine + _toothHeight + 2*_base;
_tubeHeight = 2*_conic + 2*_base + 2*_toothHeight;

module hirth_concave() {
zrot(_toothAngle/2)
union() {
difference() {
intersection() {
union() {
for (i = [0 : 360/_n : 359])
zrot(i)
_profileF();
}
tube(ir = _ir, or = _or, anchor = CENTER, l =
_tubeHeight, orient = UP, $fn = 16360);
}
_chamferF();
zcyl(r = _ir, h = _tubeHeight, $fn = 16360, anchor = CENTER);
}
_baseF();
}
} // hirth_concave

module _profileF() {
IR = _ir * .5;
OR = _or * 1.5;
tI = [spherical_to_xyz(IR,  0,            _grooveAngle + 90),
spherical_to_xyz(IR,  _toothAngle/2, _ridgeAngle + 90),
spherical_to_xyz(IR, -_toothAngle/2, _ridgeAngle + 90)];

 tO = [spherical_to_xyz(OR,   0,            _grooveAngle + 90),
       spherical_to_xyz(OR,  _toothAngle/2, _ridgeAngle + 90),
       spherical_to_xyz(OR, -_toothAngle/2, _ridgeAngle + 90)];
 up(_conic)
 skin([tI, tO], slices = 0);

}

module _chamferF() {
A = -_toothHeight/2 - .1;
B = -_toothHeight/2 + _chamferHeight;

 pts = [[0, _conic],
        [_or+tiny, A],
        [_or+tiny, B]];

 rotate_extrude(angle = 360, $fn = 16*360)
 polygon(pts);

}

module _baseF() {
A = _base + _irRatio*_toothHeight/2 + _innerCenterLine;
B = _irRatio * (_toothHeight/2 - _chamferHeight/2) + _innerCenterLine;
C = _toothHeight/2 - _chamferHeight/2;
pts = [
[_ir, A],
[_ir, B],
[_or, C],
[_or, A]
];

 rotate_extrude(angle = 360, $fn = 360*16)
 polygon(pts);

}

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<PastedGraphic-1.png>

This second picture shows some visual aids I have in my code to show the
center lines, the origin and the groove and ridge angles.

<PastedGraphic-2.png>