---
jupyter:
  description: 'A gear pump gasket as a 2D CAD exercise: a dimensioned drawing to model in planex2d or any CAD system, a
    solution from the two gear chambers united, the outline and the bolt line as their offsets and ten holes equally
    spaced along the bolt line, and the exact area to check a model against.'
---

# Gear pump gasket

A gear pump moves oil between two meshing gears that turn in two chambers of its housing, so the housing's opening is
two overlapping circles, and the gasket that seals its cover follows that figure eight. The gasket's outline and the
line of its bolt holes keep their distance from the opening all around, since they are offsets of it. The holes are
spaced equally along the bolt line, a pattern along a curve rather than on a circle. Model the gasket from the drawing
in planex2d or in any CAD system, then check its area against the value at the end of the page. A solution in planex2d
follows.

![The gear pump gasket exercise: a dimensioned drawing of a figure-eight gasket with two overlapping chambers and ten bolt holes along a bolt line](../_static/gallery/pump-gasket.png){.result}

```python
import math
import planex as px
```

```python nbsphinx="hidden"
from docs_figures import Row
```

## The opening

The [rotors](pump-rotors.md) are two gears of module 2 with 17 teeth, 38 across their tips, and their centers are 34
apart, their pitch diameter. Each chamber is a circle of 38.5 around a gear's center, a radial clearance of 0.25, on the
$x$ axis 17 to either side of the origin. `px.Unify` unites the two discs into the figure eight, with two corners where the
circles cross (Figure 1):

```python
opening = px.Unify([px.Circle((-17, 0), 38.5 / 2).Face(), px.Circle((17, 0), 38.5 / 2).Face()]).Face()
print("corners", [(round(v.x, 4), round(v.y, 4)) for v in opening.vertices])
```

```python nbsphinx="hide-input"
Row((px.Compound([px.Circle((-17, 0), 38.5 / 2), px.Circle((17, 0), 38.5 / 2)]), "The two chambers", {"fill": False}),
    (opening, "United: the opening"), caption="The opening")
```

## The outline and the bolt line

The gasket reaches 16 beyond the opening all around, and the bolt holes lie on a line 8 beyond it, in the middle of the
gasket's width. Both are offsets of the opening's face, outward. An offset of a face is a shape, here of one face.
`Face()` takes that face, and its outer wire is the bolt line (Figure 2):

```python
outline = opening.Offset(16, inplace=False).Face()  # a copy, 16 outward
bolt_line = opening.Offset(8, inplace=False).Face().outer
print("bolt line", len(bolt_line.edges), "arcs,", round(bolt_line.mass, 4), "long")
```

```python nbsphinx="hide-input"
Row((outline, "The outline, the opening and the bolt line, dashed", {"fill": False, "dashed": [bolt_line, opening.outer]}),
    caption="The outline and the bolt line")
```

## The holes

The ten holes of 6.6 lie equally spaced along the bolt line, one pitch of its length apart, the first half a pitch from
the corner where the bolt line starts. `PointAt` gives the point at a fraction of an edge's length, so walking along the
bolt line's two arcs gives each hole's center (Figure 3):

```python
pitch = bolt_line.mass / 10
targets = [pitch / 2 + k * pitch for k in range(10)]
centers, walked = [], 0.0
for e in bolt_line.edges:
    length = e.Measure()["length"]
    for t in targets:
        if walked <= t < walked + length:
            p = e.PointAt((t - walked) / length)  # the point at that fraction of the edge's length
            centers.append((p.x, p.y))
    walked += length
gasket = outline - opening
for c in centers:
    gasket = gasket - px.Circle(c, 6.6 / 2).Face()
gasket = gasket.Face()
print("pitch", round(pitch, 4))
```

```python nbsphinx="hide-input"
Row((gasket, "The gasket, the bolt line dashed", {"dashed": [bolt_line]}), caption="The holes")
```

## The checks

This is the area, to compare with your own model. Offsetting the union of two discs outward gives the union of the
larger discs, so the outline is the union of two circles of 35.25 around the same centers. Two circles of radius $r$
whose centers are $d$ apart overlap in a lens of
$2 r^2 \arccos\left(\tfrac{d}{2r}\right) - \tfrac{d}{2} \sqrt{4 r^2 - d^2}$, and their union is $2 \pi r^2$ less the
lens. The gasket is the outline's union less the opening's and the ten holes. It is symmetric about both axes, so its
centroid is the origin (Figure 4):

```python
def union(r, d=34):
    return 2 * math.pi * r**2 - (2 * r**2 * math.acos(d / (2 * r)) - d / 2 * math.sqrt(4 * r**2 - d**2))


exact = union(35.25) - union(19.25) - 10 * math.pi * 3.3**2
print("area", round(gasket.mass, 4), "mm², exact", round(exact, 4))
print("centroid", tuple(round(v, 6) for v in gasket.center[:2]))
```

```python nbsphinx="hide-input"
Row((gasket, "The gear pump gasket", {}), caption="The gear pump gasket")
```

The gasket as DXF in millimeters, for a cutting plotter or a laser cutter:

```python
gasket.WriteDXF("pump_gasket.dxf", unit=px.mm, scale=1)  # scale=1: the numbers as they are
```

The gasket as DXF: [pump_gasket.dxf](pump_gasket.dxf)

```python nbsphinx="hidden"
import sys
sys.path.insert(0, "..")
import matplotlib.pyplot as plt
from drawings import SheetFigure, View
from gallery import save_figure

# the bolt line as a center line; the chambers', the outline's and the bolt line's radii around the gears' centers
view = View([], lines=gasket.Polylines(1e-3),
            axes=[bolt_line.Polylines(1e-3)[0], ((-58, 0), (58, 0)), ((-17, -8), (-17, 8)), ((17, -8), (17, 8)),
                  ((0, -42), (0, 42))],
            radii=[((-17, 0), 19.25, (200, "inside")), ((17, 0), 35.25, (30, "outside")), ((-17, 0), 27.25, (150, "outside"))],
            diameters=[(centers[0], 3.3, (100, "leader", 10, "EQS"))],
            measures=[((-17, 0), (17, 0), -48, 34)])
drawing = SheetFigure([view], margin=22, notes=("planex2d.org", "2D exercise 22"))
print("scale", drawing.planex_scale, "| paper", [round(v * 25.4) for v in drawing.get_size_inches()], "mm")
save_figure(drawing, "pump-gasket", dpi=200)
plt.close(drawing)
```