---
jupyter:
  description: 'Symmetry in a planex2d sketch: half of a part drawn and mirrored with Mirrored(ids, axis), with the
    axis from SymmetryLine. A lifting lug, checked against its exact area.'
---

# Symmetry

A symmetric part can be drawn as one half and mirrored. `Mirrored(ids, axis)` adds mirrored copies of the given
elements under `Mirrored.1`, each named after its original, such as `Mirrored.1.base`, and with the original's
properties, including its name. Ends on the axis stay exactly on it, so the half and its mirror bound one face.
`Mirror(axis)` mirrors the whole sketch instead. `SymmetryLine(a, b)` adds the axis between two points, two lines, or
two circles or arcs, as an infinite construction line.

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

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

## A lifting lug

The example is a lifting lug cut from flat bar 80 wide, with an eye of radius 30 whose center lies 70 above the base,
and a pin hole of 26 (Figure 1). The lug is drawn as its right half in two steps.

First the construction. The two edges of the bar are construction lines, and `SymmetryLine` adds the center line
between them as the axis. The eye is a point with a construction circle around it.

```python
lug = px.QuickSketch()
lug.Move((-40, -10)).Line((-40, 110), id="left edge", construction=True)
lug.Move((40, -10)).Line((40, 110), id="right edge", construction=True)
lug.SymmetryLine("left edge", "right edge", id="axis")
lug.Vertex("eye", (0, 70))
lug.Circle(30, center="eye", id="eye circle", construction=True);
```

Then the right half, from the axis along the base until the right edge, tangent to the eye, and around the eye back
to the axis (Figure 1a). `Mirrored` adds its mirror image, and the pin hole completes the lug (Figure 1b).

```python
lug.Move((0, 0), dir=(1, 0))
lug.Line(until="right edge", id="base")
lug.Line(tangent=("eye circle", (30, 70)), id="flank", name="flank")
lug.Arc("axis", congruent="eye circle", id="crown")
half = lug.Copy()

lug.Mirrored(["base", "flank", "crown"], "axis")
lug.Circle(13, center="eye", id="pin hole")
face = lug.Face()
```

```python nbsphinx="hide-input"
axis = px.Line((0, -10), (0, 110))
Row((half, "The right half", {"fill": False, "dashed": [axis]}),
    (lug, "Mirrored, with the pin hole", {"dashed": [axis]}), caption="A lifting lug from its right half")
```

The mirrored flank also has the name `flank`, so `lug(name="flank")` selects both. Each flank is a tangent from a
base corner to the eye, with a length of √((B/2)² + He² − Re²), where B = 80 is the width, He = 70 the height of the
eye and Re = 30 its radius. The area is that of the polygon through the base corners, the touch points and the eye's
center, plus the sector over the top, minus the hole:

```python
B, He, Re = 80.0, 70.0, 30.0
L0 = math.hypot(B / 2, He)
for e in lug(name="flank"):
    print(e.id, e.Measure()["length"], "exact", math.sqrt(L0**2 - Re**2))

theta = math.atan2(-He, B / 2) + math.acos(Re / L0)
T = (Re * math.cos(theta), He + Re * math.sin(theta))
polygon = [(-B / 2, 0), (B / 2, 0), T, (0, He), (-T[0], T[1])]
pairs = zip(polygon, polygon[1:] + polygon[:1])
shoelace = 0.5 * sum(p[0] * q[1] - q[0] * p[1] for p, q in pairs)
print("area", face.mass, "exact", shoelace + 0.5 * Re**2 * (math.pi - 2 * theta) - math.pi * 13**2)
```

## What SymmetryLine accepts

| a, b | Symmetry line |
|---|---|
| Two points | The perpendicular bisector |
| Two parallel lines | The midline |
| Two crossing lines | The bisector that maps a onto b best; `reverse=True` selects the other |
| Two circles or arcs | The perpendicular bisector of their centers |

The symmetry line is an infinite construction line with the id given by `id=`. In a constraint sketch, it stays
symmetric to its references when they move. See [Symmetry and patterns](../constraints/symmetry.md) for details.