---
jupyter:
  description: 'Arcs from references in a planex2d sketch: on a given circle, between two targets, or touching two
    edges. An angle section with its root fillet and an adjustment plate with a curved slot.'
---

# Arcs from references

An arc needs a circle and two ends. Each can come from the geometry that is already drawn. The circle can have a
given center (`center=`), share another arc's center (`concentric=`), lie on another arc's circle (`congruent=`), or
touch two edges (`tangent=`). The ends can be targets, so the arc runs from one edge to the next.

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

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

## An arc touching two edges: a root fillet

The example is an equal angle section of 40 × 40 × 5 with a root radius of 6 (Figure 1). The inner faces of the two
legs are construction lines, and the root is an arc of radius 6 tangent to both. Two lines meet in four corners, so
the point (40, 40) given with each face picks the corner the arc sits in. The outline then runs around the section
and ends on the root.

```python
s = px.QuickSketch()
s.Move((-1, 5)).Line((41, 5), id="leg face", construction=True)
s.Move((5, -1)).Line((5, 41), id="web face", construction=True)

s.Arc(6, tangent=(("leg face", (40, 40)), ("web face", (40, 40))), id="root")
s.Line((5, 40), id="web").Line((0, 40)).Line((0, 0)).Line((40, 0)).Line((40, 5))
s.Line("root", congruent="leg face", id="leg")
angle = s.Face()
```

```python nbsphinx="hide-input"
Row((s, "The sketch, the inner faces dashed", {"fill": False}), (angle, "The angle section"),
    caption="An angle section whose root fillet touches both legs")
```

The root adds (1 − π/4) r² to the two legs. On a finished corner, `Fillet` does the same in one call (see
[Fillets and chamfers](fillets-chamfers.md)).

```python
print("area", angle.mass, "exact", 2 * 40 * 5 - 5**2 + (1 - math.pi / 4) * 6**2)
```

## Arcs between targets: a curved slot

The example is an adjustment plate of 62 × 62 with a pivot hole of radius 5 and a curved slot around the pivot,
8 wide, on a radius of 40, from 20° to 70° (Figure 2). The slot is built from references in three steps.

First, two construction rays from the pivot at 20° and 70° bound the slot, and its center line is a construction
arc of radius 40 from ray to ray. `Arc("ray0", "ray1", 40, center="pivot")` is an arc of radius 40 around the pivot
that runs from where it meets the first ray to where it meets the second.

```python
s = px.QuickSketch().Vertex("pivot", (0, 0))
s.Move("pivot").Line(80, angle=20, id="ray0", construction=True)
s.Move("pivot").Line(80, angle=70, id="ray1", construction=True)
s.Arc("ray0", "ray1", 40, center="pivot", id="centerline", construction=True);
```

Then the outline of the slot: the outer arc of radius 44 from ray to ray, a half circle around the end of the center
line, the inner arc back to the first ray, concentric with the pivot, and a half circle around the start of the center
line. Last come the plate and the pivot hole.

```python
s.Arc("ray0", "ray1", 44, center="pivot", id="outer")
s.Arc(center="centerline.end", sweep=180, id="cap1")
s.Arc("ray0", concentric="pivot", id="inner")
s.Arc(center="centerline.start", sweep=180, id="cap0")

s.Move((-12, -12), dir=(1, 0)).Rectangle(62, 62, id="plate")
s.Circle(0, 0, 5, id="pivot hole")
plate = s.Face()
```

```python nbsphinx="hide-input"
Row((s, "The sketch, rays and center line dashed", {"fill": False}), (plate, "The adjustment plate"),
    caption="An adjustment plate with a curved slot")
```

The slot's area is the length of its center line times its width, plus the two half circles at its ends:

```python
slot = math.radians(70 - 20) * 40 * 8 + math.pi * 8**2 / 4
print("area", plate.mass, "exact", 62**2 - math.pi * 5**2 - slot)
```

`ArcSlot` draws the same slot in one call. See [Slots, keyholes and polygons](slots-polygons.md) for details. The
end arcs of the [link](tangent.md) are a third form, arcs on another circle with `Arc(from, to, congruent=...)`, and
[Choosing between solutions](choosing.md) shows the four corners in which an arc touching two lines can sit.