---
jupyter:
  description: 'Lines and arcs in a planex2d sketch: by length, by offset or to a point, the pen turned with Turn, and
    arcs that continue the pen or leave along a direction. Four parts, each checked against its area formula.'
---

# Lines and arcs

A sketch has a pen with a position and a direction. Every line and arc starts at the pen and leaves the pen at its
end, pointing along the end of what it drew, so the next element continues from there. A line is given in one of
three ways:

- `Line(l)` draws a length l along the pen's direction;
- `Line(dh, dv)` draws by an offset from the pen;
- `Line((h, v))` draws to a point.

`Turn(angle)` turns the pen by an angle in degrees, counterclockwise for a positive angle. An arc either continues the
pen's direction, as in `Arc(r, sweep)`, or runs to a point and leaves the pen along a given direction, as in
`Arc(to, tangent=(dh, dv))`. `Close()` joins the outline to its start. The four parts below show these forms, each
checked against its area formula.

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

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

## By length, offset and point

The example is a plate of 40 × 25 with one corner chamfered by 5 (Figure 1). It starts with a length along the pen's
direction, continues with two offsets, up and then diagonally across the chamfer, and goes to a point. `Close` draws
the last side. The corners are marked.

```python
plate = px.QuickSketch().Line(40).Line(0, 20).Line(-5, 5).Line((0, 25)).Close().Face()
```

```python nbsphinx="hide-input"
Row((plate, "A chamfered plate", {"marks": Vertices(plate)}))
```

```python
print("plate", plate.mass, "exact", 40 * 25 - 5**2 / 2)
```

## With Turn

The example is an M10 hex nut, 16 across flats, with a tapped hole of 8.5 (Figure 2). Its side is 16/√3 long. Five
sides are drawn by length and a turn of 60°, and `Close` draws the sixth. `Hexagon(16)` draws the same nut in one call
(see [Slots, keyholes and polygons](slots-polygons.md)). Here it is drawn side by side to show the pen.

```python
side = 16 / math.sqrt(3)
nut = px.QuickSketch((-side / 2, -8))
for _ in range(5):
    nut.Line(side).Turn(60)
nut.Close().Circle(0, 0, 4.25)
hexagon = nut.Face()
```

```python nbsphinx="hide-input"
Row((hexagon, "A hex nut"))
```

```python
print("nut", hexagon.mass, "exact", math.sqrt(3) / 2 * 16**2 - math.pi * 8.5**2 / 4)
```

## Arcs that continue the pen

`Arc(r, sweep)` continues the pen's direction on a circle of radius r and turns left for a positive sweep. The
example is a slot of 30 × 10: a line of 30, a half turn on a radius of 5, the line back and the second half turn
(Figure 3). `Slot(30, 10)` draws it in one call.

```python
slot = px.QuickSketch((-15, -5)).Line(30).Arc(5, 180).Line(30).Arc(5, 180).Face()
```

```python nbsphinx="hide-input"
Row((slot, "A slot"))
```

```python
print("slot", slot.mass, "exact", 30 * 10 + math.pi * 5**2)
```

## Arcs to a point

`Arc(to, tangent=(dh, dv))` runs to a point, starting along the given direction. The example is an arched window,
12 wide and 10 high to the start of its arch (Figure 4). The arch starts straight up from the right side, which makes
it a half circle over the width.

```python
window = px.QuickSketch().Line(12).Line(0, 10).Arc((0, 10), tangent=(0, 1)).Close().Face()
```

```python nbsphinx="hide-input"
Row((window, "An arched window"))
```

```python
print("window", window.mass, "exact", 12 * 10 + math.pi * 6**2 / 2)
```

See [To, from and until](until-from-to.md) for lines that end on other edges, and
[Arcs from references](arc-references.md) for arcs on other circles or touching other edges.