---
jupyter:
  description: 'Tangent lines and arcs in a planex2d sketch: touching circles, arcs and splines, with the touch points
    computed from the references. A link from two bosses, checked against its exact area.'
---

# Tangents

A line or an arc can be given what it touches instead of where it ends. `tangent=` names the circle, arc or spline,
and planex2d computes the touch point, so no point of a tangent has to be worked out by hand. With two targets, as in
`tangent=("a", "b")`, a line touches both.

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

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

## A link from two bosses

The example is a link, a flat lever with a boss of radius 1.5 at one end and a boss of radius 1 at the other, 6
apart (Figure 1). The bosses are drawn as construction circles. Two lines are tangent to both, the upper one and, with
`reverse=True`, the lower one. The ends of the link are arcs on the bosses' circles, from one line to the other, and
each boss has a hole. `Face()` then makes the face of the link from the drawn edges (b).

```python
link = px.QuickSketch()
link.Circle(0, 0, 1.5, id="boss1", construction=True)
link.Circle(6, 0, 1.0, id="boss2", construction=True)
link.Line(tangent=("boss1", "boss2"), id="upper")
link.Line(tangent=("boss1", "boss2"), reverse=True, id="lower")
link.Arc("lower", "upper", congruent="boss2", id="end2")
link.Arc("upper", "lower", congruent="boss1", id="end1")
link.Circle(0, 0, 0.6, id="hole1").Circle(6, 0, 0.4, id="hole2")
face = link.Face()
```

```python nbsphinx="hide-input"
Row((link, "The sketch", {"fill": False}), (face, "The face"), caption="A link from two bosses and two tangents")
```

The area of the link is the area of the belt around two circles, which has a closed form, minus the two holes:

```python
r1, r2, d = 1.5, 1.0, 6.0
alpha = math.asin((r1 - r2) / d)
belt = (r1 + r2) * math.sqrt(d * d - (r1 - r2) ** 2) + r1**2 * (math.pi / 2 + alpha) + r2**2 * (math.pi / 2 - alpha)
print("area", face.mass, "exact", belt - math.pi * (0.6**2 + 0.4**2))
```

## From a point, along a circle, to a spline

A tangent can start at a point, leave a circle, or touch a spline (Figure 2).

- From a point, `tangent="circle"` draws one of the two tangents to the circle, and `reverse=True` the other (a).
- From a point on a circle, `Line(3, tangent="c")` leaves the circle along its tangent there, 3 long (b).
- From a point outside a spline, `tangent="curve"` touches the spline (c).

Where a line could touch a target at several points, a point given with the target, as in `tangent=("curve", p)`,
selects the touch point nearest to p.

```python
point = px.QuickSketch().Circle(5, 0, 1.5, id="circle", construction=True)
point.Move((0, 0)).Line(tangent="circle")
point.Move((0, 0)).Line(tangent="circle", reverse=True)

leaving = px.QuickSketch().Circle(0, 0, 1, id="c")
leaving.Move((0, 1)).Line(3, tangent="c")

spline = px.QuickSketch((0, 2)).Spline([(1, 1), (2, 0.8), (3, 1.2), (4, 2)], id="curve")
spline.Move((2, -1)).Line(tangent="curve");
```

```python nbsphinx="hide-input"
Row((point, "From a point", {"fill": False}), (leaving, "Leaving a circle", {"fill": False}),
    (spline, "Touching a spline", {"fill": False}), caption="Tangents from a point, along a circle and to a spline")
```

An arc can be tangent too. See [Arcs from references](arc-references.md) for arcs that touch two edges, and
[Choosing between solutions](choosing.md) for picking one of several tangents.