---
jupyter:
  description: 'Inside or outside in planex2d: Contains tests points against a face, with or without the boundary
    points, and a grid of 9600 points estimates the area of a link.'
---

# Inside or outside

`Contains` tells whether a point lies in a face, for one point or for an array of points at once. Points on the
boundary count as inside, unless `boundary=False` is given. All examples on this page use the link from
[Tangents](../sketch/tangent.md), two bosses of radius 1.5 and 1 with their centers 6 apart, with a hole in each.

```python
import numpy as np
import planex as px

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

## Three points

The example tests three points (Figure 1). The center of the large hole is outside, and a point on the web between
the bosses is inside. The far end of the small boss, at (7, 0), lies on the boundary, so it counts as inside by
default and as outside with `boundary=False`.

```python
points = [(0, 0), (3, 0), (7, 0)]
print([face.Contains(p) for p in points], face.Contains((7, 0), boundary=False))
```

```python nbsphinx="hide-input"
Row((face, "The three points", {"marks": points, "labels": [(str(face.Contains(p)), (p[0], p[1] + 0.45)) for p in points]}),
    caption="Contains for three points: in the hole, on the web, on the boundary")
```

## Counting points

An array of points is tested in one call, which returns an array of booleans. The example covers the bounding box
with a grid of 160 × 60 points, at the middles of its cells. The share of points inside, times the area of the box,
estimates the area of the face.

```python
(h0, v0), (h1, v1) = face.bounding_box
h, v = np.meshgrid(np.linspace(h0, h1, 161)[:-1], np.linspace(v0, v1, 61)[:-1])
cells = np.column_stack([h.ravel() + (h1 - h0) / 320, v.ravel() + (v1 - v0) / 120])
inside = face.Contains(cells)
print("estimate", inside.mean() * (h1 - h0) * (v1 - v0), "area", face.mass)
```