---
jupyter:
  description: 'Places along an edge in planex2d: Measure(at=...) gives the point, tangent, normal and curvature at an
    arc length or nearest a point, on the exact curves. The leading edge of a NACA 0012, marks every 50 along a heater
    track, and the place on it nearest a sensor.'
---

# Along an edge

`Measure(at=...)` finds a place along an edge, a wire, a selection of edges, or a sketch element or chain, and returns
what the geometry is like there. The place is given in one of two ways:

| Call | The place |
|---|---|
| `Measure(at=s)` | At the arc length $s$ from the start |
| `Measure(at=(h, v))` | Nearest the point; of places equally near, the one with the smallest arc length |

The result is a dict:

- `point` and `point_hv`, the place in 3D and in the plane;
- `tangent` and `tangent_hv`, the unit tangent along the direction of the chain;
- `normal` and `normal_hv`, the unit normal to its left;
- `curvature`, positive where the chain turns left;
- `arc_length` from the start, and `distance` from the point given, 0 for an arc length;
- `edge`, the edge the place lies on.

Everything is taken on the exact curves, splines included. `wire.Reverse()` runs a wire the other way, in place, and
`wire.Reversed()` returns a reversed copy, so the arc length can count from either end.

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

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

## The leading edge of an airfoil

The leading edge of a NACA four-digit airfoil is round. Near the nose, the half thickness grows as
$5t \cdot 0.2969\,\sqrt{x c}$ for a thickness $t$ as a fraction of the chord $c$, so the nose is a circle of radius
$r = (5t \cdot 0.2969)^2\,c/2$. The example is a NACA 0012 with a chord of 100. `FromNACA` draws its profile from
the formula, and the curvature measured at the nose is $1/r$ (Figure 1):

```python
naca = px.FromNACA("0012", chord=100)
nose = naca.outer.Measure(at=(0, 0))
r = (5 * 0.12 * 0.2969) ** 2 * 100 / 2
print("nose", nose["point_hv"], "tangent", nose["tangent_hv"])
print("curvature", nose["curvature"], "1/r", 1 / r)
```

```python nbsphinx="hide-input"
close = naca * px.QuickSketch((-1, -2.5)).Rectangle(6, 5).Face()
Row((naca, "A NACA 0012", {"marks": [(0, 0)]}),
    (close, "The nose and the circle of radius r", {"dashed": [px.Circle((r, 0), r)], "marks": [(0, 0)]}),
    caption="The curvature at the leading edge of a NACA 0012")
```

## Marks along a path

A place at a given arc length puts marks along a path. The example is the meander heater track from
[Offsets](../sketch/offsets.md), six runs of 40 joined by half circles of radius 3, with a mark every 50 along its
center line for a row of temperature sensors (Figure 2a). Each mark lies on the path, at the arc length asked for:

```python
s = px.QuickSketch()
for k in range(5):
    s.Line(40).Arc(3, 180 if k % 2 == 0 else -180)
s.Line(40)
track = s.Wire()
marks = [track.Measure(at=d) for d in range(0, 288, 50)]
for mark in marks:
    print("at", mark["arc_length"], "point", mark["point_hv"], "off the path", track.Distance(mark["point_hv"]))
print("length", track.mass, "exact", 6 * 40 + 5 * math.pi * 3)
```

## The place nearest a point

A point gives the place nearest it. The example is a sensor at (60, 12), beside the track. The nearest place lies as
far away as `Distance` measures, and its arc length says how far along the track it is (Figure 2b). On the reversed
track, the arc length counts from the other end:

```python
near = track.Measure(at=(60, 12))
print("nearest", near["point_hv"], "distance", near["distance"], "Distance", track.Distance((60, 12)))
print("arc length", near["arc_length"], "normal", near["normal_hv"])
back = track.Reversed().Measure(at=(60, 12))
print("from the other end", back["arc_length"], "exact", track.mass - near["arc_length"])
```

```python nbsphinx="hide-input"
Row((track, "A mark every 50 along the track", {"fill": False, "marks": [m["point_hv"] for m in marks]}),
    (track, "The place nearest the sensor at (60, 12)", {"fill": False, "marks": [(60, 12), near["point_hv"]],
                                                         "dashed": [px.Line((60, 12), near["point_hv"])]}),
    caption="Places along a heater track")
```

See [Distances and clearances](distances.md) for distances between objects, and
[Parametric curves](../basics/curves.md) for curves drawn from their formula.