---
jupyter:
  description: 'Parametric curves in a planex2d constraint sketch: formulas that name dimensions and parameters and
    are fitted again when these change, the curve as a rigid body fixed where its formula puts it, and placed by
    constraints once its fix is removed. A wavy channel and an arch.'
---

# Curves driven by dimensions

In a constraint sketch, the formula and the range of a parametric curve may name the ids of driving dimensions and
the names of parameters. When one of them changes, the curve is fitted again, and the elements joined to it follow.
The curve itself is a rigid body, fixed where its formula puts it, so the rest of the sketch can be built against it.
See [Parametric curves](../sketch/parametric-curves.md) for the formulas, and an involute whose base radius is a
dimension.

```python
import planex as px
```

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

## A wavy channel

The example is a channel with a sinusoidal wall, as in the wavy fins of a heat exchanger. The wall runs four waves of
wavelength $\lambda = 20$ and amplitude $a = 2$ about a mean height of 10. The shape of the wave is a pair of
parameters, `a` and `lam` for $\lambda$, so the fins of other sketches can share it (see [Parameters shared by sketches](parameters.md)), and the depth of the
channel is a dimension of its inlet:

```python
wave = px.Parameters(a=2, lam=20)
sk = px.Sketch().Use(wave)
sk.Move((0, 10)).Curve(("t", "10 + a * sin(360 * t / lam)"), (0, "4 * lam"), id="wall")
sk.Line((80, 0), px.Vertical, id="outlet").Line((0, 0), px.Horizontal, id="bottom").Close(px.Vertical, id="inlet")
sk.Constrain("inlet", px.Distance, 10, id="depth")
print(sk.status, {kind: list(ids) for kind, ids in sk.constraints.items()})
```

The wall is drawn first. Its fix `Fix.1` holds it where its formula puts it, and the outlet, the bottom and the inlet
run from its ends, so the vertical and horizontal flags and the depth leave nothing free (Figure 1a).

## Changing the wave

Over whole waves, the sine adds as much area above the mean height as it takes away below, so the channel's area is
its depth times its length, $10 \cdot 4\lambda$, whatever the amplitude. A larger amplitude refits the wall and keeps the area, a
longer wavelength makes the channel longer, and a new depth moves the bottom (Figure 1b):

```python
print("a 2, lam 20:", sk.Face().mass, "exact", 10 * 4 * 20)
wave.Update(a=4)
print("a 4, lam 20:", sk.Face().mass, "exact", 10 * 4 * 20)
wave.Update(lam=25)
print("a 4, lam 25:", sk.Face().mass, "exact", 10 * 4 * 25, "| wall end", sk.Location("wall.end")[:2])
sk.Update({"depth": 12})
print("depth 12:", sk.Face().mass, "exact", 12 * 4 * 25)
```

```python nbsphinx="hide-input"
first = px.Sketch().Use(px.Parameters(a=2, lam=20))
first.Move((0, 10)).Curve(("t", "10 + a * sin(360 * t / lam)"), (0, "4 * lam"), id="wall")
first.Line((80, 0), px.Vertical).Line((0, 0), px.Horizontal).Close(px.Vertical)
Row((first.Face(), "a = 2, λ = 20, depth 10"), (sk.Face(), "a = 4, λ = 25, depth 12"),
    caption="A wavy channel before and after its wave and depth change")
```

## Placing a curve

A curve that should sit against other geometry, rather than where its formula puts it, is freed by removing its fix.
It then moves as a rigid body, with its shape kept, and constraints on `.start`, `.end` or along it place it. The
example is an arch of height 5 over a span of 180, set onto a base line that starts at (40, 25):

```python
arch = px.Sketch()
arch.Curve(("t", "5 * sin(t)"), (0, 180), id="arch")
arch.Remove("Fix.1")
arch.Move((40, 25)).Line((60, 25), id="base")
arch.Constrain("base.start", px.Fix).Constrain("base", px.Horizontal).Constrain("base", px.Distance, 220, id="L")
arch.Constrain("arch.start", "base.start", px.Coincident)
arch.Constrain("arch.end", "base", px.Coincident)
print(arch.status, "start", arch.Location("arch.start")[:2], "end", arch.Location("arch.end")[:2])
```

## The rules

- A formula and a range name driving dimensions by their ids and parameters by their names, and the curve is fitted
  again whenever one of them changes.
- A dimension that a formula uses cannot be removed or driven while the curve exists. Its error says which curve uses
  it.
- The curve is fixed by default. Without its fix, it is a rigid body that constraints on its ends, or on points along
  it, place.
- `h`, `v` and `t` are no ids anywhere in a sketch, since the formulas use them.