# Helix

Curve · (x, y, z) = f(u, t)

`(2 · cos(u),  u/5 − 2.5,  2 · sin(u))`

[Open in the app](https://www.wavelace.com/app#p=37) · [This page](https://www.wavelace.com/presets/helix)

### What it draws

`x = 2cos(u)` and `z = 2sin(u)` hold the point on a circle of radius `2` about the vertical axis: `hypot(x, z) = 2` for every `u`. The second expression, `u/5 − 2.5`, is the height, and it climbs at a fixed rate of a fifth of a unit per unit of `u`. One turn therefore lifts the point by `2π/5 ≈ 1.26`. That rise per turn is the *pitch*.

At `Turns of u` 4 the parameter runs to `8π ≈ 25.13`. The height then climbs from `−2.5` to `2.53`, a rise of `8π/5 ≈ 5.03`. The `−2.5` sits those four turns evenly about the ground plane. No expression mentions `t`, so nothing moves.

### Curvature and torsion

One turn covers `2π·2 ≈ 12.57` of circle and `1.26` of climb. That is an arc length of `√(12.57² + 1.26²) ≈ 12.63`, and a climb angle of `atan(1.26/12.57) ≈ 5.7°`, the same everywhere. A circular helix of radius `a` and slope `b` has curvature `a/(a² + b²)` and torsion `b/(a² + b²)`. With `a = 2` and `b = 1/5` the curvature is `2/4.04 ≈ 0.495`. The torsion is `0.2/4.04 ≈ 0.0495`. Both are constant along the whole curve, and apart from the straight line and the circle the helix is the only curve in space for which that is true. Their ratio, `b/a = 0.1`, is the tangent of the climb angle. A curve is a general helix exactly when torsion over curvature is constant.

### Try

- `Top`: from directly above, the four turns collapse onto the one circle of radius `2`, since only the height separates them.
- `Side`: with the climb up the screen and one horizontal coordinate across it, the helix flattens to a sine wave.
- Halve the pitch. Write the height as `u/10 − 2.5`: the four turns crowd into a rise of `4π/5 ≈ 2.51`, and the climb angle drops to `2.9°`.
- Set it spinning: `2cos(u + t)` and `2sin(u + t)`. The helix turns rigidly about the vertical axis at one radian per second, a full turn every `2π ≈ 6.28` seconds at Speed 1.
- Raise `Turns of u` to 6: two more turns, each adding `2π/5` of rise, so the spring climbs past the top of the plot.

### Read more

- [Helix](https://en.wikipedia.org/wiki/Helix)
- [Curvature](https://en.wikipedia.org/wiki/Curvature)
- [Torsion of a curve](https://en.wikipedia.org/wiki/Torsion_of_a_curve)
- [Frenet–Serret formulas](https://en.wikipedia.org/wiki/Frenet%E2%80%93Serret_formulas)