# Quadric Family

Shape · (x, y, z) = f(u, v, t)

`(sqrt(max(k, 0) +(3v/π − 1.5)²) · cos(u),  sign(3v/π − 1.5) · sqrt((3v/π − 1.5)² − min(k, 0)) /(3v/π − 1.5 != 0 || k >= 0),  sqrt(max(k, 0) +(3v/π − 1.5)²) · sin(u))`

[Open in the app](https://www.wavelace.com/app#p=159) · [This page](https://www.wavelace.com/presets/quadric-family)

### What it draws

Every point of this surface satisfies `x² + z² − y² = k`, with `k` on its own slider. Wavelace draws `y` upward, so this is the textbook `x² + y² − z² = k` standing on its axis. The parameter `u` goes once around the axis, and `w = 3v/π − 1.5` runs from −1.5 to 1.5 as `v` runs from 0 to π.

For `k` above zero, `w` is the height and the radius is `√(k + w²)`. At the opening `k = 1` the waist has radius 1 and the two rims have radius `√3.25 ≈ 1.80`. For `k` below zero the roles swap: `w` is the radius and the height is `±√(w² − k)`, so each sheet reaches its tip at `w = 0`.

The last factor of the middle expression is a cut. `(w ≠ 0 || k ≥ 0)` is 1 almost everywhere, and 0 on the one row where two sheets would otherwise be joined across the gap. Dividing by it leaves that row undefined, so nothing is drawn there.

### One slider, three surfaces

At `k = 1` the surface is a *hyperboloid of one sheet*, a single waisted tube. Lower `k` and the waist, of radius `√k`, narrows. At `k = 0` it closes to a point, and the surface is the double cone `x² + z² = y²`, its sides at 45°.

Below zero the cone tears apart into a *hyperboloid of two sheets*, two bowls facing away from each other. Their tips sit at heights `±√(−k)`, so at `k = −1` they are 2 apart. The cone is the asymptote of every member of the family: far from the axis each surface hugs it, whatever `k` is.

The one-sheet hyperboloid is also doubly ruled. Through every point of it pass two straight lines that lie wholly in the surface, although it curves in every direction.

### History

The ruling made the shape buildable. Vladimir Shukhov put up the first hyperboloid tower, a water tower for the All-Russian Exhibition at Nizhny Novgorod in 1896, as a lattice of straight steel bars. The cooling towers of power stations use the same shape for the same reason.

### Try

- Drag `k` slowly from 1 down to −1. The waist closes at 0 and the tube splits into two sheets.
- Press `Front` in the deck. The outline is the hyperbola `x² − y² = k`, and at `k = 0` it is a pair of crossing lines.
- Put a 2 in front of the first expression, `2·sqrt(max(k, 0) + (3v/π − 1.5)²)·cos u`. The circles become ellipses twice as wide in `x`, the elliptic hyperboloid of the textbooks.
- Set `Span of v (×π)` to 0.5 and only the lower half is drawn, with `w` from −1.5 to 0. Below zero that is one sheet alone.

### Read more

- [Hyperboloid](https://en.wikipedia.org/wiki/Hyperboloid)
- [Quadric](https://en.wikipedia.org/wiki/Quadric)
- [Ruled surface](https://en.wikipedia.org/wiki/Ruled_surface)
- [Hyperboloid structure](https://en.wikipedia.org/wiki/Hyperboloid_structure)