# Paraboloid Family

Surface · z = f(x, y, t)

`z = x²/9 + s · y²/4`

[Open in the app](https://www.wavelace.com/app#p=160) · [This page](https://www.wavelace.com/presets/paraboloid-family)

### What it draws

The height is `x²/9 + s·y²/4` over a square 6 on a side, with `s` on its own slider. The first term is a parabola along `x` that reaches 1 at the edges `x = ±3`. The second is a parabola along `y`, scaled by `s`, which at `s = 1` reaches 2.25 at `y = ±3`.

Nothing moves on its own. The surface changes only when `s` does, and the one thing that decides its kind is the sign of `s`.

### Bowl, trough, saddle

For `s` above zero both parabolas open upward and the surface is an *elliptic paraboloid*, a bowl. Its level curves `x²/9 + s·y²/4 = c` are ellipses, and at `s = 1` the corners stand at 3.25.

At `s = 0` the second term vanishes. The height no longer depends on `y`, and every slice across the square is the same parabola: a *parabolic cylinder*, a trough.

For `s` below zero the parabola along `y` turns over and the surface is a *hyperbolic paraboloid*, a saddle. It rises along `x` and falls along `y`, and at `s = −1` the height is zero on the two lines `y = ±2x/3`.

This is the second derivative test drawn out. At the origin the determinant of the second derivatives is `(2/9)·(s/2) = s/9`. Positive means a minimum, zero means the test is silent, and negative means a saddle point.

### Try

- Drag `s` from 1 down to −1 and watch the bowl flatten into a trough at 0, then turn into a saddle.
- Set `s` to 0.4. Then `0.4/4 = 0.1` is close to `1/9 ≈ 0.11`, so the bowl is nearly round, a paraboloid of revolution.
- Replace `s` with `sin(t)`, as in `x²/9 + sin(t)·y²/4`. The surface passes through all three kinds every `2π ≈ 6.28` seconds at `Speed` 1.
- Press `Top` in the deck at `s` = −1. The colours split the square into two rising and two falling wedges, parted by the lines where the height is zero.

### Read more

- [Paraboloid](https://en.wikipedia.org/wiki/Paraboloid)
- [Parabolic cylinder](https://en.wikipedia.org/wiki/Parabolic_cylinder)
- [Second partial derivative test](https://en.wikipedia.org/wiki/Second_partial_derivative_test)
- [Quadric](https://en.wikipedia.org/wiki/Quadric)