Official websites use .gov
A
.gov website belongs to an official
government organization in the United States.
Secure .gov websites use HTTPS
A lock (
) or https:// means you've safely
connected to the .gov website. Share sensitive
information only on official, secure websites.
| Name | Co-dimension | Co-rank | Universal unfolding | Parametric Representation |
|---|---|---|---|---|
| Fold | 1 | 1 | x3 + ux |  |
| Cusp | 2 | 1 | x4 + ux2 + vx |  |
| Swallow tail | 3 | 1 | x5 + ux3 + vx2 + wx |  |
| Butterfly | 4 | 1 | x6 + ux4 + vx3 + wx2 + tx |  |
| Hyperbolic umbilic | 3 | 2 | x3 + y3 + uxy + vx + wy |  |
| Elliptic umbilic | 3 | 2 | x3 − xy2 + u(x2 + y2 ) + vx + wy |  |
| Parabolic umbilic | 4 | 2 | x2y + y4 + ux2 + vy2 + wx + ty |  |
