Example 1: The Möbius Strip
The Möbius strip is the simplest geometric shape which has only one surface and only one edge. It can be created by taking a strip of paper, giving it a half twist along its long axis, and then joining the two narrow ends together. The Möbius strip in 3 dimensions can be represented parametrically f(s,t) as follows:
Paul Bourke | |
Paul Bourke | |
M.C. Escher | |
M.C. Escher | |
M.C. Escher | |
Example 2: The Klein Bottle
Most containers have an inside and an outside, a Klein bottle is a closed surface with no interior and only one surface. It is not able to be constructed in 3 dimensions without intersecting surfaces. It can be realized in 4 dimensions. The classical representation is shown below. The Klein Bottle can be represented parametrically f(u, v) as follows:
Paul Bourke | |
Paul Bourke | |
Example 3: Apple
The Apple Surface can be represented parametrically f(u,v) as follows:
Paul Bourke | |
Paul Bourke | |
Example 4: The Triaxial Tritorus
The Triaxial Tritorus is defined parametrically as follows:
Paul Bourke | |
Paul Bourke | |
Paul Bourke | |