Surfaces
This applet takes a closed a polygon
in R2, with pairwise identification of the edges,
and returns a closed surface in standard form, specifying its orientability,
genus, and Euler characteristic.
About the applet
I am not sure that I have
tested this program thoroughly, and what is more - I think it's normal
if there are some bugs. Anyway, it seems to work. If someone finds a bug,
please feel free to tell me on inikolov@lynx.neu.edu
.
Here are some points on the applet:
-
Please,
enter the sides in the format: 1, 2, -1, 2...
-
Normally
in the final result the labels on the picture will have nothing to do with
your labels.
-
The applet will be working only if a correct closed (without any boundary) surface is entered. This is valid only if all of the sides entered are pairwise identified. e.g. if you enter '1' as a side of the polygon, you must enter once again (exactly once) '1' or '-1'.
-
Note that
in the result S stands for S2, P stands
for RP2 and T stands for T2
-
Check Show
if you want to see the calculation step by step.
-
The blue
labels are the vertices and one can see them only if Show is checked.
-
In the
final drawing the yellow passages are toruses and the blue - projective
planes.
-
The algorithm
for calculating is a six step one. The Euler characteristic is calculated
by the formula X = 2(1-g) for orientable and 1-h for non-orientable surfaces.
A description of the algorithm is to be placed here.
The source.