Parametrizing the Surface

For drawing Costa's surface, we will exploit first its symmetries. First, the vertical coordinate planes are symmetry planes, and the 180°rotation about the y=x axes becomes also a reflective surface symmetry. These together decompose the surfaces into eigth congruent pieces, one of which is represented by the first quadrant.

Within this quadrant, three points are of special importance: The origin will be a corner of the surface patch, 1 will be mapped to the planar end, and ∞ to one of the catenoid ends. To get a parametrization of the surface patch adapted to this geometry, we need to foliate the first quadrant by coordinate lines which look like polar coordinates near the ends. This can be done using

[Graphics:../Images/index_gr_19.gif]

This map maps an infinite parallel strip to the first quadrant so that the ends of the strip are mapped to 1 and ∞.

[Graphics:../Images/index_gr_20.gif]
[Graphics:../Images/index_gr_21.gif]

[Graphics:../Images/index_gr_22.gif]

Finally, we need to set up the coordinate lines of the strip so that the image grid looks pretty. This is best done manually:

[Graphics:../Images/index_gr_23.gif]

The negative values are mapped to the planar end, the positive to the catenoid end.

[Graphics:../Images/index_gr_24.gif]

This defines a step function, taking the wanted x-values near integral points. Similarly for the horizontal lines:

[Graphics:../Images/index_gr_25.gif]
[Graphics:../Images/index_gr_26.gif]

Now we produce the fundamental piece:

[Graphics:../Images/index_gr_27.gif]


Converted by Mathematica      January 2, 2000