In topology, a smooth deformation from one surface in an other is called homotopy.The parameterization of this homotopy can be found on wolfram's page.
The Boy Surface has been found 1901 by Werner Boy. One obtains a topologicaly equivalent surface by attaching a Möbius strip to a disk along its boundary. Like the Möbius strip this surface is nonorientable.
This beautiful parameterization, based on complex numbers, was discovered in the late 1980s by Rob Kusner and Robert Bryant. In this post the boy surface has already been mentioned on eat-a-bug.
On this blog I documented some of my findings, ideas, and projects from the years 2009-2013. Its a private and non-commercial website, and the content isn´t updated any more, so please forgive broken links and outdated information. I keep it as a personal archive.