A monoid surface M is a surface of degree d with only isolated singularities and which has a singularity of multiplicity d-1. If we place this singularity at the origin O, then M has an affine equation of the form: M = f_d + f_{d-1}, where the f_i are homogeneous polynomials of degree i.
It is clear that monoid surfaces are very special. E.g., it is easy to see that any monoid surface is rational, which makes it potentially useful for applications in geometric design.
Moreover, the very difficult question on the maximum number μ(d) of singularities on a surface of degree d which was mentioned in door no. 06 can be answered completely for monoid surfaces: μ(d)=d*(d-1)/2+1. This result was known to Rohn around 1900 at least for the case of quartics. In general, we first found it in a paper of P.H. Johansen, M. Loberg, R. Piene (2006, to appear in the Compass II Proceedings) on (real) monoid hypersurfaces.
The proof for one implication is not difficult and can be found in the paper mentioned above. First, one notes that a point (p0:p1:p2:p3) other than O is singular if and only if the plane curves fd and fd-1 intersect with multiplicity at least 2 at (p1:p2:p3). By Bezout's theorem, this can only happen at d*(d-1)/2 points.
Conversely, the authors of the paper give the construction which our film illustrates to show that this number is attained. In the film, f_5 is the black curve which is a deformation of a regular pentagon, and f_4 is the product of the two cirles. In the film one can see that the 10 singularities occur when the deformed pentagon touches the two circles in 10 points.
This film was made by Oliver Labs using surfex.
No comments:
Post a Comment