Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Transformations involving singular points

  1. Nov 6, 2008 #1
    Can a triangle be smoothly transformed to a circle?
     
  2. jcsd
  3. Nov 16, 2008 #2
    Either this question is too easy, too dumb or too hard. I'd just like an answer. I've found two two possibly relevant theorems:

    Hironaka: Every analytic space Y admits a resolution of singularities: there is a smooth manifold X and a proper map f:X->Y such that f is an isomorphism except over singular points.

    Castelnuovo, Enriques: Every singular surface has a unique minimal resolution of singularities.

    What exactly is a "minimal resolution"? It suggests to me that the singular points of a triangle cannot be "eliminated" by a smooth transformation, but only arbitrarily "minimized". Is this correct?
     
    Last edited: Nov 16, 2008
  4. Nov 20, 2008 #3
    People are reading, but not answering. I'll give a very simple special case example of how I think a singular point might be eliminated, not "minimally resolved" by means of a topological transformation:

    Consider an equalateral triangle circumscribed by a circle such that the three singular points lie on the circle. If the triangle is transformed to coincide with the circle (each line segment is transformed into a 2pi/3 radian arc), the singular points are eliminated.

    Can this idea be applied to the more general case of functions that are analytic almost everywhere, except for singular points?
     
    Last edited: Nov 20, 2008
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Transformations involving singular points
  1. About Singular Simplex (Replies: 3)

  2. Singular vector fields (Replies: 0)

Loading...