The parametric equation is not technically correct as the umbrella should include a handle. Hence it needs to be defined by its implicit form.
I've copied the old parametric equation below.
--Pfafrich 21:31, 15 January 2006 (UTC)
Doing some research, people refer to the map as the Whitney umbrella, which is what is given by the parametric equations (the point being that the Whitney umbrella is a model of a singularity of a map). I've thus included the parametric version and the implicit version, with annotation. Nbarth 16:07, 21 December 2006 (UTC)
-- In the implicit equation x and y have to be interchanged. This can be verified by substituting x,y.z using the parametric equations. I made the correction. —Preceding unsigned comment added by 18.104.22.168 (talk) 18:19, 19 November 2007 (UTC)
- Swapping x and y just performes an orientation reversing linear transformation on the ambient space. The Whitney umbrella point is an diffeomorphism invariant, i.e. you can take a local diffeomorphism (close to the origin) of the ambient space and still get a Whitney umbrella point. Why was the variable exchange needed? Were the axes labels wrong in the original picture? Declan Davis (talk) 19:25, 26 August 2008 (UTC)
- Yes, however they are non-local. The Whitney umbrella is a local singularity mapping an open neighbourhood of a point into an open neighbourhood. A transverse intersection maps a pair of open neighbourhood, into an open neighbourhood. --Salix alba (talk) 23:43, 26 August 2008 (UTC)