Talk:Whitney umbrella

Untitled

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.

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parametric equations

$x\quad =\quad uv$

$y\quad =\quad u$

$z\quad =\quad v^{2}$

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--Pfafrich 21:31, 15 January 2006 (UTC)

Doing some research, people refer to the map $R^{2}\to R^{3}$ 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 80.132.237.166 (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)

Stable Maps

Aren't transverse self intersections stable? The Whitney umbrella contains a line of transverse self intersections. Declan Davis (talk) 16:39, 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)

Is the Whitney umbrella a developable surface ? --DavidCary (talk) 03:12, 4 April 2012 (UTC)