TOPICS
Search

Swallowtail Catastrophe

Swallowtail
SwallowtailCatastrophe2

A swallowtail catastrophe is a catastrophe which can occur for three control factors and one behavior axis. The swallowtail catastrophe is the universal unfolding of singularity f(x)=x^5 with codimension 3, i.e., in three unfolding parameters, and is of the form F(x,u,v,w)=x^5+ux^3+vx^2+wx. The equations

x=uv^2+3v^4
(1)
y=-2uv-4v^3
(2)
z=u
(3)

display such a catastrophe (von Seggern 1993, Nordstrand). The above surface uses u in [-2,2] and v in [-0.8,0.8].

After relabeling the coordinates as (X,Y,Z)=(z,y,x), the parametrization satisfies the implicit equation

256Z^3-128X^2Z^2+16X^4Z+144XY^2Z-4X^3Y^2-27Y^4=0.
(4)

The left side is the polynomial discriminant of t^4+Xt^2+Yt+Z with respect to t. Hauser calls the implicit surface the Taube surface, whose German name means "dove." Its real zero set also contains points absent from the real parametrized image, such as (X,Y,Z)=(2,0,1), for which the quartic is (t^2+1)^2.

Explore with Wolfram|Alpha

References

Hauser, H. "Gallery of Singular Algebraic Surfaces: Taube." https://homepage.univie.ac.at/herwig.hauser/gallery.html.Nordstrand, T. "Swallowtail." http://jalape.no/math/stltxt.htm.Sanns, W. Catastrophe Theory with Mathematica: A Geometric Approach. Germany: DAV, 2000.von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 94, 1993.

Referenced on Wolfram|Alpha

Swallowtail Catastrophe

Cite this as:

Weisstein, Eric W. "Swallowtail Catastrophe." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SwallowtailCatastrophe.html

Subject classifications