On the classification and bifurcation of multigerms of maps from surfaces to 3-space

Abstract

The A -classification of multigerm singularities is discussed, based on the theory of complete transversals. An A -classification of r-multigerms from the plane to 3-space of A -codimension ≤ 6 − r is carried out and the bifurcation geometry of these singularities analysed. This work has applications to the study of two-dimensional spatial motions, giving local models for the singularities which appear on general trajectories of rigid body motions from the plane to 3-space. In addition, our classification is extensive enough to give the full list of simple multigerm singularities from the plane to 3-space.

On geometry of the midlocus associated to a smooth curve in plane and space

“…Precisely, we give the condition for this map to be A-equivalent to B ± 2 , C ± 3 , F 4 , and C ± 4 with normal forms (x, y 2 , x 2 y ± y 5 ), (x, y 2 , xy 3 ± x 3 y), (x, y 2 , x 3 y + y 5 ), and (x, y 2 , xy 3 ± x 4 y) respectively. Recall that C ± 3 is 4-determined, and the others are 5-determined and for more details in this subject we refer the reader to ( [11,13]). Case 1 a 21 0 If a 21 0, then after suitable coordinates change in the target j 5 M can be transformed to j 5 M = (x, y 2 , a 21 x 2 y+ a 13 xy 3 + a 05 y 5 ).…”

On the Simplicity of Multigerms

“…v) When (n, p) = (2, 3), a cross-cap and two immersions or a quintuple point are not simple ( [7], [25]). Example 4.21 shows that simple multigerms h = {f, g} where f is a primitive monogerm and g is a prism on a Morse function or an immersion are exceptional.…”

Encomplexed Brown invariant of real algebraic surfaces in ℝP3

“…Remark 2.10. For a closer description of these singularities (and others of higher codimension) see Hobbs and Kirk [4], more specifically table 1. There B, K correspond to S ± k , H, E correspond to A 2 0 |A ± k , C correspond to (A 0 S 0 ) k , T to A 3 0 |A k and Q to A 4 0 .…”