Frame and direction mappings for surfaces in ℝ3

Abstract

We study frames in ℝ3 and mapping from a surface M in ℝ3 to the space of frames. We consider in detail mapping frames determined by a unit tangent principal or asymptotic direction field U and the normal field N. We obtain their generic local singularities as well as the generic singularities of the direction field itself. We show, for instance, that the cross-cap singularities of the principal frame map occur precisely at the intersection points of the parabolic and subparabilic curves of different colours. We study the images of the asymptotic and principal foliations on the unit sphere by their associated unit direction fields. We show that these curves are solutions of certain first order differential equations and point out a duality in the unit sphere between some of their configurations.

On k-folding map-germs and hidden symmetries of surfaces in the Euclidean 3-space

“…[27]) as they can be used to distinguish two shapes (surfaces) from each other and, in some cases, reconstruct the surface. X 3 4 : p is on the sub-parabolic curve associated to v; the principal map with value v at p has a beaks singularity at p [7]. W 3,1 4 : p is on the sub-parabolic curve associated to the other principal direction v ⊥ ; it is on the closure of the H 3 -curve; the frame map has a cross-cap singularity at p [7].…”

“…(c) If k = 5p and 4 k, then ϑ 5p = 0, ϑ 6p = 1, ϑ 7p = ϑ 2p , so γ p (t) = −a 66 /a 21 t 5 + ϑ 2p CndRm 5 /a 2 21 t 6 + O (7). The expressions γ sp , s = 2, 3, 4, are obtaining by substituting ϑ 2p by ϑ 2(sp) in γ p .…”

On k-folding map-germs and hidden symmetries of surfaces in the Euclidean 3-space

“…U 3 4 : p is an A * 2 -point (see [11,12]); it is on the closure of the H 3 -curve. X 3 4 : p is on the sub-parabolic curve associated to v; the principal map with value v at p has a beaks singularity at p ( [9]). W 3,1 4 : p is on the sub-parabolic curve associated to the other principal direction v ⊥ ; it is on the closure of the H 3 -curve; the frame map has a cross-cap singularity at p ( [9]).…”

“…W k,j 4 : p is a special point on the parabolic curve. W 3q,q 4 : p is on the sub-parabolic curve associated to v ⊥ ; it is on the closure of the H 3 -curve associated to v; the frame map has a cross-cap singularity at p ( [9]). X k 4 : p is on the intersection of the parabolic and sub-parabolic curves associated to v; the principal map with value v at p has a beaks singularity at p ( [9]).…”

Blaschke's asymptotic lines of surfaces in R3

“…The Gauss-Kronecker curvature and the affine mean curvature are defined, respectively, as By direct calculations using (14) we have…”