Parallel and dual surfaces of cuspidal edges

Abstract

Abstract. We study parallel surfaces and dual surfaces of cuspidal edges. We give concrete forms of principal curvature and principal direction for cuspidal edges. Moreover, we define ridge points for cuspidal edges by using those. We clarify relations between singularities of parallel and dual surfaces and differential geometric properties of initial cuspidal edges.

Tubular surfaces of center curves on spacelike surfaces in Lorentz‐Minkowski 3‐space

“…Under the condition, the surface   at   (s 0 , 0 ) is locally diffeomorphic to the swallowtail, the curve d ± (s) at d ± (s 0 ) is locally diffeomorphic to C (2,3,4).…”

“…We remark that all of diffeomorphisms in the above assertions are diffeomorphism germs. We respectively call C(2, a (2, 3)-cusp (see Figure 5), C (2,3,4) Figure 8). Proof.…”

“…In differential geometry, curves and surfaces are formulated as the maps with one‐parameter and two‐parameter, respectively. Applying the singularity theory to the study of curves and surfaces has always been an interesting area for many geometers, these applications result in many interesting results in some fields such as computer vision and medical imaging.…”

“…2. By the definition of v = (v 0 , v 1 , v 2 ) ∈ R 3 1 and (s) = (x 0 (s), x 1 (s), x 2 (s)) ∈ R 3 1 . Hence, we have D D (s, v) = −(v 0 − x 0 (s)) 2 + (v 1 − x 1 (s)) 2 + (v 2 − x 2 (s)) 2 − 1 and…”

Focal Surfaces of Wave Fronts in the Euclidean 3-Space

“…Then κ = κ ν holds along the u-axis (cf. [36,37]). Moreover, we assume that κ = 0 on U in what follows, when we treat cuspidal edges.…”