Abstract
We study the singularity on principal normal and binormal surfaces generated by smooth curves with singular points in the Euclidean 3-space. We discover the existence of singular points on such binormal surfaces and study these singularities by the method of singularity theory. By using structure functions, we can characterize the ruled surface generated by special curves.
Highlights
As the easiest parameterized surfaces, ruled surfaces are widely used in project practices, architecture, and computer-aided design [1,2]
From [3,4,5,6,7], we know that the generic singularities of a developable surface are the cuspidal edge, the swallowtail, and the cuspidal cross-cap
For singular curves, the situation is different, and we will study the character of the singular points on such a binormal surface
Summary
As the easiest parameterized surfaces, ruled surfaces are widely used in project practices, architecture, and computer-aided design [1,2]. We know that the cross-cap is the only singular point of existence on the principal normal surface of regular space curves (see [8,9]). In [12], Liu et al defined structure functions, which are invariants of non-developable surfaces. They used these functions to characterize the properties of surfaces. In. Sections 3 and 4, we give the notations of the principal normal and binormal surfaces of a Frenet-type framed base curve in Euclidean 3-space and investigate the character of singular points on these surfaces.
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