Abstract
Developable surfaces, which are important objects of study, have attracted a lot of attention from many mathematicians. In this paper, we study the geometric properties of one-parameter developable surfaces associated with regular curves. According to singularity theory, the generic singularities of these developable surfaces are classified—they are swallowtails and cuspidal edges. In addition, we give some examples of developable surfaces which have symmetric singularity models.
Highlights
The study of developable surfaces has many practical applications
Since developable surfaces may be constructed by bending flat sheets, they are important in manufacturing objects from cardboard, plywood, and sheet metal
There is some literature about developable surfaces of space curves from the viewpoint of singularity theory [1,2]
Summary
The study of developable surfaces has many practical applications. There is much literature about developable surfaces, (see, e.g., [1,2,3,4,5]). The geometric invariants can characterize the contact between a space curve and a helix In this sense, the study of the singularities of developable surfaces is an interesting subject. At least to the best of our knowledge, there exists little literature concerning the singularities of one-parameter developable surfaces related to regular space curves in Euclidean space. We study this problem in the present paper. We define the one-parameter support functions on regular space curves, which can be used to study the geometric properties of one-parameter developable surfaces. We give some examples to illustrate the main results in this paper
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