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.Symmetry 2019, 11, 108 2 of 15 surfaces are sections of one-parameter developable surfaces. We also define the one-parameter support functions on regular space curves, which can be used to study the geometric properties of one-parameter developable surfaces. In fact, one-parameter developable surfaces are the discriminant sets of these functions. The main result, Theorem 2, shows that the singularities of developable surfaces are A k -singularities (k = 2, 3) of these functions.The organization of this paper is as follows: We review the concepts of ruled surfaces in Euclidean space in Section 2. In Section 3, the one-parameter developable surfaces of a space curve are defined, and we obtain two geometric invariants of the curve. We also get singularities of one-parameter developable surfaces (Theorem 1), and Theorem 2 gives the classification of these singularities in this section. The preparations for the proof of Theorem 2 are in Sections 4 and 5. In the last section, we give some examples to illustrate the main results in this paper.
Basic NotationLet R 3 be 3-dimensional Euclidean space and x = (x 1 , x 2 , x 3 ), y = (y 1 , y 2 , y 3 ) ∈ R 3 . We denote their standard inner product by x · y, and the norm of x is denoted by x . Let γ : I → R 3 be a curve and the tangent vector respect to t isγ(t) = dγ/dt(t). The arc-length is s(t) = t t 0