We study ribbons of vanishing Gaussian curvature, i.e. flat ribbons, constructed along a curve in $\mathbb {R}^{3}$. In particular, we first investigate to which extent the ruled structure determines a flat ribbon: in other words, we ask whether for a given curve $\gamma$ and ruling angle (angle between the ruling line and the curve's tangent) there exists a well-defined flat ribbon. It turns out that the answer is positive only up to an initial condition, expressed by a choice of normal vector at a point. We then study the set of infinitely narrow flat ribbons along a fixed curve $\gamma$ in terms of energy. By extending a well-known formula for the bending energy of the rectifying developable, introduced in the literature by Sadowsky in 1930, we obtain an upper bound for the difference between the bending energies of two solutions of the initial value problem. We finally draw further conclusions under some additional assumptions on the ruling angle and the curve $\gamma$.

Recent improvements in robotic arms have increased their interest in many areas such as the industry and biomedical sectors. Path planning is an essential part of the robotic arm, since most automated factories seek to move things from one place to another with obstacles providing the shortest route. This paper presents a novel optimal path planning algorithm based on the 3D cubic Bézier curve with three shape parameters and its geometric properties and hierarchical clustering. The proposed method utilizes a feature vector which is obtained from curvature, torsion, and path length of candidate curves. A hierarchical clustering is applied to determine curve pairs. Then, a multi-objective function is used to determine the best curve pair, which gives the best curve for the robotic arm. Besides forming the optimal 3D cubic Bézier path, the optimal ruled and developable path surfaces are obtained. In addition to presenting theoretical results, this work also demonstrates the proposed method on several Kinova Gen3 robotic arm cases.

The classical notion of a two-dimensional develpable surface in Euclidean three-space is extended to the case of arbitrary dimension and codimension. A collection of characteristic properties is presented. The theorems are stated with the minimal possible integer smoothness. The main tool of the investigation is Cartan's moving frame method.