G3 Shape Adjustable GHT-Bézier Developable Surfaces and Their Applications

In this article, we proposed a novel method for the construction of generalized hybrid trigonometric (GHT-Bézier) developable surfaces to tackle the issue of modeling and shape designing in engineering. The GHT-Bézier developable surface is obtained by using the duality principle between the points and planes with GHT-Bézier curve. With different shape control parameters in their domain, a class of GHT-Bézier developable surfaces can be established (such as enveloping developable GHT-Bézier surfaces, spine curve developable GHT-Bézier surfaces, geodesic interpolating surfaces for GHT-Bézier surface and developable GHT-Bézier canal surfaces), which possess many properties of GHT-Bézier surfaces. By changing the values of shape parameters the effect on the developable surface is obvious. In addition, some useful geometric properties of GHT-Bézier developable surface and the G1, G2 (Farin-Boehm and Beta) and G3 continuity conditions between any two GHT-Bézier developable surfaces are derived. Furthermore, various useful and representative numerical examples demonstrate the convenience and efficiency of the proposed method.

Non-Parametric Shape Design of Free-Form Shells Using Fairness Measures and Discrete Differential Geometry

A non-parametric approach is proposed for shape design of free-form shells discretized into triangular mesh. The discretized forms of curvatures are used for computing the fairness measures of the surface. The measures are defined as the area of the offset surface and the generalized form of the Gauss map. Gaussian curvature and mean curvature are computed using the angle defect and the cotangent formula, respectively, defined in the field of discrete differential geometry. Optimization problems are formulated for minimizing various fairness measures for shells with specified boundary conditions. A piecewise developable surface can be obtained without a priori assignment of the internal boundary. Effectiveness of the proposed method for generating various surface shapes is demonstrated in the numerical examples.

An overview of developable surfaces in geometric modeling

Background: A developable surface is a special ruled surface with vanishing Gaussian curvature. The study of developable surfaces is of interest in both academia and industry. The application of developable surfaces ranges from ship hulls, architecture to origami, clothes, and others, as they are suitable for the modeling of surfaces with materials that are not amenable to stretch like leather, paper, fiber, and sheet metal. Objective: We survey techniques and patents of developable surfaces in the field of geometric modeling. The theory, algorithms, and applications are discussed to provide a comprehensive summary for modeling developable surfaces. Methods: Prior methods that model smooth and discrete developable surfaces in diverse disciplines are collected and reviewed. In particular, our review focuses on C^2, C^1 and C^0 developable surfaces, which are driven by the problems and challenges in the industry. Results: Many papers and patents of developable surface modeling are classified in this review paper. It is found that remarkable developments and improvements have been achieved in both analytical computations and practical applications. Conclusion: Global piecewise-developable surfaces, exploration of shape space of developable surfaces, joint optimization of geometry and physics, and other fundamental problems should be further studied.

Deployable Telescopic Tubular Mechanisms With a Steerable Tongue Depressor Towards Self-Administered Oral Swab

Swabbing tests have proved to be an effective method of diagnosis for a wide range of diseases. Potential occupational health hazards and reliance on healthcare workers during traditional swabbing procedures can be mitigated by self-administered swabs. Hence, we report possible methods to apply closed kinematic chain theory to develop a self-administered viral swab to collect respiratory specimens. The proposed sensorized swab models utilizing hollow polypropylene tubes possess mechanical compliance, simple construction, and inexpensive components. In detail, the adaptation of the slider-crank mechanism combined with concepts of a deployable telescopic tubular mechanical system is explored through four different oral swab designs. A closed kinematic chain on suitable material to create a developable surface allows the translation of simple two-dimensional motion into more complex multi-dimensional motion. These foldable telescopic straws with multiple kirigami cuts minimize components involved in the system as the characteristics are built directly into the material. Further, it offers a possibility to include soft stretchable sensors for realtime performance monitoring. A variety of features were constructed and tested using the concepts above, including 1) tongue depressor and cough/gag reflex deflector; 2) changing the position and orientation of the oral swab when sample collection is in the process; 3) protective cover for the swabbing bud; 4) a combination of the features mentioned above.

Bridging the gap between rectifying developables and tangent developables: a family of developable surfaces associated with a space curve

There are two familiar constructions of a developable surface from a space curve. The tangent developable is a ruled surface for which the rulings are tangent to the curve at each point and relative to this surface the absolute value of the geodesic curvature κ g of the curve equals the curvature κ . The alternative construction is the rectifying developable. The geodesic curvature of the curve relative to any such surface vanishes. We show that there is a family of developable surfaces that can be generated from a curve, one surface for each function k that is defined on the curve and satisfies | k | ≤ κ , and that the geodesic curvature of the curve relative to each such constructed surface satisfies κ g = k .

Limits of Extramobile and Intramobile Motion of Cylindrical Developable Mechanisms

Mechanisms that can both deploy and provide motions to perform desired tasks offer a multifunctional advantage over traditional mechanisms. Developable mechanisms (DMs) are devices capable of conforming to a predetermined developable surface and deploying from that surface to achieve specific motions. This paper builds on the previously identified behaviors of extramobility and intramobility by introducing the terminology of extramobile and intramobile motions, which define the motion of developable mechanisms while interior and exterior to a developable surface. The limits of motion are identified using defined conditions. It is shown that the more difficult of these conditions to kinematically predict may be treated as a non-factor during the design of cylindrical developable mechanisms given certain assumptions. The impact of toggle positions for each case is discussed. Physical prototypes demonstrate the results.

Sweeping surfaces with Natural mate curve of a spatial curve in Euclidean 3-Space

In this paper, we introduce the notion of sweeping surfaces with Natural mate curve of a spatial curve in Euclidean 3-space E3 . We also show that the parametric curves on these surfaces are lines of curvature. Then, we derive the necessary and sufficient condition for the sweeping surface to become a developable ruled surface. In particular, we analyze the necessary and sufficient conditions when the resulting developable surface is a cylinder, cone or tangent surface. Finally, some representative curves are chosen to construct the corresponding developable surfaces which possessing these curves as lines of curvature.

Timelike Sweeping Surfaces According to Type-2 Bishop Frame in Minkowski 3–space

In this paper, we express timelike sweeping surfaces using rotation minimizing frames in Minkowski 3–Space E3 1 . Necessary and sufficient conditions for timelike sweeping surfaces to be developable ruled surfaces are derived. Using these, we analyze the conditions when the resulting timelike developable surface is a cylinder, cone or tangential surface.