In this study, we present a novel method for modeling the canopy surface of an umbrella. Our approach involves representing the area between the ribs on the canopy as a trimmed bilinear patch. Furthermore, we conduct an in-depth exploration of various differential geometric properties of the umbrella surface. We introduce a method for unfolding the canopy surface onto a plane, which serves as a valuable technique for fabricating a cardboard template to accurately cut canopy fabrics. To validate the effectiveness of our geometric modeling method, we apply it to several umbrella models, showcasing its practical application and benefits.

We present a novel approach to generate developable strips along a space curve. The key idea of the new method is to use the rotation angle between the Frenet frame of the input space curve, and its Darboux frame of the curve on the resulting developable strip as a free design parameter, thereby revolving the strip around the tangential axis of the input space curve. This angle is not restricted to be constant but it can be any differentiable function defined on the curve, thereby creating a large design space of developable strips that share a common directrix curve. The range of possibilities for choosing the rotation angle is diverse, encompassing constant angles, linearly varying angles, sinusoidal patterns, and even solutions derived from initial value problems involving ordinary differential equations. This enables the potential of the proposed method to be used for a wide range of practical applications, spanning fields such as architectural design, industrial design, and papercraft modeling. In our computational and physical examples, we demonstrate the flexibility of the method by constructing, among others, toroidal and helical windmill blades for papercraft models, curved foldings, triply orthogonal structures, and developable strips featuring a log-aesthetic directrix curve.

This paper presents a robotic 3D printer: a robot system that can assemble toy brick sculptures from their 3D CAD models. In this system, a 3D CAD model is automatically converted to a block model consisting of primitive toy blocks. Then an assembly plan of the block model is automatically generated, if feasible. According to the plan, an industrial robot assembles a brick sculpture layer by layer from bottom to top. We demonstrate successful assembly of several brick sculptures.

Current CAD modeling techniques enable the design of objects with aesthetically pleasing smooth freeform surfaces. However, the fabrication of these freeform shapes remains challenging. Our novel method uses orthogonal principal strips to fabricate objects whose boundary consists of freeform surfaces. This approach not only lends an artistic touch to the appearance of objects, but also provides directions for reinforcement, as the surface is mostly bent along the lines of curvature. Moreover, it is unnecessary to adjust the bending of these orthogonal strips during the construction process, which automatically reforms the design shape as if it is memorized, provided the strips possess bending rigidity. Our method relies on semi-isometric mapping, which preserves the length of boundary curves, and approximates angles between boundary curves under local minimization. Applications include the fabrication of paper and sheet metal craft, and architectural models using plastic plates. We applied our technique to several freeform objects to demonstrate the effectiveness of our algorithms.

This paper presents a geometric algorithm for the generation of uniform cubic B-spline curves interpolating a sequence of data points under tangent and curvature vectors constraints. To satisfy these constraints, knot insertion is used to generate additional control points which are progressively repositioned using corresponding geometric rules. Compared to existing schemes, our approach is capable of handling plane as well as space curves, has local control, and avoids the solution of the typical linear system. The effectiveness of the proposed algorithm is illustrated through several comparative examples. Applications of the method in NC machining and shape design are also outlined.

We present an unstructured triangular mesh generation algorithm that approximates a set of mutually non-intersecting simple trimmed rational B-spline surface patches within a user specified geometric tolerance. The proposed method uses numerically robust interval geometric representations/computations and also addresses the problem of topological consistency (homeomorphism) between the exact geometry and its approximation. Those are among the most important outstanding issues in geometry approximation problems. Our surface tessellation algorithm is based on the unstructured Delaunay mesh approach which leads to an efficient adaptive triangulation. A robust decision criterion is utilized to prevent possible failures in the conventional Delaunay triangulation. To satisfy the prescribed geometric tolerance, an adaptive node insertion algorithm is employed. Unstructured triangular meshes for free-form surfaces frequently involve triangles with high aspect ratio and accordingly, result in ill-conditioned meshing. Our proposed algorithm constructs 2D triangulation domains which sufficiently preserve the shape of triangles when mapped into 2D space and furthermore, the algorithm provides an efficient method that explicitly controls the aspect ratio of the triangular elements.

We present an efficient and reliable method for computing the unit-in-the-last-place (ulp) of a double-precision floating-point number, taking advantage of the standard binary representation for floatingpoint numbers defined by IEEE Std 754-1985. The ulp is necessary to perform software rounding for robust rounded-interval arithmetic (RIA) operations. Hardware rounding, using two of the standard rounding modes defined by IEEE-754, may be more efficient. RIA has been used to produce robust software systems for the solution of systems of nonlinear equations, interrogation of geometric and differential properties of curves and surfaces, curve and surface intersections, and solid modeling.