Characterizing envelopes of moving rotational cones and applications in CNC machining

Motivated by applications in CNC machining, we provide a characterization of surfaces which are enveloped by a one-parametric family of congruent rotational cones. As limit cases, we also address ruled surfaces and their offsets. The characterizations are higher order nonlinear PDEs generalizing the ones by Gauss and Monge for developable surfaces and ruled surfaces, respectively. The derivation includes results on local approximations of a surface by cones of revolution, which are expressed by contact order in the space of planes. To this purpose, the isotropic model of Laguerre geometry is used as there rotational cones correspond to curves (isotropic circles) and higher order contact is computed with respect to the image of the input surface in the isotropic model. Therefore, one studies curve-surface contact that is conceptually simpler than the surface-surface case. We show that, in a generic case, there exist at most six positions of a fixed rotational cone that have third order contact with the input surface. These results are themselves of interest in geometric computing, for example in cutter selection and positioning for flank CNC machining.RYC-2017-2264

Automatic fitting of conical envelopes to free-form surfaces for flank CNC machining

We propose a new algorithm to detect patches of free-form surfaces that can be well approximated by envelopes of a rotational cone under a rigid body motion. These conical envelopes are a preferable choice from the manufacturing point of view as they are, by-definition, manufacturable by computer numerically controlled (CNC) machining using the efficient flank (peripheral) method with standard conical tools. Our geometric approach exploits multi-valued vector fields that consist of vectors in which the point-surface distance changes linearly. Integrating such vector fields gives rise to a family of integral curves, and, among them, linear segments that further serve as conical axes are quickly extracted. The lines that additionally admit tangential motion of the associated cone along the reference geometry form a set of candidate lines that are sequentially clustered and ordered to initialize motions of a rigid truncated cone. We validate our method by applying it on synthetic examples with exact envelopes, recovering correctly the exact solutions, and by testing it on several benchmark industrial datasets, detecting manufacturable conical envelope patches within fine tolerances

FREE-FORM TOOLS DESIGN AND FABRICATION FOR FLANK SUPER ABRASIVE MACHINING (FSAM) NON DEVELOPABLE SURFACES

Manufacturing improvements are becoming a real need in industry. In order to satisfy these industrial requirements, they should be targeted in two different directions: new manufacturing processes and surface optimization through algorithms. On the one hand, Super Abrasive Machining (SAM) is presented as a new manufacturing process combining benefits from milling and grinding technologies. On the other hand, there is a tendency to manufacture non developable surfaces by flank milling and to achieve final dimensional and roughness requirements, by calculating mathematically-optimized tool trajectories. This work presents a design and manufacturing of a free form tool to be used for the manufacturing of a complex surface through Flank SAM (FSAM). Based on the tool requirements, it will cover the following stages: tool geometry design, tool core manufacturing, and electroplating for final abrasive tool generation

Augmented graphic thinking in Geometry: developable architectural surfaces in experimental pavilions

We will analyze how the incorporation of digital manufacturing in our schools is motivating a deep reflection about the need to be familiar with both the foundations of geometry as well as with more advanced knowledge. The reinterpretation of our inherited graphic design discipline in light of current digital tools can open up new fields of study and work, such as is occurring in the field of developable surfaces, warped surfaces and many others. In addition, through non-linear graphic processes and digital tools of parametric design, we can arrive at an "expanded graphic thinking" that we can place at the service of production and morphological research. Thus, the old descriptive geometry - geometry based on graphics – comes to serve a cybernetically enlarged mind. We will present four experimental pavilions resulting from several workshops on geometry and digital manufacturing carried out in collaboration between the University of Seville and several Ibero-American Universities. Based on the deep geometric knowledge of developable helical surfaces and surfaces of equal slope, a guided exercise is proposed to approach the design, manufacturing and assembly phases of these architectural installations on a real scale

Approximating a Target Surface with 1-DOF Rigid Origami

We develop some design examples for approximating a target surface at the final rigidly folded state of a developable quadrilateral creased paper, which is folded with a 1-DOF rigid folding motion from the planar state. The final rigidly folded state is reached due to the clashing of panels. Now we can approximate some specific types of non-developable surfaces, but we do not yet fully understand how to approximate an arbitrary surface with a developable creased paper that has limited DOFs. Our designs might have applications in areas related to the formation of a shell structure from a planar region

Surface discretisation with rectifying strips on Geodesics

The use of geodesic curves of surfaces has enormous potential in architecture due to their multiple properties and easy geometric control using digital graphic tools. Among their numerous properties, the geodesic curves of a surface are the paths along which straight strips can be placed tangentially to the surface. On this basis, a graphical method is proposed to discretize surfaces using straight strips, which optimizes material consumption since rectangular straight strips take advantage of 100% of the material in the cutting process. The contribution of the article consists of presenting the geometric constraints that characterize this type of panelling from the idea of “rectifying surface”, considering the material inextensible. Experimental prototypes that have been part of the research are also described and the final theoretical results are presented

Geometry, mechanics and actuation of intrinsically curved folds

We combine theory and experiments to explore the kinematics and actuation of intrinsically curved folds (ICFs) in otherwise developable shells. Unlike origami folds, ICFs are not bending isometries of flat sheets, but arise via non-isometric processes (growth/moulding) or by joining sheets along curved boundaries. Experimentally, we implement both, first making joined ICFs from paper, then fabricating flat liquid crystal elastomer (LCE) sheets that morph into ICFs upon heating/swelling via programmed metric changes. Theoretically, an ICF's intrinsic geometry is defined by the geodesic curvatures on either side, $\kappa_{g_i}$. Given these, and a target 3D fold-line, one can construct the entire surface isometrically, and compute the bending energy. This construction shows ICFs are bending mechanisms, with a continuous family of isometries trading fold angle against fold-line curvature. In ICFs with symmetric $\kappa_{g_i}$, straightening the fold-line culminates in a fully-folded flat state that is deployable but weak, while asymmetric ICFs ultimately lock with a mechanically strong finite-angle. When unloaded, freely-hinged ICFs simply adopt the (thickness $t$ independent) isometry that minimizes the bend energy. In contrast, in LCE ICFs a competition between flank and ridge selects a ridge curvature that, unusually, scales as $t^{-1/7}$. Finally, we demonstrate how multiple ICFs can be combined in one LCE sheet, to create a versatile stretch-strong gripper that lifts $\sim$40x its own weight.Comment: The supplemental movies are available at https://drive.google.com/drive/folders/1CR5TdbZNhveHiDYt0_a20O7_nQYS6xZ