Free-form Curves Research Articles

Extending the Hough transform to recognize and approximate space curves in 3D models

Feature curves are space curves identified by color or curvature variations in a shape, which are crucial for human perception (Biederman, 1995). Detecting these characteristic lines in 3D digital models becomes important for recognition and representation processes. For recognizing plane curves in images, the Hough transform (HT) provided a very good solution to the problem. It selects the best-fitting curve in a dictionary of families of curves through a voting procedure that makes it robust to noise and missing parts. Since 3D digital models are often obtained by scanning real objects and may have many defects, the HT has been extended to recognize and approximate space curves in 3D models that correspond to relevant featuresThis work overviews three HT-based different approaches for identifying and approximating spatial profiles of points extracted from point clouds or meshes. A first attempt at this extension involved projecting the spatial points onto the regression plane, thus reducing the problem to planar recognition and using families of plane curves. A second approach has been proposed to recognize spatial profiles that cannot be projected onto the regression plane, using two types of space curve families. Unfortunately, the main drawback of methods based on traditional HT is that it requires prior knowledge of which family of curves to look for.To overcome this limitation, a third method has been developed that provides a piecewise space curve approximation using specific parametric polynomial curves. Additionally, free-form curves that a parametric or implicit form cannot express can be represented using this technique.In the paper, we also analyze the pros and cons of the various approaches and how they managed and reduced the HT's computational cost, given the large number of parameters introduced when families of space curves are considered.

An Effective Method for Slicing Triangle Meshes Using a Freeform Curve

Slicing 3D polygonal meshes is a fundamental operation in various applications such as virtual surgery, garment simulation, and game development. Existing methods primarily slice meshes using either a single line or a set of line segments approximating a smooth curve. This paper introduces a novel approach to freely slice a triangle mesh using a freeform curve without discretizing it into line segments. The user draws a stroke on the screen, defining the desired cutting trajectory. Subsequently, a freeform curve approximating this stroke is generated and extended into a ruled surface in the user’s viewing direction. To efficiently compute intersections between the ruled surface and a triangle mesh, the Line–Surface Intersection (LSI) problem is broken down into two subproblems: Plane–Curve Intersection (PCI) followed by Line–Line Intersection (LLI). Intersection points are then connected to form polylines, effectively cutting the mesh into multiple submeshes. To ensure the solidity of the submeshes, cross-sections are generated by trimming the ruled surface along the polylines and merged with the corresponding submeshes. Our method empowers users to slice triangle meshes along arbitrary trajectories encompassing both straight and freely curved paths while preserving efficiency and accuracy. The effectiveness of the proposed approach is demonstrated through experimental results showing various examples of mesh slicing.

Sharp bend and large FSR ring resonator based on the free-form curves on a thin-film lithium niobate platform.

Sharp bends are crucial for large-scale and high-density photonics integration on thin-film lithium niobate platform. In this study, we demonstrate low-loss (<0.05 dB) and sharp bends (Reff = 30 µm) using free-form curves with a 200-nm-thick slab and a rib height of 200 nm on x-cut lithium niobate. Employing the same design method, we successfully realize a compact fully-etched ring resonator with a remarkably large free spectral range of 10.36 nm experimentally. Notably, the equivalent radius of the ring resonator is a mere 15 µm, with a loaded Q factor reaching 2.2 × 104.

Design of ultrahigh-Q silicon microring resonators based on free-form curves.

A design method for ultrahigh-Q microring resonators (MRRs) based on Bezier free-form curves was proposed and demonstrated. An MRR consisting of a specially designed 180° waveguide bend, a directional coupler, and two low-loss multi-mode strip waveguides was designed. The free-form curves were used to increase the degree of freedom in the design, shaping the waveguide bend with a gradient width and curvature. This design effectively reduced the propagation loss caused by the roughness of waveguide sidewalls and the mode mismatch loss caused by the excitation of high order modes. The small effective radius of only 20µm enabled the MRR to have a large free spectral range (FSR) and a compact and flexible structure. The MRR was manufactured using a standard process provided by foundry and measured to have an ultrahigh loaded Q factor of 1.86 × 106 and a FSR of about 1 nm.

Real-time arc length parameter-based integrated control strategy of contour error compensation for free-form curve CNC machining

Real-time arc length parameter-based integrated control strategy of contour error compensation for free-form curve CNC machining

Ultra-Compact Multimode Waveguide Routings Based on the Free-form Curves

Ultra-Compact Multimode Waveguide Routings Based on the Free-form Curves

Generalization of the Shape Uniqueness Theorem for Free-form Curves

Generalization of the Shape Uniqueness Theorem for Free-form Curves

Ultra-compact silicon multimode waveguide bends based on special curves for dual polarizations

Multimode waveguide bends (MWBs) with very compact sizes are the key building blocks in the applications of different mode-division multiplexing (MDM) systems. To further increase the transmission capacity, the silicon MWBs for dual polarizations are of particular interest considering the very distinct mode behaviors under different polarizations in the silicon waveguides. Few silicon MWBs suitable for both polarizations have been studied. In this paper, we analyze several dual-polarization-MWBs based on different bending curve functions. These special curve-based silicon MWBs have the advantages of easy fabrication and low loss compared with other structures based on subwavelength structures, such as gratings. A comparison is made between the free-form curve (FFC), Bezier curve, and Euler curve, which are used in the bending region instead of a conventional arc. The transmission spectra of the first three TE and TM modes in the silicon multimode waveguide with a core thickness of 340 nm are investigated. The simulation results indicate that, with the premise of having the same effective radius, which is only 10 μm in this paper, the six-mode MWB based on the FFC has the optimal performances, including an extremely low loss <0.052 dB and low crosstalk below −25.97 dB for all six modes in the wide band of 1500 to 1600 nm. The MWBs based on the Bezier and Euler curve have degraded performances in terms of the loss and crosstalk. The results of this paper provide an efficient design method of the polarization insensitive silicon MWBs, which may leverage research for establishing complicated optical transmission systems that incorporate both the MDM and polarization-DM technology.

Kernel-based construction operators for Boolean sum and ruled geometry

Boolean sum and ruling are two well-known construction operators for both parametric surfaces and trivariates. In many cases, the input freeform curves in ▪ or surfaces in ▪ are complex, and as a result, these construction operators might fail to build the parametric geometry so that it has a positive Jacobian throughout the domain.In this work, we focus on cases in which those constructors fail to build parametric geometries with a positive Jacobian throughout while the freeform input has a kernel point. We show that in the limit, for high enough degree raising or enough refinement, our construction scheme must succeed if a kernel exists. In practice, our experiments, on quadratic, cubic and quartic Bézier and B-spline curves and surfaces show that for a reasonable degree raising and/or refinement, the vast majority of construction examples are successful.

A simplified isogeometric approach for vibrational analysis of nanocomposite panels with a free-form curve

This paper investigates, the free vibration behavior of nanocomposite panels with a free-form curve slides on a straight line. It is assumed that the laminated composite panel is fabricated from two phases: polymeric matrix and graphene platelets (GPLs) as a reinforcement in a way that the weight fraction of GPLs varies from layer-to-layer. The equation of motions of the panel are presented using the curvilinear coordinate system in conjunction with the Kirchhoff–Love shell assumptions. In the next step, a novel and simplified procedure have been introduced to discretize the weak form equation by using B-spline basis functions. In other words, there is no need to calculate weight coefficients in the presented methodology. Consequently, the discretization procedure yields simple transformation equations which can be employed for a general singly curved panel with an arbitrary profile. The numerical results are validated for elliptic, parabolic and quartic panels with those generated by ABAQUS as well as cylindrical panels with those available in the open literature. After that, an extensive set of results are obtained as benchmark solutions to study the influences of geometrical and material parameters on the vibrational behavior of the nanocomposite panels.

Generalized Riemann-Liouville fractional Bézier curve and its applications in engineering surface

Modeling of free-form curves and surfaces is vital in the manufacturing industry and engineering. Curves and surfaces that have flexible shapes and adjustable lengths and sizes are a necessity to fulfill the needs of the manufacturing industry. Hence, researchers develop numerous aesthetic Bézier curves and surfaces to have such flexibility and adjustability. In this paper, the generalized Riemann–Liouville fractional Bézier curves and surfaces are proposed in the modeling of complex surfaces. The generalized Riemann–Liouville fractional Bézier curves and surfaces have two outstanding parameters: shape and fractional parameters. Shape parameters are utilized to change the shape of the curves and surfaces without altering the control points hence, adding flexibility in controlling the shape. While fractional parameters are utilized in curves and surfaces’ length and size adjustability. By adjusting the sizes of surfaces via fractional parameters, surfaces with optimal sizes can be modeled. In this paper, five types of engineering surfaces will be modeled, namely ruled surface, swept surface, swung surface, rotation surface, and coons patch. The geometric effects of the implementation of shape and fractional parameters to these engineering surfaces will also be analyzed, thus proving the generalized Riemann–Liouville fractional Bézier surfaces is an excellent tool in designing complex surfaces.

Quasi-Developable B-Spline Surface Design with Control Rulings

An intuitive design method is proposed for generating developable ruled B-spline surfaces from a sequence of straight line segments indicating the surface shape. The first and last line segments are enforced to be the head and tail ruling lines of the resulting surface while the interior lines are required to approximate rulings on the resulting surface as much as possible. This manner of developable surface design is conceptually similar to the popular way of the freeform curve and surface design in the CAD community, observing that a developable ruled surface is a single parameter family of straight lines. This new design mode of the developable surface also provides more flexibility than the widely employed way of developable surface design from two boundary curves of the surface. The problem is treated by numerical optimization methods with which a particular level of distance error is allowed. We thus provide an effective tool for creating surfaces with a high degree of developability when the input control rulings do not lie in exact developable surfaces. We consider this ability as the superiority over analytical methods in that it can deal with arbitrary design inputs and find practically useful results.

Cross-coupled control for contour tracking error of free-form curve based on fuzzy PID optimized by improved PSO algorithm

Aiming at the free-form curve contour tracking error problem of CNC machining, a cross-coupled controller based on Fuzzy PID using an improved PSO algorithm is proposed. First, the exact formulas of the linear and circular arc contour errors under a plane trajectory are derived, and an estimation method for the free curve profile error under a plane trajectory based on the principle of approximate approximation of preferred points is designed. Second, a cross-coupled controller based on Fuzzy PID using an improved PSO algorithm is achieved. An improved PSO algorithm, whose learning factor is adjusted with the inertia weight, is used to optimize the controller parameters and achieve contour tracking of the free curve. Finally, a two-dimensional biaxial experimental platform is used to test the proposed method. The results demonstrate that the optimal gain parameters obtained by the proposed strategy can significantly improve contour control accuracy in biaxial contour tracking tasks. The proposed contour error control scheme can achieve not only a nearly perfect contouring error estimation, but also an obvious promotion of contouring accuracy.

Ultra-sharp silicon multimode waveguide bends based on double free-form curves

Mode-division multiplexing (MDM) can greatly improve the capacity of information transmission. The multimode waveguide bend (MWB) with small size and high performance is of great significance for the on-chip MDM integrated system. In this paper, an MWB with high performance based on double free-form curves (DFFCs) is proposed and realized. The DFFC is a combination of a series of arcs optimized by the inverse design method. The fabrication of this MWB only needs one-step lithography and plasma etching and has a large fabrication tolerance. MWBs with effective radii of 6 μm and 10 μm are designed to support three modes and four modes, respectively. The proposed method gives the best overall performance considering both the effective bending radius and the transmission efficiency. The fabricated MWB with four mode channels has low excess losses and crosstalks below − 21 dB in the wavelength range from 1520 to 1580 nm. It is expected that this design can play an important role in promoting the dense integration of multimode transmission systems.

A quadratic trigonometric B-Spline as an alternate to cubic B-spline

The idea of the quadratic trigonometric spline (QTS) for the curve modeling approach inspired this paper using a quadratic trigonometric function presented in it. By creating B-spline-like local support basis functions, a competent freeform curve design approach is created. Local support, unity partition, and positivity are satisfied by the built-in basis functions. These generated local support basis functions give the freeform designed curve technique geometric features like convex hull and variation diminishing. The shape features of point tension, interval tension, and global tension are also observed by employing data point repetitions in a freeform curve scheme. A converted Bernstein Bézier form is used to display the non-uniform freeform curve for consideration in the proposed approach. The recommended method is C2 as a benefit. The proposed method is an addition to the existing family of spline procedures, and it gives a superior quadratic equivalent to the cubic method.

Relevant integrals of NURBS and its application in hull line element design

NURBS are a powerful mathematical tool for representing free-form curves and surfaces and have been the international industrial standards for the representation, design, data exchange of geometric information. In this paper, starting from the NURBS matrix form, a piecewise-integration is used to derive the formula of relevant integrals of NURBS curve based on the properties of NURBS. A benefit of matrix representation for NURBS is that the explicit mathematical expression of each curve segment is obtained. The area and perimeter of a circle and an ellipse are evaluated from the matrix representation for NURBS. The integrand function from the matrix is solved by composite Newton-Cotes formula and composite Gauss-Legendre formula respectively and the relative error is evaluated with the analytical solution. Results show that three-point composite Gauss-Legendre rule has fast convergence, high accuracy and low evaluation cost. Subsequently, the formula of relevant integrals of NURBS curve is applied to the hull line elements. Taking the hull sectional area curve (SAC) as an example, the evaluation of prismatic coefficient (CP) and longitudinal center of buoyancy (LCB) is carried out; taking the design waterline (DWL) as an example, the evaluation of waterplane coefficient (CW) and transverse center of floation (Xf) is carried out; taking the middle cross section as an example, the evaluation of midship section area coefficient (CM) is carried out. Finally, taking the revolution as an example, the surface area and volume are evaluated. Through these real ship examples, the effectiveness and universality of the engineering application in hull line element design are verified.

An Analytical Study of the Latest Trends of Free-Form Molds

With the development of technology, the number of free-form structures—as well as their value—is increasing. In order to construct such free-form structures, a number of studies are being conducted on free-form molds from multifaceted perspectives. However, it is difficult to identify the progress of studies related to free-form molds, as the scope of the studies is redundant or similar in many cases. Therefore, the current study focused on the identification of the trends of preceding studies on free-form molds using the PRISMA technique. The study classified the studies into three topics in order to identify the trends: ‘free-form curve fabrication technology’, ‘free-form mold fabrication technology’, and the ‘analysis of free-form panel forms.’ Each topic was further categorized into two tiers for more in-depth analysis. The whole process was adopted in order to suggest the trends of studies on free-form molds. The findings are expected to be used to provide fundamental data for future studies on free-form molds, and to set the directions for new studies.

Detection and computation of conservative kernels of models consisting of freeform curves and surfaces, using inequality constraints

We present an algorithm to compute a tight-as-needed conservative approximation of the kernel domain of freeform curves in R2 and freeform surfaces in R3. Inequality constraints to detect the interior of the kernel domain are formulated as multivariates, and solved with a subdivision-based approach to find the domains in R2 or R3 that satisfy the inequalities and are in the kernel. The convex hull of the computed domains is also included in the kernel, and adopted as the approximated kernel domain. We can apply the presented algorithm to detect the kernel domain of not only C1 continuous closed regular curves and surfaces, but also the kernel domains of multiple piecewise C1 continuous regular freeform curves and surfaces. Further, the presented algorithm can be applied to find the gamma-kernel as well as the kernel domain of open curves and surfaces, under some assumptions. We demonstrate our experimental result using various freeform curves and surfaces, and compare it with the kernel computation algorithm presented in Elber et al. (2006).

A Novel Generalization of Bézier-like Curves and Surfaces with Shape Parameters

Bézier curves and surfaces with shape parameters have received more attention in the field of engineering and technology in recent years because of their useful geometric properties as compared to classical Bézier curves, as well as traditional Bernstein basis functions. In this study, the generalized Bézier-like curves (gBC) are constructed based on new generalized Bernstein-like basis functions (gBBF) with two shape parameters. The geometric properties of both gBBF and gBC are studied, and it is found that they are similar to the classical Bernstein basis and Bézier curve, respectively. Some free form curves can be modeled using the proposed gBC and surfaces as the applications.

Machining of elliptical curve profile for application in manufacturing industries

Manufacture of products with freeform curves is the need of the day in manufacturing industries like turbine blades, automobile bodies, aerofoils etc. Such profiles are usually manufactured on high precision machines. High precision (CNC) machines are required to enhance product quality at a reasonable cost. CNC programs with preparatory codes (G01, G02, or G03) provide automation, but the machines are restricted to the ability of linear and circular interpolators. Hence, due to the unavailability of appropriate preparatory function, machining of these curved shaped components causes difficulty on CNC machines. The CAD / CAM software divides the profile into the numbers of small linear segments. The transmission of the linear segments from software to CNC machine is subjected to errors. It results in noise and lost data. The discontinuity of the freeform toolpath deteriorates the surface quality and shape accuracy due to linear segments. To enhance the shape accuracy and surface quality, the numbers of linear segments are to be increased. Each linear segment to be machined is adding a single block in the part program. It will increase the size of the part program and program memory size at the cost of the surface quality. To achieve better surface quality and shape accuracy, MACRO programming is required. In the present work, the freeform regular axis-symmetrical profile (i.e. ellipse) is machined on mild steel (MS) and aluminum material using NURBS as well as Hermite interpolators. The curved profiles machined by MACRO and conventional programming methods are examined to measure surface quality using surface roughness tester (i.e., Mitutoyo SJ-210®), while toolpath deviation is examined with the actual profile plotted with CAD software (i.e., AutoCAD®). Radial and chordal errors are the least in the case of NURBS interpolation compared to Hermite interpolation. A better surface finish is observed while the elliptical profile is machined on mild steel and aluminium with MACRO programming