Free-form Surface Representation Research Articles

Additive manufacturing of non-planar layers with variable layer height

Abstract The freedom and potential gained by creating objects layer-by-layer using additive manufacturing technologies is offset by the limitations the 2.5D process holds. Currently, surface quality and mechanical properties of parts are limited by path planning algorithms. This study presents an algorithm for non-planar path planning with variable layer height, enabling the accurate representation of freeform surfaces and introducing the potential for increasing the parts’ mechanical properties by tailoring the layers to the load-case. By using mathematical functions to describe non-planar layers and calculating the amount of material to be extruded, layer thickness may vary inside the same layer, enabling transitions between different curved layers without introducing air gaps. In addition, means for selecting a beneficial nozzle geometry for non-planar additive manufacturing are presented. Finally, manufactured parts are evaluated, showing the benefits of the presented method. Specifically, it is shown that surface roughness can be reduced by 76% for curved top surfaces and indications for improved mechanical properties can be derived from performed tests.

Designing double freeform surfaces for point source and extended sources using a construction, iteration and optimization process

In this paper, a construction-iteration-optimization method is developed to design a double freeform surface lens for point sources as well as extended sources to achieve uniform illuminance. In the construction process, the initial double freeform surfaces are first calculated by the variable separation method with preset source-target ray mapping to obtain the maximum utilization ratio with Fresnel loss. Then, the double freeform surfaces are fitted into a fourth-order XY polynomial surface considering both the coordinates and surface normals of the data points, which ensures the smoothness of freeform surfaces. To reduce the relative standard deviation of the illuminance distribution, an iteration process is required to reconstruct the two freeform surfaces by turns, and this process can be repeated several times to further decrease the deviation of the rays. The system after the construction-iteration process can be taken as a good starting point for further optimization with optical design software. Although the construction-iteration-optimization process is for a point source at first, with the advantage of multi-parameter representation of freeform surfaces we can further optimize the parameters for an extended source by replacing the point source. Simulation results for illuminance uniformity are demonstrated to verify the effectiveness of this method. The energy utilization ratios are above 86.5% and 82.3% with Fresnel loss, while the relative standard deviations of the illuminance distribution are less than 0.036 and 0.063 for a point source and an extended source, respectively.

Vectorial aberrations of biconic surfaces.

For non-rotationally symmetric systems, spherical surfaces provide a non-sufficient effect in correcting aberrations. In anamorphic systems, biconic surfaces are used to provide different focal power in tangential and sagittal directions. The biconic surface shape is also used as the basic shape of the freeform surface representation. The different focal power in tangential and sagittal planes generates large field-constant astigmatism, which is decoupled from a coma generated by the biconic surface. In this paper, the biconic surface sag is expanded up to the fourth order and decomposed into the spherical part, the aspherical part, and the freeform part. The vectorial aberrations of the biconic surface are derived based on the extended nodal aberration theory. When biconic surfaces are used to obtain the initial setup of off-axis systems, it is beneficial due to the decoupling of astigmatism and coma. For instance, in the initial system design of reflective camera systems, biconic surfaces provide the possibility to obtain the nodal point in the center of the field of view, which is beneficial for systems with large field of view. This paper provides the insight of aberration analysis of biconic surfaces and the conversion of biconic surfaces into Zernike fringe freeform surfaces.

Zonal wavefront reconstruction of Shack–Hartmann and Hartmann patterns with hexagonal cells

In this article we will develop a method to integrate Shack–Hartmann and Hartmann pattern with hexagonal cells, using a polynomial representation (modal integration) over each hexagonal cell. Since each hexagonal has six sampling points, one at each vertex, instead of the typical four sampling points in square cells, it is possible to have a different representation of the wavefront in each cell, each with different aberration terms. The local curvatures and low order aberrations in each cell are calculated more accurately than for square cells. All the analytical functions over each hexagonal cell have a different unknown piston term, that is calculated with a method to be described here. As a result, wavefront retrieval and representation of freeform optical surfaces for some optical systems can be made, due to the calculation of aberrations in each hexagonal cell.

3D representation and CNC machining of 2D digital images

In this paper a new paradigm for CNC carving of digital images in the form of lookalike three-dimensional (3D) surfaces on wooden or metallic plaques is discussed. This work is focused on development of a single page windows based console application for conversion of a two-dimensional (2D) digital image into a 3D freeform surface representation in the form of a point cloud data and STL format. Subsequently, the 3D surface data is then used for generating an efficient toolpath data for 3-axis CNC finish machining using a ball end mill tool. The results from the developed algorithm are validated using a machining simulation in the virtual environment of an Open-GL based 3D graphical simulator ToolSim [1].

Review of optical freeform surface representation technique and its application

Modern advanced manufacturing and testing technologies allow the application of freeform optical elements. Compared with traditional spherical surfaces, an optical freeform surface has more degrees of freedom in optical design and provides substantially improved imaging performance. In freeform optics, the representation technique of a freeform surface has been a fundamental and key research topic in recent years. Moreover, it has a close relationship with other aspects of the design, manufacturing, testing, and application of optical freeform surfaces. Improvements in freeform surface representation techniques will make a significant contribution to the further development of freeform optics. We present a detailed review of the different types of optical freeform surface representation techniques and their applications and discuss their properties and differences. Additionally, we analyze the future trends of optical freeform surface representation techniques.

Conformal freeform surfaces

The conformality of freeform surfaces highly affects their rendering and tessellation results. To improve the conformality of freeform surfaces, a novel freeform surface representation named hierarchical freeform surfaces is presented in this paper. The conformality energy of hierarchical freeform surfaces is first formulated and its numerical approximation is then constructed using the composite Simpson’s rule. By constructing the parameterization of the initial freeform transformation using the Ricci flow method, the optimal freeform transformation is obtained by the Levenberg–Marquardt method, which is further interleaved with the freeform refinement procedure to generate a hierarchical freeform surface with bounded conformality deviations. Examples are given to show the performance of our algorithm for rendering and tessellation applications.

Five-axis tool path generation in CNC machining of [formula omitted]-spline surfaces

Because of its flexible topology and robust data structure, the T-spline surface has become the trend of free-form surfaces representation in the realm of CAD design, animation and CAE. Yet its application in manufacturing has not been fully explored. In this work, the possibility of direct tool path generation on the T-spline surface has been discussed. An improved space-filling curve (ISFC) tool path planning algorithm has been proposed to exploit the advantage of T-spline as a mathematical representation of free-form surfaces in CAM process, as well as to overcome its disadvantages such as irregular boundaries and holes in the pre-image. The turning problem in traditional SFC has been tackled using Hermite compensation curves. Finally, a prototype system has been developed to implement the proposed algorithm and actual machining has been conducted. The result shows the feasibility as well as the efficiency of the proposed method for T-spline surfaces tool path generation compared to commercial CAM system.

Tomography-Based Customized IOL Calculation Model

Purpose: To provide a mathematical calculation scheme for customized intraocular lens (IOL) design based on high resolution anterior segment optical coherence tomography (AS-OCT) of anterior eye segment and axial length data.Material and Methods: We use the corneal and anterior segment data from the high resolution AS-OCT and the axial length data from the IOLMaster to create a pseudophakic eye model. An inverse calculation algorithm for the IOL back surface optimization is introduced. We employ free form surface representation (bi-cubic spline) for the corneal and IOL surface. The merit of this strategy is demonstrated by comparing with a standard spherical model and quadratic function. Four models are calculated: (1) quadratic cornea + quadratic IOL; (2) spline cornea + quadratic IOL; (3) spline cornea + spline IOL; and (4) spherical cornea + spherical IOL. The IOL optimization process for the pseudophakic eye is performed by numerical ray-tracing method within a 6-mm zone. The spot diagram on the fovea (forward ray-tracing) and wavefront at the spectacle plane (backward ray-tracing) are compared for different models respectively.Results: The models with quadratic (1) or spline (3) surface representation showed superior image performance than the spherical model 4. The residual wavefront errors (peak to valley) of models 1, 2, and 3 are below one micron scale. Model 4 showed max wavefront error of about 15 µm peak to valley. However, the combination of quadratic best fit IOL with the free form cornea (model 2) showed one magnitude smaller wavefront error than the spherical representation of both surfaces (model 3). This results from higher order terms in cornea height profile.Conclusions: A four-surface eye model using a numerical ray-tracing method is proposed for customized IOL calculation. High resolution OCT data can be used as a sufficient base for a customized IOL characterization.

Design of a freeform varifocal panoramic optical system with specified annular center of field of view

A new zoom mechanism was proposed for the realization of a freeform varifocal panoramic annular lens (PAL) with a specified annular center of the field of view (FOV). The zooming effect was achieved through a rotation of the varifoal PAL around an optical axis, which is different from a conventional zooming method by moving lenses back and forth. This method solves the problem of FOV deviation from the target scope during the zooming process, since the optical axis was not taken as the zooming center of the FOV. The conical surface corresponding to a certain acceptance angle was specified as the annular center of the FOV, and it was adopted as the reference surface of zooming for the FOV. As an example, the design principle and optimization process of a freeform varifocal PAL was discussed in detail. The annular center of the FOV was specified at the acceptance angle of 90°. The absolute FOV in the direction of acceptance angles is relative to the specified annular center, with cosine deviation from ±20° at 0° rotational angle to ±10° at ±180° rotational angle on both sides around optical axis. An X-Y polynomial (XYP) was used for the representation of freeform surfaces for its simple form and convergence efficiency. The correction for irregular astigmatism and distortion and the position offset of an entrance pupil caused by an irregular aperture spherical aberration are also discussed. The results from the analysis of the modulus of the optical transfer function (MTF) and f-theta distortion show that the zooming method by a rotation of the varifocal freeform PAL is feasible.

Real-time simulation and visualization of robotic belt grinding processes

Real time simulation and visualization are important for robot programmers to verify and optimize the path planning for the robotic belt grinding process. A new free-form surface representation based on discrete surfel element is developed to facilitate the system implementation, which exploits the advantage of the new development of point-based rendering technology in computer graphics. A local process model is integrated to calculate the material removal rate by considering the local geometry information and non-uniform force distribution. The final surface grinding error is easy to be assessed and visualized for quality evaluation. The experiments show that the simulation error is below 15%, even for a non-uniform contact under stable cutting conditions.

Tunnel-free voxelisation of rational Bézier surfaces

To synthesize natural and artificial objects into a hybrid graphics scene represented by a set of voxels, voxelisation of geometric models is necessary. Rational parametric surfaces have been widely used in the representation of free-form surfaces. Voxelisation of these surfaces is therefore of great importance in the development of a voxel-based modeling system. A key issue is to develop a tunnel-free voxelisation algorithm for these continuous surfaces. In this paper, we propose such an algorithm for a rational Bezier surface. We derive the bound of the parametric steps to ensure that the voxelised rational Bezier surface is, by our algorithm, 6-tunnel-free, and we give the mathematical proof of this property. For efficient computation, we employ the forward difference technique in homogeneous form in the implementation of the algorithm. For more general applications, we show that voxelisation of a NURBS surface can be realised by first converting it into a piecewise rational Bezier surface and then voxelising each of the rational Bezier surfaces. We indicate the advantages carrying out this procedure.

Surface signatures: an orientation independent free-form surface representation scheme for the purpose of objects registration and matching

This paper introduces anew free-form surface representation scheme for the purpose of fast and accurate registration and matching. Accurate registration of surfaces is a common task in computer vision. The proposed representation scheme captures the surface curvature information (seen from certain points) and produces images, called "surface signatures," at these points. Matching signatures of different surfaces enables the recovery of the transformation parameters between these surfaces. We propose using template matching to compare the signature images. To enable partial matching, another criterion, the overlap ratio is used. This representation scheme can be used as a global representation of the surface as well as a local one and performs near real-time registration. We show that the signature representation can be used to recover scaling transformation as well as matching objects in 3D scenes in the presence of clutter and occlusion. Applications presented include: free-form object matching, multimodal medical volumes registration, and dental teeth reconstruction from intraoral images.

Interactive Virtual Tools for Manipulating NURBS Surfaces in a Virtual Environment

DN-Edit is a virtual environment developed to allow the manipulation of non-uniform rational b-spline (NURBS) surfaces using virtual shaping tools. NURBS have become the industry standard for representation of free-form curves and surfaces. The contribution of the work presented here is in the development of shaping tools which are used to operate directly on the NURBS data and change the shape of the surfaces in a virtual environment. These shaping tools allow surface manipulations to be made using methods that match the shaping of real malleable objects. The virtual shaping tools are three-dimensional shapes that are controlled through a six degree-of-freedom tracking system that converts user hand motions into computer input. The NURBS surface updates itself in real time due to the effect of the tools on the surface. The new shape of the surface is dependent on the position and orientation of the shaping tool relative to the surface. Constraint-based surface manipulation is used to obtain multiple point direct manipulation of NURBS surfaces. In addition, computational methods to allow the user to have direct control of the first derivative of the surface over an area are implemented. This application has been developed using the C2 software libraries and Iowa State University’s C2 surround screen virtual environment.

Discrete fairing and variational subdivision for freeform surface design

The representation of freeform surfaces by sufficiently refined polygonal meshes has become common in many geometric modeling applications where complicated objects have to be handled. While working with triangle meshes is flexible and efficient, there are difficulties arising prominently from the lack of infinitesimal smoothness and the prohibitive complexity of highly detailed 3Dmodels. In this paper we discuss the generation of fair triangle meshes which are optimal with respect to some discretized curvature energy functional. The key issues are the proper definition of discrete curvature, the smoothing of high resolution meshes by filter operators, and the efficient generation of optimal meshes by solving a sparse linear system that characterizes the global minimum of an energy functional. Results and techniques from differential geometry, variational surface design (fairing), and numerical analysis are combined to find efficient and robust algorithms that generate smooth meshes of arbitrary topology which interpolate or approximate a given set of data points.

Computing moments of objects enclosed by piecewise polynomial surfaces

Combining a polynomial free-form surface representation with Gauss' divergence theorem allows efficient and exact calculation of the moments of the enclosed objects. For example, for an cubic representation, volume, center of mass, and the inertia tensor can be computed in seconds even for complex objects with serval thousand patches while change due to local modification of the surface geometry can be computed in real-time as feedback for animation or design. Speed and simplicity of the approach allow solving the inverse problem of modeling to match prescribed moments.

C-curves: An extension of cubic curves

A linearly parametrized set of curves, named C-curves, is suggested with basis sin t, cos t, t, and 1. C-curves are an extension of cubic curves, they depend on a parameter α > 0, and their limiting case for α → 0 is a cubic curve. They can deal with free form curves and surfaces, and provide exact reproduction of circles and cylinders. So, they could be used to unify the representation and processing of both free and normal form curves and surfaces in engineering.

Constructing face octrees from voxel-based volume representations

Boolean operations between solids can be performed efficiently by operating their respective octree encodings. Face octrees belong to the group of octree-based representation models. They are approximate representations that are a good compromise between the need to save storage and the simplicity of the algorithms involved. They are also a good choice for the representation of smooth freeform surfaces. The construction of face octrees from voxel-based volume representations yields a more compact, smoother and further operable encoding. A method of performing such a conversion is presented. The method consists of two steps. First, a network of points that represents the volume data is extracted. The extraction is based on the geometrically deformed models technique. Then, the network of points is transformed into a face octree. In the transformation process, face octree nodes are compacted as much as possible while the volume data precision is preserved.

Cubicoids: modeling and visualization

The objective of this paper is to develop a new implicit representation of free-form surfaces which is well suited for modeling as well as for high-quality rendering purposes. The core of our approach is a scheme for constructing tangent plane continuous surfaces passing through an arbitrary set of points in R 3 according to some prescribed topology which is supposed to be given in terms of a piecewise triangular interpolant of the points. The constructed surface consists of algebraic patches of degree three and matches given normal directions at the data points. In contrast to an earlier scheme based on quadric patches it requires a significantly less complicated patch structure, is completely local, and involves a robust default choice for shape parameters which, in particular, supports selecting the “correct” zero sheets of the third-order patches.

Feature-based design and finite element mesh generation for functional surfaces

This paper describes an approach to feature-based design and feature-based mesh generation for multi-featured functional surfaces. Unlike standard free-form parametric surface representation where the parametric surface patch plays the key role in both surface design and finite element mesh generation, we propose an approach to these two tasks which proceeds from the level of a complete feature (for example, a pocket or channel). The result is a more direct method for modeling functional surface characteristics and a more efficient feature-based implementation of Delaunay surface triangulation.