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Design for Manufacturing Using B-Spline Developable Surfaces
Abstract
A developable surface can be formed by bending or rolling a planar surface without stretching or tearing; in other words, it can be developed or unrolled isometrically onto a plane. Developable surfaces are widely used in the manufacture of items that use materials that are not amenable to stretching such as the formation of ducts, shoes, clothing and automobile parts including upholstery and body panels (Frey & Bindschadler 1993). Designing a ship hull entirely of developable surfaces would allow production of the hull using only rolling or bending. Heat treatment would only be required for removal of distortion, thus greatly reducing the labor required to form the hull. Although developable surfaces play an important role in various manufacturing applications, little attention has been paid to implementing developable surfaces from the onset of a design. This paper investigates novel, user friendly methods to design complex objects using B-spline developable surfaces based on optimization techniques. Illustrative examples show the substantial improvements this method achieves over previously developed methods.
... 12 To overcome the limitation in surface shape, there has been a great effort in studying parameterized developable surfaces, see, for example, those using Bézier surfaces. [13][14][15][16] However, the developability condition for a parameterized surface is highly nonlinear. To deal with this difficulty, some of the existing studies find the developable surfaces by interpolating the specified boundary curves. ...
... The new conformal factors are used to update the radii of circles in Equation (14), and then the member lengths in Equation (6). The Ricci flow algorithm is simple and converges fast. ...
... Step 3: Update the radii r i using Equation (14). Update the member lengths l ij using Equation (6) with the fixed intersection angle φ ij , and return to Step 1. ...
This paper presents an approach for the design of discrete architectural surfaces that are globally developable; that is, having zero Gaussian curvature at every interior node. This kind of architectural surface is particularly suitable for fast fabrication at a low cost, since their curved geometry can be developed into a plane. This highly non‐linear design problem is broken down into two sub‐problems: (1) find the member lengths of a triangular mesh that lead to zero Gaussian curvature, by employing the discrete surface Ricci flow developed in the field of discrete differential geometry; (2) realize the final geometry by solving an optimization problem, subject to the constraints on member lengths as well as the given boundary. It is demonstrated by the numerical examples that both of these two sub‐problems can be solved with small computational costs and sufficient accuracy. In addition, the Ricci flow algorithm has an attractive feature—the final design is conformal to the initial one. Conformality could result in higher structural performance, because the shape of each panel is kept as close as possible to its initial design, suppressing possible distortion of the panels. This paper further presents an improved circle packing scheme implemented in the discrete surface Ricci flow to achieve better conformality, while keeping its simplicity in algorithm implementation as in the existing Thurston's scheme .
... Reference [11] used B-spline surfaces, but they started from one directrix curve and a pair of rulings. The second directrix is created between the end points of the rulings assuring developable surface with the use of certain constraints. ...
... Reference [12] used the same design principles as [11] with the use of a normal directrix to the surface and Catmull-Rom and Beta-splines to model the curves. Reference [13] uses conical and cylindrical surfaces in shipyard applications. ...
... This example shows the lines of a hard chine craft. The chine, sheer and centre line can be found in reference [11] and are presented in Table 2. ...
The use of developable surfaces in ship design is of engineering importance because they can be easily manufactured without stretching or tearing, or without the use of heat treatment. In some cases, a ship hull can be entirely designed with the use of developable surfaces. In this paper, a method to create a quasi-developable B-spline surface between two limit curves is presented. The centreline, chines and sheer lines of a vessel are modelled as B-spline curves. Between each pair of these boundary curves or directrix lines, the generator lines or rulings are created and a quasi-developable B-spline surface containing the rulings is defined. A procedure based on multiconic development is used to modify the directrix lines in case the rulings intersect inside the boundary curves, avoiding non-developable portions of the surface. B-spline curves and surfaces are widely used today in practically all the design and naval architecture computer programs. Some examples of ship hulls entirely created with developable surfaces are presented.
... Chalfant and Maekawa [7] studied design of a developable B-spline surface by joining m developable Bézier patches along their end rulings with C 2 continuity, but their method restricts the two boundary curves to lie in parallel planes. Their later work [8] developed a method for design of developable surfaces with general 3D boundary curves. Optimization techniques are employed to compute the remaining control points after the user has designated the first curve and the end points of the other, but the solved surface may not always be precisely developable. ...
... Co-planarity can be represented in terms of the triple scalar product of the two tangent vectors and the ruling vector A(w)−B(w): Substituting the Bézier representation of both curves into the above equation leads to a complicated system of equations that must be fulfilled by the Bézier control points to ensure its developability. All previous studies [4][5][6][7][8][9][10] impose Eqn. (1). ...
... Each patch must satisfy its seven developability constraints, Eqns. (5)(6)(7)(8)(9)(10)(11). Therefore, the number of the remaining DOF's is 8(3) − 7 − 7 = 10. ...
This paper studies geometric design of developable surfaces that consist of consecutive Bézier patches. It is shown that the number of degrees of freedom (DOF) for the surface design is independent of the degree of the surface. With a first boundary curve freely specified, (2m+3), (m+4), and five DOF’s are available for a second boundary curve of a developable surface containing m patches, when the surface is G0, G1, and G2, respectively. There remain five and (7−2m) DOF’s for C1 and C2 continuity. Four and three DOF’s are left for the patch design when the end ruling vanishes on one and both sides. Design examples are presented that fully utilize the corresponding DOF’s subject to various continuity conditions. This work provides the foundation for systematic implementation of a CAGD system for developable composite Bézier surfaces.
... In current CAD/CAM systems, there are basically two approaches to deal with developable surfaces. On the one hand, a developable surface can be expressed as a tensor product surface in the Euclidean space [2][3][4][5][6][7][8][9][10][11]; on the other hand, a developable surface can be based on the dual representation, in the sense of projective geometry [12][13][14][15][16][17][18]. The first approach treats developable surface as ruled surfaces and derives conditions that have to be additionally imposed in order to achieve developability. ...
... The first is constructed with a given directrix and a given generator direction, and the second is constructed by interpolating two boundary curves. Developable free-form surfaces, in terms of Bézier surfaces, were studied in [4][5][6][7][8] and later were extended to B-spline form in [9][10][11]. All these representations are not direct or explicit. ...
... The intersection of the three planes, that is, the solution of Eqs. (6), (9) and (10), is the characteristic point for the given value of t. Therefore, the coordinates of the characteristic point are computed by In this section, a developable surface is considered the surface composed of the tangent lines to a spine curve. ...
This paper presents two direct explicit methods of computer-aided design for developable surfaces. The developable surfaces are designed by using control planes with C-Bézier basis functions. The shape of developable surfaces can be adjusted by using a control parameter. When the parameter takes on different values, a family of developable surfaces can be constructed and they keep the characteristics of Bézier surfaces. The thesis also discusses the properties of designed developable surfaces and presents geometric construction algorithms, including the de Casteljau algorithm, the Farin–Boehm construction for G2 continuity, and the G2 Beta restricted condition algorithm. The techniques for the geometric design of developable surfaces in this paper have all the characteristics of existing approaches for curves design, but go beyond the limitations of traditional approaches in designing developable surfaces and resolve problems frequently encountered in engineering by adjusting the position and shape of developable surfaces.
... Substituting the Bézier representation of both curves into the above equation leads to a complicated system of equations that must be fulfilled by the Bézier control points to ensure its developability. All previous studies [4] [5] [6] [7] [8] [9] [10] impose Eqn. (1). ...
... Chalfant and Maekawa [7] studied design of a developable B-spline surface by joining m developable Bézier patches along their end rulings with C 2 continuity, but their method restricts the two boundary curves to lie in parallel planes. Their later work [8] developed a method for design of developable surfaces with general 3D boundary curves. Optimization techniques are employed to compute the remaining control points after the user has designated the first curve and the end points of the other, but the solved surface may not always be precisely developable. ...
... Co-planarity can be represented in terms of the triple scalar product of the two tangent vectors and the ruling vector A(w)−B(w): Substituting the Bézier representation of both curves into the above equation leads to a complicated system of equations that must be fulfilled by the Bézier control points to ensure its developability. All previous studies45678910 impose Eqn. (1). ...
This paper studies geometric design of developable composite Bézier surfaces from two boundary curves. The number of degrees of freedom (DOF) is characterized for the surface design by deriving and counting the developability constraints imposed on the surface control points. With a first boundary curve freely chosen, (2m+3), (m+4), and five DOFs are available for a second boundary curve of a developable composite Bézier surface that is G0, G1, and G2, respectively, and consists of m consecutive patches, regardless of the surface degree. There remain five and (7-2m) DOFs for the surface with C1 and C2 continuity. Allowing the end control points to superimpose produces Degenerated triangular patches with four and three DOFs left, when the end ruling vanishes on one and both sides, respectively. Examples are illustrated to demonstrate various design methods for each continuity condition. The construction of a yacht hull with a patterned sheet of paper unrolled from 3D developable surfaces validates practicality of these methods in complex shape design. This work serves as a theoretical foundation for applications of developable composite Bézier surfaces in product design and manufacturing.
... A developable surface is a special type of ruled surface that can be flattened into a plane without stretching or tearing. Due to their simplicity and ease of fabrication, they are used in a variety of fields [6][7][8] (interested readers can watch this YouTube video https://www.youtube.com/watch?v=-jZfFgfGj1M (accessed on 22 October 2023) about the applications of ruled surfaces in architecture.). ...
... Discussing the importance and applications of developable surfaces, the paper in [14] reviews techniques and patents in geometric modelling and highlights developments in this field and concludes that further research is needed in several key areas of developable surfaces. Some special curves construct ruled surfaces with the help of the Frenet frame and there are also very good papers on this subject, such as [15][16][17][18][19][20][21][22][23]. ...
Ruled surfaces play an important role in various types of design, architecture, manufacturing, art, and sculpture. They can be created in a variety of ways, which is a topic that has been the subject of a lot of discussion in mathematics and engineering journals. In geometric modelling, ideas are successful if they are not too complex for engineers and practitioners to understand and not too difficult to implement, because these specialists put mathematical theories into practice by implementing them in CAD/CAM systems. Some of these popular systems such as AutoCAD, Solidworks, CATIA, Rhinoceros 3D, and others are based on simple polynomial or rational splines and many other beautiful mathematical theories that have not yet been implemented due to their complexity. Based on this philosophy, in the present work, we investigate a simple method of generating ruled surfaces whose generators are the curvature axes of curves. We show that this type of ruled surface is a developable surface and that there is at least one curve whose curvature axis is a line on the given developable surface. In addition, we discuss the classifications of developable surfaces corresponding to space curves with singularities, as these curves and surfaces are most often avoided in practical design. Our research also contributes to the understanding of the singularities of developable surfaces and, in their visualisation, proposes the use of environmental maps with a circular pattern that creates flower-like structures around the singularities.
... Developable surfaces model the way the pages of a book are folded [1], the forms of facades in architecture [2] or the shapes adopted by garments [3] with plane patterns. They are also useful in industries related to building with sheets of steel or wood, such as naval industry [4,5,6], or even automobile industry [7]. In the case of steel this means that parts of the hull of a ship can be modeled with developable surfaces and can be produced by folding machines without application of heat, reducing costs and modifications of the metallic structure. ...
... according to corollary 3.If we perform a Möbius transformation (5.1) with b = 2 on both bounding curves, so that the respective new sets of weights are {8, 4, 6/5, 5/6} and {8, 4, 22/15, 43/45}, the new parametrisation for the developable surface has new constants given by(5 ...
In this paper we provide a characterisation of rational developable surfaces in terms of the blossoms of the bounding curves and three rational functions $\Lambda$, $M$, $\nu$. Properties of developable surfaces are revised in this framework. In particular, a closed algebraic formula for the edge of regression of the surface is obtained in terms of the functions $\Lambda$, $M$, $\nu$, which are closely related to the ones that appear in the standard decomposition of the derivative of the parametrisation of one of the bounding curves in terms of the director vector of the rulings and its derivative. It is also shown that all rational developable surfaces can be described as the set of developable surfaces which can be constructed with a constant $\Lambda$, $M$, $\nu$ . The results are readily extended to rational spline developable surfaces.
... Developable surfaces model the way the pages of a book are folded [1], the forms of facades in architecture [2] or the shapes adopted by garments [3] with plane patterns. They are also useful in industries related to building with sheets of steel or wood, such as naval industry [4,5,6], or even automobile industry [7]. In the case of steel this means that parts of the hull of a ship can be modeled with developable surfaces and can be produced by folding machines without application of heat, reducing costs and modifications of the metallic structure. ...
... according to corollary 3.If we perform a Möbius transformation (5.1) with b = 2 on both bounding curves, so that the respective new sets of weights are {8, 4, 6/5, 5/6} and {8, 4, 22/15, 43/45}, the new parametrisation for the developable surface has new constants given by(5 ...
In this paper we provide a characterisation of rational developable surfaces in terms of the blossoms of the bounding curves and three rational functions Λ, M , ν. Properties of developable surfaces are revised in this framework. In particular, a closed algebraic formula for the edge of regression of the surface is obtained in terms of the functions Λ, M , ν, which are closely related to the ones that appear in the standard decomposition of the derivative of the parametrisation of one of the bounding curves in terms of the director vector of the rulings and its derivative. It is also shown that all rational developable surfaces can be described as the set of developable surfaces which can be constructed with a constant Λ, M , ν. The results are readily extended to rational spline developable surfaces. Mathematics subject classification: 65D17, 68U07.
... Concerning the applications in industry, quasi-developable surfaces are constructed in [11] and [12]. In [13] developable surfaces for designing ship hulls are constructed by graphical methods. ...
... This example is borrowed from[11] and corresponds to the hull of a boat : We have three curves named sheer, chine and center line, with respective B-spline polygonsc 0 = (0.00, 0.00, 9.00), c 1 = (6.86, 7.10, 8.22), c 2 = (21.6, ...
In this paper we construct developable surface patches which are bounded by two rational or NURBS curves, though the resulting patch is not a rational or NURBS surface in general. This is accomplished by reparameterizing one of the boundary curves. The reparameterization function is the solution of an algebraic equation. For the relevant case of cubic or cubic spline curves, this equation is quartic at most, quadratic if the curves are Bezier or splines and lie on parallel planes, and hence it may be solved either by standard analytical or numerical methods.
... Their system requires solving non-linear system equations to ®nd the Be Âzier control points. Very recently Chalfant and Maekawa [10,11] developed a method to design developable surfaces where boundary curves do not necessarily lie in parallel planes. Optimization techniques were employed to construct a developable surface subject to error bounds placed on the control points by the user. ...
... For w 0, it coincides with A 0 B 0 B 1 A 1 and for w 1 with A 1 B 1 B 2 A 2 . The same conclusion was obtained, using optimization techniques, by Chalfant and Maekawa [10]. ...
... Their survey [10] summarizes the properties, representations, and methods of designing developable surfaces. Chalfant and Maekawa [11,12] proposed a method to design developable surfaces where boundary curves do not necessarily lie in parallel planes. Optimization techniques were employed to construct a developable surface subject to error bounds placed on the control points. ...
... Co-planarity can be represented in terms of the triple scalar product of the two tangent vectors and the ruling vector A(w)−B(w) in between: Substituting the Bézier representation of both curves into Equation (2-1) leads to a complicated system of equations that must be imposed on the Bézier control points to ensure developability of the surface. Typically the solution involves a non-linear system of equations [7,9,11]. ...
This paper investigates the number of degrees of freedom for geometric design of developable Bézier surfaces. The conditions for developability are derived geometrically from the de Casteljau algorithm and expressed as a set of equations that must be fulfilled by the Bézier control points. This set of equations enables us to infer important properties of developable Bézier patches that characterize the patch design and simplify its solution process. With one boundary curve freely specified in 3D space, five more degrees of freedom are available for the second boundary curve of the same degree. Imposing parametric or geometric continuities across the boundary of two adjacent developable Bézier patches results in a composite developable Bézier surface that has fewer degrees of freedom. This work provides the foundation for a systematic implementation of a computer-aided design system for developable Bézier surfaces.
... The above review shows that very little research has been done on the geometric design of developable B-spline surfaces. Chalfant and Maekawa [3] converted a developable Bézier surface into its equivalent B-spline surface by knot insertion, but their method was still subject to the limitations of the previous methods [1,7,9]. To overcome this deficiency, this study examines the geometric design of developable B-spline surfaces defined by two boundary curves. ...
... To ensure the developability of the surface, its control points must satisfy the constraints that are imposed by these equations. The solution process usually involves a highly nonlinear system of equations and it often yields surfaces that are not exactly developable because of numerical instability [3]. ...
This study investigates the geometric design of B-spline surfaces constructed by two boundary curves. The developability constraints are geometrically derived from the de Boor algorithm and expressed as a set of equations that must be fulfilled by the B-spline control points. These equations specify the number of degrees of freedom (DOFs) for the surface design. For a cubic B-spline surface with a freely selected first boundary curve, five more DOFs are available for a second boundary curve when both curves are defined by four control points. There remain (7–2m) DOFs for designing a cubic surface that consists of m consecutive patches. The results are consistent with previous findings for equivalent composite Bézier surfaces. A test example demonstrates design methods that fully use all of the DOFs without generating over-constrained systems in the solution process. This work provides a preliminary foundation for applications of developable B-spline surfaces in product design and manufacture.
... However, the complex system of coupled equations is very difficult for the design of developable Bézier surfaces in a CAD system. Chalfant and Maekawa [4] investigated novel, useful methods to design complex objects using B-spline developable surfaces based on optimization techniques. Another approach to design developable surfaces is a direct surface represented in terms of geometric duality between points and planes in 3D projective space by Chu and Séquin [5]. ...
... In the same way, for the segment R 2 (r) = (− r 2 2 + 5r − 17 2 , − r 2 2 + 7r − 19 2 , 0), r ∈ [3,4], τ 2 = 0. In order to smoothly connect the adjacent developable surface, we assume θ 2 = π. ...
Developable surface and line of curvature play an important role in geometric design and surface analysis. This paper proposes a new method to construct a developable surface possessing a given curve as the line of curvature of it. We analyze the necessary and sufficient conditions when the resulting developable surface is a cylinder, cone or tangent surface. Finally, we illustrate the convenience and efficiency of this method by some representative examples.
... Their system requires solving non-linear system equations to ®nd the Be Âzier control points. Very recently Chalfant and Maekawa [10] [11] developed a method to design developable surfaces where boundary curves do not necessarily lie in parallel planes. Optimization techniques were employed to construct a developable surface subject to error bounds placed on the control points by the user. ...
... The same conclusion was obtained, using optimization techniques, by Chalfant and Maekawa [10] ...
Geometric design of quadratic and cubic developable Bézier patches from two boundary curves is studied in this paper. The conditions for developability are derived geometrically from the de Casteljau algorithm and expressed as a set of equations that must be fulfilled by the Bézier control points. This set of equations allows us to infer important properties of developable Bézier patches that provide useful parameters and simplify the solution process for the patch design. With one boundary curve freely specified, five more degrees of freedom are available for a second boundary curve of the same degree. Various methods are introduced that fully utilize these five degrees of freedom for the design of general quadratic and cubic developable Bézier patches in 3D space. A more restricted generalized conical model or cylindrical model provides simple solutions for higher-order developable patches.
... For instance, in naval architecture sheets of steel are adapted to fit into the hull of a ship [3,4,5]. If these sheets are combed just with a folding machine, the costs are lower than if they require the use of heat. ...
In this talk we review the problem of constructing a developable surface patch bounded by two rational or NURBS (Non-Uniform Rational B-spline) curves.
... Sheet metal and wooden panels are widely used materials for building ship-hulls and freeform architectures, which possess the physical property of high bendability and low stretchability. The developable surface is a suitable mathematical model for these materials and thus has important industrial applications [1][2][3]. Developable surfaces have also been investigated in Computer Numerical Control (CNC) flank milling since they possess zero twist at the ruling lines and can be accurately machined with conical cutting tools [4,5]. However, current commercial CAD software does not have flexible and effective capabilities for model-ing developable surfaces or quasi-developable surfaces. ...
We propose a method for generating a ruled B-spline surface fitting to a sequence of pre-defined ruling lines and the generated surface is required to be as-developable-as-possible. Specifically, the terminal ruling lines are treated as hard constraints. Different from existing methods that compute a quasi-developable surface from two boundary curves and cannot achieve explicit ruling control, our method controls ruling lines in an intuitive way and serves as an effective tool for computing quasi-developable surfaces from freely designed rulings. We treat this problem from the point of view of numerical optimization and solve for surfaces meeting the distance error tolerance allowed in applications. The performance and efficacy of the proposed method are demonstrated by the experiments on a variety of models including an application of the method for path planning in 5-axis Computer Numerical Control (CNC) flank milling.
... In other words, it is a 3D surface that is constructed by isometrically bending and gluing a 2D panel. The design of such surfaces has ergo been for long of interest in the several őelds of industrial manufacturing (Ferris, 1968;Tang and Wang, 2005;Chalfant and Maekawa, 1998) as they can be used to design surfaces made of materials that hardly stretch such as metal sheets for instance. ...
This thesis deals with the direct simulation and inverse design of garments in the presence of frictional contact.The shape of draped garments results from the slenderness of the fabric, which can be represented in mechanics by a thin elastic plate or shell, and from its interaction with the body through dry friction. This interaction, necessary to reproduce the threshold friction occuring in such contacts, is described by a non smooth law, which, in general, makes its integration complex. In a first contribution, we modify the so-called Projective Dynamics algorithm to incorporate this dry frictional contact law in a simple way. Projective Dynamics is a popular method in Computer Graphics that quickly simulates deformable objects such as plates with moderate accuracy, yet without including frictional contact. The rationale of this algorithm is to solve the integration of the dynamics by successively calculating estimates of the shape of the object at the next timestep. We take up the same idea to incorporate a procedure for estimating the frictional contact law that robustly captures the threshold phenomenon. In addition it is interesting to note that simulators developed in Computer Graphics, originally targeted at visual animation, have become increasingly accurate over the years. They are now being used in more "critical" applications such as architecture, robotics or medicine, which are more demanding in terms of accuracy. In collaboration with mechanicists and experimental physicists, we introduce into the Computer Graphics community protocols to verify the correctness of simulators, and we present in this manuscript our contributions related to plate and shell simulators. Finally, in a last part, we focus on garment inverse design. The interest of this process is twofold. Firstly, for simulations, solving the inverse problem provides a "force-free" and possibly curved version of the input (called the rest or natural shape), whether it comes from a 3D design or a 3D capture, that allows to start the simulation with the input as the initial deformed shape. To this end, we propose an algorithm for the inverse design of clothes represented by thin shells that also accounts for dry frictional contact. Within our framework, the input shape is considered to be a mechanical equilibrium subject to gravity and contact forces. Then our algorithm computes a rest shape such that this input shape can be simulated without any sagging. Secondly, it is also appealing to use these rest shapes for a real life application to manufacture the designed garments without sagging. However, the traditional cloth fabrication process is based on patterns, that is sets of flat panels sewn together. In this regard, we present in our more prospective part our results on the adaptation of the previous algorithm to include geometric constraints, namely surface developability, in order to get flattenable rest shapes.
... However, this method does not apply to our case where only sparse design curves are available. In (Chalfant et al., 1998), the method for generating quasi-developable B-spline surfaces based on optimization techniques is presented. However, the method can only generate quasi-developable Bézier surfaces with two boundary curves being of the same degree. ...
In this paper, we present a new method for generating high-accuracy developable surface from two design curves via crow search algorithm (CSA) based modification of design curves. For achieving high-accuracy developability, we allow perturbation of the control points of design curves within a allowed bound to expand the solution space of developable surfaces, and search within the allowed perturbation bound for the set of optimal control points with which the resultant surface has the highest degree of developability via CSA. As a result, design curves are automatically and intelligently modified when that is necessary for achieving a high-accuracy developability. Modeling examples demonstrate that the method is effective and easy to implement to obtain high-accuracy developables.
... However, this method does not apply to our case where only sparse design curves are available. In (Chalfant et al., 1998), the method for generating quasi-developable B-spline surfaces based on optimization techniques is presented. However, the method can only generate quasi-developable Bézier surfaces with two boundary curves being of the same degree. ...
In this paper, we present a new method for generating high-accuracy developable surface from two design curves via crow search algorithm (CSA) based modification of design curves. For achieving high-accuracy developability, we allow perturbation of the control points of design curves within a allowed bound to expand the solution space of developable surfaces, and search within the allowed perturbation bound for the set of optimal control points with which the resultant surface has the highest degree of developability via CSA. As a result, design curves are automatically and intelligently modified when that is necessary for achieving a high-accuracy developability. Modeling examples demonstrate that the method is effective and easy to implement to obtain high-accuracy developables.
... Ruling lines are considered as design guidance in some works. Chalfant et al. propose a quasi-developable design method based on given boundary curves and boundary rulings, and discuss its applications in ship-hull surface modeling [8]. Park at el. give the direction of a set of ruling lines and two endpoints of the boundary curve, and obtain developable surfaces by optimal control [23]. ...
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.
... One method is point geometry. We can refer to the papers [1][2][3][4][5][6][7]. Aumann [1] proposed the conditions to construct a developable Bézier surface with two boundary curves that are restricted in parallel planes. ...
Developable surface plays an important role in geometric design, architectural design, and manufacturing of material. Bézier curve and surface are the main tools in the modeling of curve and surface. Since polynomial representations can not express conics exactly and have few shape handles, one may want to use rational Bézier curves and surfaces whose weights control the shape. If we vary a weight of rational Bézier curve or surface, then all of the rational basis functions will be changed. The derivation and integration of the rational curve will yield a high degree curve, which means that the shape of rational Bézier curve and surface is not easy to control. To solve this problem of shape controlling for a developable surface, we construct C-Bézier developable surfaces with some parameters using a dual geometric method. This yields properties similar to Bézier surfaces so that it is easy to design. Since C-Bézier basis functions have only two parameters in every basis, we can control the shape of the surface locally. Moreover, we derive the conditions for C-Bézier developable surface interpolating a geodesic.
... . Julie [15] , . Konesky [16] , , ...
A novel method is proposed for quasi-developable surface construction between two boundary curves. Different from existing methods which construct a developable surface by connecting finite sampling points of two curves, the present method allows continuous mapping between boundary curves, which enhances the ability of searching for better developable surface. This method directly generates a developable B-spline surface whose boundaries interpolate target profile curves, as a contrast to existing approaches which construct smooth devel-opable surfaces by joining surface patches. Moreover, with the present method, it is easy to control the overall smoothness of surface and avoid emergence of regression curves inside concerned surface region. The method is applied to several ship hull data and quasi-developable surfaces which meet manufacturing requirements are successfully generated. © 2018, Beijing China Science Journal Publishing Co. Ltd. All right reserved.
... Another approach is the contruction of approximately developable surfaces which may be useful for the industry [11]. One way to accomplish this is the use of spline cones [12]. ...
In this paper we address the issue of designing developable surfaces with Bezier patches. We show that developable surfaces with a polynomial edge of regression are the set of developable surfaces which can be constructed with Aumann's algorithm. We also obtain the set of polynomial developable surfaces which can be constructed using general polynomial curves. The conclusions can be extended to spline surfaces as well.
... One approach is to prove characterizing equations for the free form surfaces to be developable. The readers can refer to papers (Aumann, 1991, Chalfant and Maekawa, 1998, Nolan, 1971. However, the complex system of coupled equations is very difficult for the design of developable Bézier surfaces in a CAD system. ...
This paper studies the problem to construct a spacelike developable surface possessing a given non-null parametric curve as the line of curvature in Minkowski 3-space. By using Frenet frame to express the surface, we derive the necessary and sufficient conditions when the resulting spacelike developable surface is cylinder, cone or tangent surface. Finally, we give some representative examples.
... One can also construct surfaces which are approximately developable instead (Chalfant and Maekawa (1998); Pottmann and Wallner (1999); Leopoldseder (2001); Peternell (2004); Liu et al. (2011); Zeng et al. (2012)). ...
In this paper we address the problem of interpolating a spline developable
patch bounded by a given spline curve and the first and the last rulings of the
developable surface. In order to complete the boundary of the patch a second
spline curve is to be given. Up to now this interpolation problem could be
solved, but without the possibility of choosing both endpoints for the rulings.
We circumvent such difficulty here by resorting to degree elevation of the
developable surface. This is useful not only to solve this problem, but also
other problems dealing with triangular developable patches.
... Most previous approaches to developable surfaces [1][2][3][4][5] treat them as ruled surfaces and derive conditions that have to be imposed additionally in order to achieve developability. All these representations are not direct or explicit. ...
In order to overcome the difficulties in representation of developable surfaces utilizing
traditional approaches, and resolve the problems in adjusting and controlling the position and
shape of developable surfaces that often faced in Engineering. In this paper, we propose a
directly explicit and efficient method of computer-aided design for developable surfaces based
on triangle-B spline. The shapes of developable surfaces can be adjusted using a control
parameter. Meanwhile, we show that the techniques for the geometric design of developable
surfaces in this paper have all the characteristics of existing approaches for curves design. The
algorithms are explained in detail, and demonstrated with the examples in the paper.
... Problem 2. This example is from shipbuilding industries, and it is a case of a patch taken on the forward part of a hard chine fishing vessel. It is well known [43], that hard chine vessels and boats are complex objects, and their computer aided design and interrogation is challenging. Let (6,7,8), and (7,8,9). ...
Design of fair surfaces over irregular domain is a fundamental problem in computer aided geometric design (CAGD), and has applications in engineering sciences (i.e. aircraft science, automobile science and ship science etc.). In design of fair surfaces over irregular domain defined over scattered data it was widely accepted till recently that one should use Delaunay triangulation because of its global optimum property. However, in recent times it has been shown that for continuous piecewise polynomial parametric surfaces improvements in the quality of fit can be achieved if the triangulation pattern is made dependent upon some topological property of the data set or is simply data dependent. The smoothness and fairness of surface’s planar cuts is important because not only it ensures favorable hydrodynamic drag, but also helps in reducing manhours during the production of the surface. In this paper we discuss a method for construction of C1 piecewise polynomial parametric fair surfaces which interpolate prescribed R3 scattered data using spaces of parametric splines defined on R3 triangulation. We show that our method is more specific to the cases when the projection on 2-D plane may consist of triangles of zero area. The proposed method is fast, numerically stable and robust, and computationally inexpensive. In the present work numerical examples dealing with surfaces approximated on standard curved plates, and ship hull surface have been presented.
... For the design of such surfaces different methods have been presented. Many papers prove (nonlinear) characterizing equations for free form surfaces to be developable (Aumann, 1991aAumann, , 1991b Lang and Röschel, 1992; Chalfant and Maekawa, 1998; Maekawa, 1998). Another approach by Pottmann and Farin (1995) have been discussed by Clements and Leon (1987), Gurunathan and Dhande (1987) or Weiß and Furtner (1988). ...
An algorithm is presented that generates developable Bézier surfaces through a Bézier curve of arbitrary degree and shape. The algorithm has two important advantages. No (nonlinear) characterizing equations have to be solved and the control of singular points is guaranteed. Further interpolation conditions can be met. 2003 Elsevier B.V. All rights reserved.
... Developable surfaces, which can be developed onto a plane without stretching and tearing, have natural applications in many areas of engineering and manufacturing, including modeling of ship hulls [1], apparel [2], automobile components [3], and so on. In computer graphics, many objects can be approximated by piecewise continuous developable surfaces. ...
This paper proposes a new method for designing a developable surface by constructing a surface pencil passing through a given curve, which is quite in accord with the practice in industry design and manufacture. By utilizing the Frenet trihedron frame, we derive the necessary and sufficient conditions to construct a developable surface through a given curve. Considering the requirements in shoemak-ing and garment-manufacture industries, we also study the special case of specifying the given curve as a geodesic. The given geodesic can be classified into three types corresponding to each type of developable surface. We also present the polynomial representation of the developable surface. The algorithm is convenient and efficient for applications in engineering.
... Many papers give the applications of developable surfaces in industry (Mancewicz and Frey, 1992; Frey and Bindschadler, 1993; Pottmann and Wallner, 2001; Chu and Séquin, 2002). Many authors constructed developable free form surfaces by nonlinear characteristic equations (Aumann, 1991; Lang and Röschel, 1992; Chalfant and Maekawa, 1998; Maekawa, 1998). A different approach to designing developable surfaces is a direct surface representation in terms of geometric duality between points and planes in 3D projective space (Bodduluri and Ravani, 1993; Pottmann and Farin, 1995). ...
A new algorithm is presented that generates developable Bézier surfaces through a Bézier curve called a directrix. The algorithm
is based on differential geometry theory on necessary and sufficient conditions for a surface which is developable, and on
degree evaluation formula for parameter curves and linear independence for Bernstein basis. No nonlinear characteristic equations
have to be solved. Moreover the vertex for a cone and the edge of regression for a tangent surface can be obtained easily.
Aumann’s algorithm for developable surfaces is a special case of this paper.
... There is quite a lot of literature on modeling with developable surfaces, see [1][2][3][4][5][6] and their references. B-spline representations and the dual representation are well known. ...
Given a set of data points as measurements from a developable surface, the present paper investigates the recognition and reconstruction of these objects. We investigate the set of estimated tangent planes of the data points and show that classical Laguerre geometry is a useful tool for recognition, classification and reconstruction of developable surfaces. These surfaces can be generated as envelopes of a one-parameter family of tangent planes. Finally we give examples and discuss the problems especially arising from the interpretation of a surface as set of tangent planes.
... That is, roll bending shapes belong to a class of developable surfaces. There is quite a lot of literature on modeling and approximation of developable surfaces and readers are referred to Pottmann and Farin [8], Chalfant and Maekawa [9], Randrup [10], Pottmann and Wallner [11,12], Chu and Séquin [13], Aumann [14], and their references. They use two approaches to interpolate or approximate rational developable surfaces. ...
This paper presents a simple and efficient method to approximate a developable surface to a compound design surface by a polynomial. It is required to predict a final shape of roll bending in the fabrication of a curved shell plate. The roll bending process usually makes the cylindrical or conical curvature from an initial flat plate. It means that the final shape is developable or the surface representation has zero Gaussian curvature. The fabrication shape is important in order to estimate process parameters of roller bending.An optimization problem is formulated to determine the polynomial surface which is in the closest proximity to the design surface or the given shell plate, which is subjected to developability. The results and the efficiency of this algorithm are verified and evaluated by applying it to some shell plates which are obtained from a real ship model. The predicted bending shape becomes fundamental information in determining more process parameters for the fabrication of a compound curved shell plate.
Composite sheet forming, especially thermo-forming, has been well researched over the last few decades. However, the concept of developable surfaces has not been implemented for the fibrous composites. These surfaces form critical and lucrative aspects in geometric designs, surface analyses and structural applications, with their roots being strongly established in origami and various shape designs around the world. The current work aims to analytically derive and determine developable (de) curvatures on a flat sheet, based on established theories on lines of curvature. The theoretically derived de curves are then validated by metallic sheet forming before implementing them for the first time on both multi-layered wood veneers and short flax-fibre reinforced polypropylene composites. The work is a novel application of the established developable surface generation in the case of composite sheet forming, the success of which can expand its usage in various consumer goods and structural components.
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.
In this paper, we proposed a new device for geometry errors measurement and coaxiality evaluation, and the corresponding methodology for coaxiality evaluation from measurement data is presented, which allows to characterize multiple holes at a time. Unlike traditional measurement system a laser sensor is mounted onto out of the holes so that multi-hole surfaces can be “seen” by the senor when it rotates around a fixed axis. First the intersections (or ellipse profiles) of the sensor’s scanning plane and holes, are computed by fitting. Then, the center coordinates and profile points of the ellipse are computed and transformed to the 3D global coordinate frame. Finally the centerline of the hole is determined from the 3D profile points by using a weighted least-squares fitting algorithm. In addition, to reduce the effect of noises on the measurement result, error analysis and compensation techniques are studied to improve the measurement accuracy. A case study is presented to validate the measurement principle and data processing approach.
In this study, we determine subplanes of Hall projective plane coordinatized
by elements of a left nearfield of order 9 by using an algorithm (implemented in C#).
For potential applications in geometric design and manufacturing of material, the connection of many pieces of developable surfaces is an important issue. In this paper, by using de casteljau algorithm we study the connection of four pieces of developable surfaces with Bézier boundary curves. We convert these surfaces to tensor form firstly, then characterize the constrains of the control points of the surfaces need to satisfy when connecting them. This method can also be extended to the case when the developable surfaces possess Bézier boundary curves with different degrees.
Both the general and rational developable surface pencils through an arbitrary parametric curve as its common asymptotic curve were analyzed. By employing the parametric representation of a developable surface pencil taking a given curve as its asymptotic curve, the expression for the case that the pencil is developed was presented and the type of the designed developable surface was discussed. The rational Bézier form of the developable surface pencil through a given Bézier curve as its asymptotic curve was given. Programming examples for the general and rational developable surfaces through a circular helix, conical helix or Bézier curve as its asymptotic curve were illustrated to verify the correctness and effectivity of the algorithm.
Developable surfaces are especially important in ship design when working with sheet materials like plywood, steel or aluminum. Developable surfaces can be formed from flat sheets without stretching or tearing and with a minimum use of heat treatments, so the forces required to form sheet materials into developable surfaces are much less than for other surfaces and the construction costs are lower. Manufacture can be improved if developable surfaces are considered from the onset of a design. In this paper, a method to create a developable NURBS (Non Uniform Rational B Spline) surfaces between two curves is presented, which is applied to the construction of hard chine hulls. The centreline, chines and sheer lines of a vessel are modeled as B Splines curves in order to apply the method. Between each pair of these boundary curves or directrix lines, the generator lines or rulings are created according to the method using a numerical approximation and then, a NURBS surface containing the rulings can be created. NURBS curves and surfaces are used in the method because they are widely accepted today in all practical designs and naval architecture computer programs. An example of ship hulls entirely created with developable surfaces is presented, and some small prototypes have been constructed in order to validate the method.
Generating visually realistic deformation of 3D objects in real-time for character body or garment is essential for the animation field, both for video games applications and movie production. Current fast deformation methods such as "skinning", or low resolution physically based simulations fail to capture some important natural behaviors. For instance constant volume deformation of a body, muscle bulging of a character, or wrinkling its garment which must be modeled by a developable surface. This thesis presents several methods making a geometrical models "active", that is to say able to maintain a certain number of intrinsic constraints of the surface linked to its bounded volume, or its developability. We develop three case studies - The addition of local volume constraints during skinning animation of a virtual creature. - The addition of wrinkles on garments to model stretch minimizing surface starting from a general low resolution animation provided as input. - The generation of folded paper looking surface based on length preservation with respect to its planar pattern. For every cases, we rely on a procedural approach based on progressive deformation, and eventually, on-the-fly refinement of the geometry just before the rendering stage.
In the metal building and garment industries, product model is required to be or closed to be developable surface (can be flattened onto planar patterns without any distortion, tear or stretch).Current work mainly focuses on interpolating only two curves with a narrow developable strip, and lose the generality of applications in industries, which often require N (N>2) curves to determine the final shape feature, and seek one interpolated developable surface. In this paper, a new developability degree criterion is introduced first and a novel lofting algorithm is proposed to model a quasi-developable mesh surface for the given characteristic curves. The curves are first adaptively sampled and then planar triangles are tiled to interpolate all sampling points. The optimal triangulation in terms of given criterion is mapped as one shortest-path finding graph problem, which can be solved by using well-known Dijkstra’s algorithm.
This paper studies the problem of designing a rational Bézier developable surface pencil with a common isogeodesic, and provides an algorithm for the representation of complicated geometric models in industrial applications which need to satisfy that the shape surface can be developed and a given curve is geodesic. By employing the local Frenet orthonormal frame, the explicit expression of the rational Bézier developable surface pencil is derived. Furthermore, the order of the rational Bézier developable surface pencil interpolating a planar or non-planar curve as its geodesic is discussed. The formulae of the control net vertices for the derived surface are presented. Finally, the effectiveness and correctness of the algorithm are verified by examples of the rational Bézier developable surface pencil through a degree 2 or 3 Bézier curve as a common geodesic.
This article presents a brief introduction to the classical geometry of ruled surfaces with emphasis on the Klein image and studies aspects which arise in connection with a computational treatment of these surfaces. As ruled surfaces are one parameter families of lines, one can apply curve theory and algorithms to the Klein image, when handling these surfaces. We study representations of rational ruled surfaces and get efficient algorithms for computation of planar intersections and contour outlines. Further, low degree boundary curves, useful for tensor product representations, are studied and illustrated at hand of several examples. Finally, we show how to compute efficiently low degree rational G 1 ruled surfaces.
To solve the problems in adjusting and controlling shapes of developable surfaces, a direct explicit and efficient methods of computer-aided design for developable surfaces with multiple local shape parameters are proposed. Firstly, a class of novel λμ-B-spline basis functions with two shape parameters is presented. And then, following the important idea of duality between points and planes in 3D projective space, the corresponding developable λμ-B-spline surfaces with multiple shape parameters are represented using control planes with λμ-B-spline basis functions. The developable λμ-B-spline surfaces inherit the outstanding properties of the B-spline surfaces, with a good performance on adjusting their local shapes by changing the two shape parameters. In addition, some properties of the developable λμ-B-spline surfaces and applications in developable surfaces design are discussed. The modeling examples illustrate that the developable λμ-B-spline surfaces provide a valuable ways for the design of developable surfaces.
In this paper, we discuss developable surfaces of ruled surfaces based on C-Bézier basis, which named C-developable surfaces. This method has three important advantages. No (nonlinear) charactering equations have to be solved and some special surfaces such as circular conical surface and circular cylindrical surface can be represented exactly. Moreover ruled surfaces based on C-Bézier basis can be computed by subdivision algorithm. It is useful to free form developable surfaces modeling with C-developable surfaces.
An explicit and efficient method of computer-aided design for adjustably developable surfaces is proposed. It is based on the important idea of duality between plane and point geometries. The shapes of developable surfaces can be adjusted using a control parameter. We show that the techniques for the geometric design of developable surfaces in this paper have all the characteristics of existing approaches for curves design. Further, we present a new method of displaying control planes and the generating plane at physically and geometrically meaningful location relative to the resulting surface. The new algorithm can not only overcome the difficulties in representation of developable surfaces utilizing traditional approaches, but also resolve the problems in adjusting and controlling the shape of developable surfaces that often faced in Engineering.
Given a closed plane curve c(t) = (c 1 , c 2)(t) ∈ R 2 and associated function values g(t) we present a geometric idea and an algorithm to solve the equation ∇ f = a = const. with respect to the boundary values g(t) along the boundary c(t). This is equivalent to finding a developable surface D of constant slope a = tan α through the spatial curve C determined by (c 1 , c 2 , g)(t). The presented method constructs level curves of the surface D. We put some emphasis on the treatment of the singularities of the solution which are D's self intersections.
An algorithm is presented that generates developable Bézier surfaces through a Bézier curve of arbitrary degree and shape, which interpolates prescribed corners and tangent planes along the boundary rulings. The control of singular points is guaranteed. The algorithm is based upon the algorithm of de Casteljau and the technique of degree elevation.
In this paper the authors present a way to study the analytical properties of a ship hull surface in order to modify it in a new one simpler to build for a shipyard but that also preserves some properties of the original surface.
The authors present a real case study of a ship with the functions used to modify the surface and with the hydrodynamic performances of the new ship.
Even if this is a “trial run” developed by the Dipartimento di Ingegneria Navale of Naples, the results are very interesting. However the main aim of this work is to optimize a numerical procedure for the above transformations.
For this reason the authors present the “tools” used to modify the surface in an algorithmic way so they can be easily implemented in a CAD software.
This paper presents a method to compute curvatures of a surface patch S obtained from a degenerate representation defined over a rectangular domain. To compute curvatures at a point q in S where the surface representation is degenerate one has to assume that the point set S has a tangent plane and well defined curvatures at the point q where the curvatures are to be computed. Our computation method employs a local height function representation of S in a neighborhood of the point q where the height function h is defined over the tangent plane of S at the point q. In our method the second order partial derivatives of the height function h at q are computed by using second order derivatives of any three surfaces curves on S which end up in q with pairwise linearly independent tangent directions. The curvature entities of S at q are then computed by using the second order partial derivatives of the height function h at q. We also show how the method is extended to compute partial derivatives of any order of the function h at q. For this purpose we show (in Theorem 1) how the partial derivatives up to nth order of a surface at a point q can be computed using derivatives up to order n of n + 1 surface curves emanating from q. We also present a definition of a concept of generalized surface curvatures under weaker assumptions appropriate for degenerate surfaces. Under those weaker assumptions one does not require a locally C2-smooth height function representation of the surface over the tangent plane at the degenerate point q, nor a locally well defined (single-valued) height function representation of the surface in a neighborhood of the point q. One only needs the existence of a unique local second order approximation of the degenerate surface with a quadratic function defined over the tangent plane at q. The surface curvatures of the corresponding approximating quadratic surface then define the curvatures of the approximated degenerate surface in the contact point q.
An algorithm mathematizes a developable surface design procedure for the digital computer. Given points on a pair of boundary curves, the algorithm generates spline-approximating polynomials to represent the boundary curves and computes a set of closely spaced rulings which lie in the surface spanning the boundary curves. Offsets to the surface are then computed at any specified transverse, vertical, or horizontal plane cutting the surface. The procedure emphasizes freedom of shape but does not guarantee the existence of a developable surface. In this case, the results will be information describing the developability violation rather than offsets describing the surface. The outstanding advantages of a computer-aided approach to developable surface design are speed and precision.
The fairbody, chine and sheer lines of a proposed vessel are represented by cubic spline functions. Between each pair of chine lines a ruled surface is generated which has the same tangent plane at all points of each generator or ruling line. A procedure based on the multiconic development of a surface is used to modify the given chine lines to ensure that no ruling lines intersect at a point within the surface. The result is a developable hull surface. A simple method is suggested for fairing the modified chine lines, and the steps necessary to generate tables of offsets from ruling-line intercepts are outlined briefly.
A constructive geometric approach to developable rational Bézier and B-spline surfaces is presented. It is based on the dual representation in the sense of projective geometry. By the principle of duality, projective algorithms for NURBS curves can be transferred to constructions for developable NURBS surfaces in dual rational B-spline form. We discuss the conversion to the usual tensor product representation of the obtained surfaces and develop algorithms for basic design problems arising in this context.
The concept of duality between points and planes in 3D projective space is used to develop a new representation for developable surfaces in terms of plane geometry. In this manner, a developable surface is designed using control planes with appropriate basis functions. The use of rational Bézier and B-spline bases is focused on, and a technique for the geometric design of developable surfaces is developed that has all the characteristics of existing methods for curve design. It is shown that some of the geometric constructions that exist for curves also generalize to the design of developable surfaces. In particular, de Casteljau- and Farin-Boehm-type construction algorithms are developed for Bézier developable surfaces.
The paper presents pseudocode algorithms, c-language code, and error analysis for removing knots from rational B-spline curves and surfaces. Efficient and easy-to-use algorithms are presented that, with one call, remove all the removable knots from a B-spline curve or surface.
Aumann, G., Interpolation with developable Bézier patches, Computer Aided Geometric Design 8 (1991) 409–420. This paper deals with the design of interpolating developable Bézier patches. Necessary and sufficient conditions are given for these patches to be free of singular points. From these conditions we deduce simple design criteria. Furthermore, G1 - and G2 -continuous connections of these patches will be studied.
Rational (1, n)-Bézier surfaces are ruled surfaces: They are generated by a one parameter set of straight lines. Among the ruled surfaces the developable ones play a special role in technical use. In this paper we give a general characterisation of developable rational (1, n)-Bézier surfaces.
This paper describes a new robust method to decompose a free-form surface into regions with specific range of curvature and provide important tools for surface analysis, tool-path generation, and tool-size selection for numerically controlled machining, tessellation of trimmed patches for surface interrogation and finite-element meshing, and fairing of free-form surfaces. The key element in these techniques is the computation ofall real roots within a finite box of systems of nonlinear equations involving polynomials and square roots of polynomials. The free-form surfaces are bivariate polynomial functions, but the analytical expressions of their principal curvatures involve polynomials and square roots of polynomials. Key components are the reduction of the problems into solutions of systems of polynomial equations of higher dimensionality through the introduction ofauxiliary variables and the use ofrounded interval arithmetic in the context of Bernstein subdivision to enhance the robustness of floating-point implementation. Examples are given that illustrate our techniques.