Analysis and avoidance of singularities for local G 1 surface interpolation of Bézier curve network with 4-valent nodes

We propose a local method of constructing piecewise G ¹ Bézier patches to span a Bézier curve network with odd- and 4-valent node points. We analyze all possible singular cases of the G ¹ condition that is to be met by the curve network interpolation and propose a new G ¹ continuity condition using linear and quartic scalar weight functions. Using this condition, a curve network can be interpolated without modification at 4-valent nodes with two collinear tangent vectors, even in the presence of singularities. We demonstrate our approach by generating G ¹ surfaces over the curve network which includes singularities at its node vertices and edges.     

参数曲面上的插值与混合

如何表示曲面上的曲线,在处理诸如数控加工中的路径设计以及CAD/CAM等领域频繁出现的曲面裁剪问题时显得日益重要.给出了数据点的切方向(切方向及曲率向量或测地曲率值)指定而G¹连续(G²连续)插值曲面上任意点列的方法.作为曲面上曲线插值问题的特例,还讨论了曲面上曲线的混合问题.基本思想是借助于微分几何的有关结论,曲面上曲线的插值问题被转化为其参数平面上类似的曲线插值问题.该方法能够用二维隐式方程来表示曲面上的插值曲线,从而把在显示该曲线时所面对的曲面求交的几何问题转化为计算隐式曲线的代数问题.实验证明该方法是可行的,而且适用于CAD/CAM及计算机图形学等领域.     

A Multi‐sided Bézier Patch with a Simple Control Structure

A new n‐sided surface scheme is presented, that generalizes tensor product Bézier patches. Boundaries and corresponding cross‐derivatives are specified as conventional Bézier surfaces of arbitrary degrees. The surface is defined over a convex polygonal domain; local coordinates are computed from generalized barycentric coordinates; control points are multiplied by weighted, biparametric Bernstein functions. A method for interpolating a middle point is also presented. This Generalized Bézier (GB) patch is based on a new displacement scheme that builds up multi‐sided patches as a combination of a base patch, n displacement patches and an interior patch; this is considered to be an alternative to the Boolean sum concept. The input ribbons may have different degrees, but the final patch representation has a uniform degree. Interior control points—other than those specified by the user—are placed automatically by a special degree elevation algorithm. GB patches connect to adjacent Bézier surfaces with G¹continuity. The control structure is simple and intuitive; the number of control points is proportional to those of quadrilateral control grids. The scheme is introduced through simple examples; suggestions for future work are also discussed.     

Sketching piecewise clothoid curves

We present a novel approach to sketching 2D curves with minimally varying curvature as piecewise clothoids. A stable and efficient algorithm fits a sketched piecewise linear curve using a number of clothoid segments with G² continuity based on a specified error tolerance. Further, adjacent clothoid segments can be locally blended to result in a G³ curve with curvature that predominantly varies linearly with arc length. We also handle intended sharp corners or G¹ discontinuities, as independent rotations of clothoid pieces. Our formulation is ideally suited to conceptual design applications where aesthetic fairness of the sketched curve takes precedence over the precise interpolation of geometric constraints. We show the effectiveness of our results within a system for sketch-based road and robot-vehicle path design, where clothoids are already widely used.     

Adaptive patch-based mesh fitting for reverse engineering

In this paper,  we propose a novel adaptive mesh fitting algorithm that fits a triangular model with G¹ smoothly stitching bi-quintic Bézier patches. Our algorithm first segments the input mesh into a set of quadrilateral patches, whose boundaries form a quadrangle mesh. For each boundary of each quadrilateral patch, we construct a normal curve and a boundary-fitting curve, which fit the normal and position of its boundary vertices respectively. By interpolating the normal and boundary-fitting curves of each quadrilateral patch with a Bézier patch, an initial G¹ smoothly stitching Bézier patches is generated. We perform this patch-based fitting scheme in an adaptive fashion by recursively subdividing the underlying quadrilateral into four sub-patches. The experimental results show that our algorithm achieves precision-ensured Bézier patches with G¹ continuity and meets the requirements of reverse engineering.     

Characterizing degrees of freedom for geometric design of developable composite Bézier surfaces

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.     

Curvature continuous interpolation of curve meshes

This paper presents a method for local construction of a curvature continuous (GC²) piecewise polynomial surface which interpolates a given rectangular curvature continuous quintic curve mesh. First, we create a C² quintic basic curve mesh, which interpolates the same grid points, preserves the tangent slopes and curvatures but not derivative vectors at the grid points. After estimating twist and higher order mixed partial derivatives for each grid point, we generate locally the so-called C² biquintic basic patches, which serve to compute the first and second order cross-derivative functions of the final interpolation surface. In order to match the tangents and second order derivative vectors of the original boundary curves at the grid points, these basic patches are reparametrized by 5 × 3 and 3 × 5 functions, which lead to vector-valued first and second order cross-derivative functions of degrees 7 and 9 of the final surface patches, and eventually lead to a GC² piecewise polynomial surface of degree 9 × 9, which is then converted to a GC² Bézier composite surface. By arranging the surface patches in a chess-board fashion, the degrees of the final surface patches can be 9 × 5 and 5 × 9. An example for the construction of a GC² ship hull, together with its color-coded curvature maps, is given to illustrate the method. This method is attractive because the final surface has a much lower degree than other similar methods, it allows flexible local modification of the original curve mesh and local editing of the interpolation surface, and it is easily integrated into state-of-the-art geometric modeling systems.     

Triangular bubble spline surfaces

We present a new method for generating a Gⁿ-surface from a triangular network of compatible surface strips. The compatible surface strips are given by a network of polynomial curves with an associated implicitly defined surface, which fulfill certain compatibility conditions. Our construction is based on a new concept, called bubble patches, to represent the single surface patches. The compatible surface strips provide a simple Gⁿ-condition between two neighboring bubble patches, which are used to construct surface patches, connected with Gⁿ-continuity. For n≤2, we describe the obtained Gⁿ-condition in detail. It can be generalized to any n≥3. The construction of a single surface patch is based on Gordon–Coons interpolation for triangles.Our method is a simple local construction scheme, which works uniformly for vertices of arbitrary valency. The resulting surface is a piecewise rational surface, which interpolates the given network of polynomial curves. Several examples of G⁰, G¹ and G²-surfaces are presented, which have been generated by using our method. The obtained surfaces are visualized with reflection lines to demonstrate the order of smoothness.     

Interpolating G Bézier surfaces over irregular curve networks for ship hull design

We propose a local method of constructing piecewise G¹ Bézier patches to span an irregular curve network, without modifying the given curves at odd- and 4-valent node points. Topologically irregular regions of the network are approximated by implicit surfaces, which are used to generate split curves, which subdivide the regions into triangular and/or rectangular sub-regions. The subdivided regions are then interpolated with Bézier patches. We analyze various singular cases of the G¹ condition that is to be met by the interpolation and propose a new G¹ continuity condition using linear and quartic scalar weight functions. Using this condition, a curve network can be interpolated without modification at 4-valent nodes with two collinear tangent vectors, even in the presence of singularities. We demonstrate our approach in a ship hull.     

G 2 interpolation and blending on surfaces

We introduce a method for curvature-continuous (G ²) interpolation of an arbitrary sequence of points on a surface (implicit or parametric) with prescribed tangent and geodesic curvature at every point. The method can also be used forG ² blending of curves on surfaces. The interpolation/blending curve is the intersection curve of the given surface with a functional spline (implicit) surface. For the construction of blending curves, we derive the necessary formulas for the curvature of the surfaces. The intermediate results areG ² interpolation/blending methods in IR².     

GC 1 multisided Bézier surfaces

This paper presents a new method for generating a tangent-plane continuous (GC¹) multisided surface with an arbitrary number of sides. The method generates piecewise biquintic tensor product Bézier patches which join each other with G¹-continuity and which interpolate the given vector-valued first order cross-derivative functions along the boundary curves. The problem of the twist-compatibility of the surface patches at the center points is solved through the construction of normal-curvature continuous starlines and by the way the twists of surface patches are generated. This avoids the inter-relationship among the starlines and the twists of surface patches at the center points. The generation of the center points and the starlines has many degrees of freedom which can be used to modify and improve the quality of the resulting surface patches. The method can be used in various geometric modeling applications such as filling n-sided holes, smoothing vertices of polyhedral solids, blending multiple surfaces, and modeling surface over irregular polyhedral line and curve meshes.     

G 2 blends of linear segments with cubics and Pythagorean-hodograph quintics

Fillets, also known as blend arcs, are used in CNC machining to round corners. Fillets are normally circular arcs, which have G ¹ contact with the straight line segments to which they are joined. Recent advances in machining technology allow NURBS, including Pythagorean-hodograph (PH) curve segments, to be incorporated in CNC tool paths. This article examines the use of cubic and PH quintic Bézier curve segments that have a single curvature extremum, and which have G ² contact with the straight line segments to which they are joined, as fillets. It is shown how the extreme circle of curvature can be determined. The point of curvature extremum and the corresponding value of the curvature can be changed by adjusting the joining points of the blending curve with the neighbouring straight lines. These blending curves can also be incorporated in computer-aided design packages for curve or surface design.     

Constructing up to G 2 continuous curve on freeform surface

This paper presents new methods for G ¹ and G ² continuous interpolation of an arbitrary sequence of points on an implicit or parametric surface with prescribed tangent direction and both tangent direction and curvature vector, respectively, at every point. We design a G ¹ or G ² continuous curve in three-dimensional space, construct a so-called directrix vector field using the space curve and then project a special straight line segment onto the given surface along the directrix vector field. With the techniques in classical differential geometry, we derive a system of differential equations for the projection curve. The desired interpolation curve is just the projection curve, which can be obtained by numerically solving the initial-value problems for a system of first-order ordinary differential equations in the parametric domain associated to the surface representation for the parametric case or in three-dimensional space for the implicit case. Several shape parameters are introduced into the resulting curve, which can be used in subsequent interactive modification such that the shape of the resulting curve meets our demand. The presented method is independent of the geometry and parameterization of the base surface, and numerical experiments demonstrate that it is effective and potentially useful in patterns design on surface.     

Curve fitting and fairing using conic splines

We present an efficient geometric algorithm for conic spline curve fitting and fairing through conic arc scaling. Given a set of planar points, we first construct a tangent continuous conic spline by interpolating the points with a quadratic Bézier spline curve or fitting the data with a smooth arc spline. The arc spline can be represented as a piecewise quadratic rational Bézier spline curve. For parts of the G¹ conic spline without an inflection, we can obtain a curvature continuous conic spline by adjusting the tangent direction at the joint point and scaling the weights for every two adjacent rational Bézier curves. The unwanted curvature extrema within conic segments or at some joint points can be removed efficiently by scaling the weights of the conic segments or moving the joint points along the normal direction of the curve at the point. In the end, a fair conic spline curve is obtained that is G² continuous at convex or concave parts and G¹ continuous at inflection points. The main advantages of the method lies in two aspects, one advantage is that we can construct a curvature continuous conic spline by a local algorithm, the other one is that the curvature plot of the conic spline can be controlled efficiently. The method can be used in the field where fair shape is desired by interpolating or approximating a given point set. Numerical examples from simulated and real data are presented to show the efficiency of the new method.     

Rational bi‐cubic G2 splines for design with basic shapes

The paper develops a rational bi‐cubic G² (curvature continuous) analogue of the non‐uniform polynomial C² cubic B‐spline paradigm. These rational splines can exactly reproduce parts of multiple basic shapes, such as cyclides and quadrics, in one by default smoothly‐connected structure. The versatility of this new tool for processing exact geometry is illustrated by conceptual design from basic shapes.     

G2 quasi-developable Bezier surface interpolation of two space curves

Surface development is used in many manufacturing planning operations, e.g., for garments, ships and automobiles. However, most freeform surfaces used in design are not developable, and therefore the developed patterns are not isometric to the original design surface. In some domains, the CAD model is created by interpolating two given space curves. In this paper, we propose a method to obtain a G² quasi-developable Bezier surface interpolating two arbitrary space curves. The given curves are first split into a number of piecewise Bezier curves and elemental Bezier patches each of which passes through four splitting points are constructed. All neighboring elemental patches are G² connected and they are assembled optimally in terms of the degree of developability (the integral Gaussian curvature). Experiments show that the final composite Bezier surface is superior to a lofted one which is defined regardless of the final surface developability.     

Transfinite surface interpolation with interior control

There are various techniques to design complex free-form shapes with general topology. In contrast to the approaches based on trimmed surfaces and control polyhedra, in curve network-based design feature curves can be directly created and edited in 3D. Multi-sided patches interpolate this curve network with slopes given by associated tangent ribbons. The patches are smoothly connected and yield a natural and predictable surface model. This paper focuses on special design techniques to adjust the interior of transfinite patches when further shape control is needed. While the boundary constraints are retained, additional vertices, curves and even interior control surfaces are supplemented to gain more design freedom. The main idea is to apply different distance-based blending functions with special parameterizations over non-regular, n-sided domains. This concept can be naturally extended to create one- and two-sided patches as well. Shape variations will be demonstrated by a few simple examples.     

Constructing G 1 continuous curve on a free-form surface with normal projection

In this paper, we develop a new method for G ¹ continuous interpolation of an arbitrary sequence of points on an implicit or parametric surface with a specified tangent direction at every point. Based on the normal projection method, we design a G ¹ continuous curve in three-dimensional space and then project orthogonally the curves onto the given surface. With the techniques in classical differential geometry, we derive a system of differential equations characterizing the projection curve. The resulting interpolation curve is obtained by numerically solving the initial-value problems for a system of first-order ordinary differential equations in the parametric domain associated to the surface representation for a parametric case or in three-dimensional space for an implicit case. Several shape parameters are introduced into the resulting curve, which can be used in subsequent interactive modification such that the shape of the resulting curve meets our demand. The presented method is independent of the geometry and parameterization of the base surface, and numerical experiments demonstrate that it is effective and potentially useful in surface trim, robot, patterns design on surface and other industrial and research fields.     

CAD-based sampling for CMM inspection of models with sculptured features

This paper addresses sampling models for trimmed sculptured surfaces, and multiple patches with boundary conditions. A CAD-based sampling system is developed and implemented in this work. The sculptured features are sampled along their isoparametric curves. These curves are then used to re-construct the model geometry using the skinning of cross section curves. We refer to the re-constructed model as the substitute geometry. The problem is to determine the sample curve locations such that the substitute geometry satisfies certain geometric conditions. These are, the form error, and the continuity of the substitute geometry across the boundaries of adjacent surfaces. Three criteria are integrated to determine the sample locations: the surface curvature change, the substitute geometry deviation from the CAD model, and the significance of trimmed portions of the surface. A boundary representation-based methodology for the sampling of trimmed surfaces is developed and implemented. This methodology is extended to handle n-sided surfaces obtained through filling n-sided regions with quadrilateral surface patches, and models that may include multiple surface patches. Furthermore, a tool to assess the sampling plans based on the continuity of the substitute geometry across boundaries of adjacent surface patches is developed. The developed algorithms, their implementations, and case studies are presented in this paper.     

A framework for quad/triangle subdivision surface fitting: Application to mechanical objects

In this paper we present a new framework for subdivision surface approximation of three‐dimensional models represented by polygonal meshes. Our approach, particularly suited for mechanical or Computer Aided Design (CAD) parts, produces a mixed quadrangle‐triangle control mesh, optimized in terms of face and vertex numbers while remaining independent of the connectivity of the input mesh. Our algorithm begins with a decomposition of the object into surface patches. The main idea is to approximate the region boundaries first and then the interior data. Thus, for each patch, a first step approximates the boundaries with subdivision curves (associated with control polygons) and creates an initial subdivision surface by linking the boundary control points with respect to the lines of curvature of the target surface. Then, a second step optimizes the initial subdivision surface by iteratively moving control points and enriching regions according to the error distribution. The final control mesh defining the whole model is then created assembling every local subdivision control meshes. This control polyhedron is much more compact than the original mesh and visually represents the same shape after several subdivision steps, hence it is particularly suitable for compression and visualization tasks. Experiments conducted on several mechanical models have proven the coherency and the efficiency of our algorithm, compared with existing methods.