Tangent Plane Continuity Research Articles

A Local Approach for Computing Smooth B-Spline Surfaces for Arbitrary Quadrilateral Base Meshes

Abstract We present a method for approximating surface data of arbitrary topology by a model of smoothly connected B-spline surfaces. Most of the existing solutions for this problem use constructions with limited degrees-of-freedom or they address smoothness between surfaces in a post-processing step, often leading to undesirable surface behavior in proximity of the boundaries. Our contribution is the design of a local method for the approximation process. We compute a smooth B-spline surface approximation without imposing restrictions on the topology of a quadrilateral base mesh defining the individual B-spline surfaces, the used B-spline knot vectors, or the number of B-spline control points. Exact tangent plane continuity can generally not be achieved for a set of B-spline surfaces for an arbitrary underlying quadrilateral base mesh. Our method generates a set of B-spline surfaces that lead to a nearly tangent plane continuous surface approximation and is watertight, i.e., continuous. The presented examples demonstrate that we can generate B-spline approximations with differences of normal vectors along shared boundary curves of less than one degree. Our approach can also be adapted to locally utilize other approximation methods leading to higher orders of continuity.

Subdivision Schemes for Quadrilateral Meshes with the Least Polar Artifact in Extraordinary Regions

AbstractThis paper presents subdivision schemes with subdivision stencils near an extraordinary vertex that are free from or with substantially reduced polar artifact in extraordinary regions while maintaining the best possible bounded curvature at extraordinary positions. The subdivision stencils are firstly constructed to meet tangent plane continuity with bounded curvature at extraordinary positions. They are further optimized towards curvature continuity at an extraordinary position with additional measures for removing or for minimizing the polar artifact in extraordinary regions. The polar artifact for subdivision stencils of lower valences is removed by applying an additional constraint to the subdominant eigenvalue to be the same as that of subdivision at regular vertices, while the polar artifact for subdivision stencils of higher valances is substantially reduced by introducing an additional thin‐plate energy function and a penalty function for maintaining the uniformity and regularity of the characteristic map. A new tuned subdivision scheme is introduced by replacing subdivision stencils of Catmull‐Clark subdivision with that from this paper for extraordinary vertices of valences up to nine. We also compare the refined meshes and limit surface quality of the resulting subdivision scheme with that of Catmull‐Clark subdivision and other tuned subdivision schemes. The results show that subdivision stencils from our method produce well behaved subdivision meshes with the least polar artifact while maintaining satisfactory limit surface quality.

Smooth Local Interpolation of Surfaces Using Normal Vectors

This paper proposes a simple surface interpolation attaining tangent‐plane continuity. It is a natural extension of the local quadratic C0 interpolator developed by the author (2005) in one of his works, which has already been applied successfully to diverse engineering problems. The methodology presented in this paper inherits most of the advantages possessed by the C0 scheme. That is, (i) The algorithm is efficient and completely local requiring only the position vectors and normals given at the nodes of a patch, and hence it is suitable for parallel processing. (ii) It converges rapidly to the given surface with the increase in the number of nodes. (iii) Singular points (apexes, sharp edges, etc.) and nonmanifolds can be treated quite easily. (iv) Because of the minimization criteria assigned to the surface coefficients, it is rather robust and amenable to computational analyses. Validity and effectiveness of the proposed technique are demonstrated through numerical examples.

Approximating tensor product Bézier surfaces with tangent plane continuity

We present a simple method for degree reduction of tensor product Bézier surfaces with tangent plane continuity in L 2 -norm. Continuity constraints at the four corners of surfaces are considered, so that the boundary curves preserve endpoints continuity of any order α . We obtain matrix representations for the control points of the degree reduced surfaces by the least-squares method. A simple optimization scheme that minimizes the perturbations of some related control points is proposed, and the surface patches after adjustment are C ∞ continuous in the interior and G 1 continuous at the common boundaries. We show that this scheme is applicable to surface patches defined on chessboard-like domains.

A new ternary interpolatory subdivision scheme for polyhedral meshes with arbitrary topology

Interpolation using ternary subdivision is an attractive characteristic in geometric modeling due to its advantage over commonly used binary subdivision. In this paper, a new ternary interpolatory subdivision scheme with one parameter based on arbitrarily triangular meshes is proposed. It can be used to any polyhedral meshes with arbitrary topology. The sufficient conditions of the convergence and the tangent plane continuity of the presented subdivision scheme are obtained using geometric methods and algebraic manipulations. It is simple, fast and has a better local property in comparison with the existing ternary interpolatory surface subdivision schemes.

Hierarchical triangular splines

Smooth parametric surfaces interpolating triangular meshes are very useful for modeling surfaces of arbitrary topology. Several interpolants based on these kind of surfaces have been developed over the last fifteen years. However, with current 3D acquisition equipments, models are becoming more and more complex. Since previous interpolation methods lack a local refinement property, there is no way to locally adapt the level of detail. In this article, we introduce a hierarchical triangular surface model. The surface is overall tangent plane continuous and is defined parametrically as a piecewise quintic polynomial. It can be adaptively refined while preserving the overall tangent plane continuity. This model enables designers to create a complex smooth surface of arbitrary topology composed of a small number of patches to which details can be added by locally refining the patches until an arbitrary small size is reached. It is implemented as a hierarchical data structure where the top layer describes a coarse, smooth base surface and the lower levels encode the details in local frame coordinates.

Polynomial surfaces interpolating arbitrary triangulations

Triangular Bezier patches are an important tool for defining smooth surfaces over arbitrary triangular meshes. The previously introduced 4-split method interpolates the vertices of a 2-manifold triangle mesh by a set of tangent plane continuous triangular Bezier patches of degree five. The resulting surface has an explicit closed form representation and is defined locally. In this paper, we introduce a new method for visually smooth interpolation of arbitrary triangle meshes based on a regular 4-split of the domain triangles. Ensuring tangent plane continuity of the surface is not enough for producing an overall fair shape. Interpolation of irregular control-polygons, be that in 1D or in 2D, often yields unwanted undulations. Note that this undulation problem is not particular to parametric interpolation, but also occurs with interpolatory subdivision surfaces. Our new method avoids unwanted undulations by relaxing the constraint of the first derivatives at the input mesh vertices: The tangent directions of the boundary curves at the mesh vertices are now completely free. Irregular triangulations can be handled much better in the sense that unwanted undulations due to flat triangles in the mesh are now avoided.

A 3D contact smoothing method using Gregory patches

AbstractIn this work, a method is developed for smoothing three‐dimensional contact surfaces. The method can be applied to both regular and irregular meshes. The algorithm employs Gregory patches to interpolate finite element nodes and provide tangent plane continuity between adjacent patches. The resulting surface interpolation is used to calculate gaps and contact forces, in a variationally consistent way, such that contact forces due to normal and frictional contact vary smoothly as slave nodes transition from one patch to the next. This eliminates the ‘chatter’ which typically occurs in a standard contact algorithm when a slave node is situated near a master facet edge. The elimination of this chatter provides a significant improvement in convergence behaviour, which is illustrated by a number of numerical examples. Furthermore, smoothed surfaces also provide a more accurate representation of the actual surface, such that resulting stresses and forces can be more accurately computed with coarse meshes in many problems. This fact is also demonstrated by the numerical examples. Published in 2002 by John Wiley & Sons, Ltd.

Continuity between Gregory-like patches

The Gregory technique seeks to construct patches on the faces formed by a net of intersecting curves in space. It uses the curves themselves and cross-boundary tangent information, and the resultant patches join together with tangent plane continuity. Extensions of the technique to give second-order continuity are investigated. It is shown that a direct generalisation fails because local coordinate systems do not interact appropriately. An alternative means for generating these coordinates is proposed. This works successfully for patches with at least four sides and gives curvature continuity across patch boundaries.

Free-Form Modeling in Bilateral Brep and CSG Representation Schemes

This paper presents a unified approach for incorporating free-form solids in bilateral Brep and CSG representation schemes, by resorting to low-degree (quadratic, cubic) algebraic surface patches. We develop a general CSG solution that represents a free-form solid as a boolean combination of a direct term and a complicated delta term. This solution gives rise to the trunctet-subshell conditions, under which the delta term computation can be obviated. We use polyhedral smoothing to construct a Brep consisting of quadratic algebraic patches that meet with tangent-plane continuity, such that the trunctet-subshell conditions are guaranteed automatically. This guarantee is not currently available for cubic patches. The general CSG solution thus applies whenever trunctet-subshell conditions are violated, e.g. sometimes for cubic patches or sometimes for patches of any degree that are subject to shape control operations. Manifold solids of arbitrary topology can be represented in our dual representation system. Ensuing CSG constructs are parallel processed on the RayCasting Engine to support a wide range of solid modeling applications, including general sweeping, Minkowski operations, NC machining, and touch-sense probing.

A Symbolic Approach to Freeform Parametric Surface Blends

This paper presents a symbolic approach to the computation of blend surfaces across piecewise polynomial and rational surfaces. Curves in the parameter space of the primary surfaces can be specified as the rail curves, also known as trimlines, of the blend. Several techniques which require various levels of user interaction are presented for defining the cross-boundary tangent curves along the rail curves. The resulting blend is represented as a polynomial surface having tangent plane continuity with the primary surfaces to an accuracy bounded only by the machine precision. Also presented is a normalization method that approximates unit vector fields, an approach that might benefit other applications such as offset approximation and animation curve construction. © 1997 by John Wiley & Sons, Ltd.

A Symbolic Approach to Freeform Parametric Surface Blends

This paper presents a symbolic approach to the computation of blend surfaces across piecewise polynomial and rational surfaces. Curves in the parameter space of the primary surfaces can be specified as the rail curves, also known as trimlines, of the blend. Several techniques which require various levels of user interaction are presented for defining the cross-boundary tangent curves along the rail curves. The resulting blend is represented as a polynomial surface having tangent plane continuity with the primary surfaces to an accuracy bounded only by the machine precision. Also presented is a normalization method that approximates unit vector fields, an approach that might benefit other applications such as offset approximation and animation curve construction. © 1997 by John Wiley & Sons, Ltd.

The Gaussian and mean curvature criteria for curvature continuity between surfaces

In this paper, the influence of the tangent-plane continuity of two surfaces along a common linkage curve on (1) the Dupin indicatrices of the surfaces in terms of squared Dupin indicatrices; and (2) the Gaussian and mean curvatures of the surfaces in terms of ratios of differences between Gaussian and mean curvatures is studied. Alternative sets of curvature continuity conditions in terms of intrinsic geometry of surfaces are developed. Relationships between the equalities of Gaussian and mean curvatures and curvature continuity of the surfaces along the linkage curve are studied. For two surfaces which are tangent-plane continuous along a linkage curve, the following two criteria individually guarantee their curvature continuity along the linkage curve: (1) The Gaussian Curvature Criterion: the Gaussian curvatures along the linkage curve are the same, if the linkage curve does not contain straight lines or binormal generators of the two surfaces; (2) The Mean Curvature Criterion: the mean curvatures are the same along the linkage curve. Also an alternative proof for the Linkage Curve Theorem is provided. These criteria can be used in shape interrogations, such as in visual detection of curvature discontinuity between two surfaces.

Surface design using cyclide patches

Cyclide surfaces have low algebraic degree, exact nurbs representations and circular lines of curvature. Geometric modelling applications of the cyclide include blending of quadric surface intersections, piping layouts and cable harnesses. In this paper, we describe a scheme for smoothly composing a topologically rectangular network of cyclide patches. It is a two-step process. First, a cyclide quadrilateral mesh is constructed to approximate the target shape. Then, cyclide patches corresponding to the quadrilateral mesh are constructed. There is tangent plane continuity across the patch boundaries. Computations are based on the geometry of the cyclide and it is easy to detect degenerate cyclide patches, i.e. parabolic cyclide, cone, cylinder and sphere. We discuss issues related to global and local shape modification, and portability. Implemented examples are also included. Finally, we describe a simple strategy for NC machining of composite cyclide surfaces. Advantages are highlighted and implemented examples included.

A triangular G 1 patch from boundary curves

For some applications it is necessary to fit an irregular surface to given data, e.g. to develop a geometric model of a human skeletal bone from computerized tomography scans. Such a surface does not always have easily distinguishable isoparametric lines. It is thus not convenient to use standard global curve fitting techniques such as those based on B-splines. A global method may also smooth away essential features. A reasonable approach is to use a composite surface where individual surface patches are locally determined. To obtain some visual smoothness it is desirable that these patches join their neighbours in a manner that preserves positional as well as tangent plane continuity. Several methods have been presented for constructing surfaces in such a manner. A common initial stage in developing the patches is to determine a network of boundary curves. This article reports on some results using boundary curves based on a recent technique for point normal interpolation.

A degree estimate for subdivision surfaces of higher regularity

Subdivision algorithms can be used to construct smooth surfaces from control meshes of arbitrary topological structure. In contrast to tangent plane continuity, which is well understood, very little is known about the generation of subdivision surfaces of higher regularity. This work presents a degree estimate for piecewise polynomial subdivision surfaces saying that curvature continuity is possible only if the bi-degree d d of the patches satisfies d ≥ 2 k + 2 d \ge 2k+2 , where k k is the order of smoothness on the regular part of the surface. This result applies to any stationary or non-stationary scheme consisting of masks of arbitrary size provided that some generic symmetry and regularity assumptions are fulfilled.

Fair Surface Reconstruction Using Quadratic Functionals

AbstractAn algorithm for surface reconstruction from a polyhedron with arbitrary topology consisting of triangular faces is presented. The first variant of the algorithm constructs a curve network consisting of cubic Bézier curves meeting with tangent plane continuity at the vertices. This curve network is extended to a smooth surface by replacing each of the networks facets with a split patch consisting of three triangular Bézier patches. The remaining degrees of freedom of the curve network and the split patches are determined by minimizing a quadratic functional. This optimization process works either for the curve network and the split patches separately or in one simultaneous step. The second variant of our algorithm is based on the construction of an optimized curve network with higher continuity. Examples demonstrate the quality of the different methods.

A note on degenerate triangular Bézier patches

In (Neamtu and Pfluger, 1994) degenerate Bézier patches were used to interpolate irregularly structured spatial data. This very interesting approach is based on the observation that singular parametrizations are no fundamental obstacle to modeling smooth spline surfaces. However, the smoothness conditions on degenerate Bézier patches provided in (Neamtu and Pfluger, 1994) are inaccurate. Moreover, proving tangent plane continuity is not sufficient to guarantee smoothness in the sense of differential geometry. What has to be shown is regularity, i.e. the existence of a regular smooth re-parametrization representing a degenerate patch locally near the singular point. Both correct sufficient smoothness conditions and the proof of regularity are presented in this paper.

Pseudoedge: nonintersected parametric quilt modeling of multiply connected objects

The paper presents a new modeling approach for the representation of physical objects. In this approach, the shape of an object is expressed by its areas of prominence or high curvature, for which the term ‘pseudoedges’ is used. In terms of traditional mechanical design, these represent fillet, chamfer and intersection lines, as well as more general shape features. The pseudoedges of the model combine with a skeletal shape that is used as a starting form, thereby creating a hierarchy of geometric dependencies that allows both global and local control. The resulting surface may be closed and multiply connected, and it is represented by a quilt of parametric patches, with tangent-plane continuity being ensured by simple rules. Considerable degrees of deformation are possible, with predictable control and small computational expense; no computation of intersections or parameter-space trimming of patches are involved.

Canopy surface technique for parametric blending

A method based on canopy surfaces is presented for blending parametric surfaces. The blend designed using this method gives freedom to the designer in selecting i) the primitive surfaces to be blended, ii) the contact curves lying on them (in which the blend meets the primitives), iii) the endtangent directions along the contact curves (which are used to ensure tangentplane continuity), and iv) the shape of the cross-sectional curve. An important feature of this method, which is not seen in earlier methods, is the use of end tangents to ensure theC 1 continuity of iso-parametric curves across the junctions between the blend and the primitives in addition to the tangent-plane continuity.