Blending Surfaces Research Articles

Smoothly Blending of Two Elliptic Cylinders Whose Axes are Non-Coplanar

This paper studies the smoothly blending problem of two elliptic cylinders whose axes are non-coplanar at sections where they are perpendicular to axes. Based on conditions of smoothly blending axes, we acquire necessary and sufficient conditions of existing blending surface by constructed elliptic cylindrical helicoid tube. And when we meet those conditions, the specific blending examples will be provided.

On modeling with rational ringed surfaces

A surface in Euclidean space is called ringed (or cyclic) if there exists a one-parameter family of planes that intersects this surface in circles. Well-known examples of ringed surfaces are the surfaces of revolution, (not only rotational) quadrics, canal surfaces, or Darboux cyclides. This paper focuses on modeling with rational ringed surfaces, mainly for blending purposes. We will deal with the question of rationality of ringed surfaces and discuss the usefulness of the so called P-curves for constructing rational ringed-surface-blends. The method of constructing blending surfaces that satisfy certain prescribed constraints, e.g. a necessity to avoid some obstacles, will be presented. The designed approach can be easily modified also for computing n-way blends. In addition, we will study the contour curves on ringed surfaces and use them for computing approximate parameterizations of implicitly given blends by ringed surfaces. The designed techniques and their implementations are verified on several examples.

Semi-structured [formula omitted]-spline for blending two [formula omitted]-spline surfaces

Surface blending is a useful operation in geometric design for rounding sharp edges or corners. Meanwhile, NURBS has already become the de facto industrial standard in existing CAD/CAM systems. Therefore, it is required to study how to blend two B-spline surfaces. However, two arbitrary B-spline surfaces (called base surfaces) are hard to be blended with a B-spline surface (called blending surface) because the knot vectors of the two base surfaces are usually mismatched. In this paper, we proposed a curve-based spline representation, i.e., the semi-structured B-spline surface, which is generated by skinning a series of B-spline curves with different knot vectors. By assigning suitable knot vectors to the head and tail skinned curves, the semi-structured B-spline surface can blend two B-spline surfaces smoothly without disturbing them at all. We formulated the B-spline surface blending problem as an optimization problem with continuity constraints, and the continuity between the base and blending surfaces can reach G2 or C2. Examples illustrated in this paper validate the effectiveness and efficiency of our method.

Blending using ODE swept surfaces with shape control and $$C^1$$ C 1 continuity

Surface blending with tangential continuity is most widely applied in computer-aided design, manufacturing systems, and geometric modeling. In this paper, we propose a new blending method to effectively control the shape of blending surfaces, which can also satisfy the blending constraints of tangent continuity exactly. This new blending method is based on the concept of swept surfaces controlled by a vector-valued fourth order ordinary differential equation (ODE). It creates blending surfaces by sweeping a generator along two trimlines and making the generator exactly satisfy the tangential constraints at the trimlines. The shape of blending surfaces is controlled by manipulating the generator with the solution to a vector-valued fourth order ODE. This new blending methods have the following advantages: (1) exact satisfaction of $$C^1$$ C 1 continuous blending boundary constraints, (2) effective shape control of blending surfaces, (3) high computing efficiency due to explicit mathematical representation of blending surfaces, and (4) ability to blend multiple (more than two) primary surfaces.

Employing Generalized Bezier Tube to Smoothly Blending Tubes Whose Axes are Non-Coplanar

Given two tubes with non-coplanar axes whose sections are circle, we acquire that axes of the tubes can be smoothly blent by Bezier curve. In this situation, the paper proposes a new method to build the blending surface by generalized Bezier tube. Meantime, we obtain smoothly blending parameter surface.

Automatic Creation of Blending Surfaces in Hydropower Generators Turbine Blades

Hydropower generators turbine blade is a one of kind product that requires specific maintenance. They are designed in very specific software tools based on features, which allow good CAD/CAE automation by simply tweaking parameters. However, such CAD/CAE automation has brought an undesired side effect: it's hard to add new features that are not considered in the original CAD. Specially in simulation-optimization applications, where parameters that are not considered in the design must be modified. This is the case in this work, where fluid-structure analysis is performed and a geometric feature that transcend the design activity (blending surface between the crown and the blade) is modified. This paper describes an ongoing project which involves the implementation of a module able to intercept the geometry generated by a CAD program using the IGES format, create/modify a new geometric feature and update the fluid volume and blade geometries. The surface boundary curves are extracted, and the adjacency relationship between the surfaces is determined by coordinate numerical comparison. The surfaces are coherently oriented and the solid models are completely determined. Using the de Boor's algorithm and the surface orientation a model with exclusive triangular faces coherently oriented is created, and then the blending surface is automatically created. The refined mesh for the models are created and fluid-structure analysis is performed for the simulation-optimization.

Employing Generalized Cylindrical Helicoid Tube to Smoothly Blending Tubes Whose Axes are Non-Coplanar

Given two tubes with non-coplanar axes whose sections are circle, we acquire that axes of the tubes can be smoothly blent by cylindrical helix. In this situation, the paper proposes a new method to build the blending surface by generalized cylindrical helicoid tube. Meantime, we obtain smoothly blending parameter surface.

Controllable C1 continuous blending of time-dependent parametric surfaces

This paper proposes the concept of blending time-dependent varying surfaces, and develops a new method to create a controllable C1 continuous blending surface between primary parametric surfaces whose position and shape change with time. We treat it as a boundary-valued problem defined by the mathematical model of a vectored dynamic fourth-order partial differential equation subjected to time-dependent C1 continuous blending boundary constraints. High performance blending surface generation is achieved through the development of an approximate analytical solution of the mathematical model. We investigate the accuracy and efficiency of the solution, study the effective shape control of the blending surfaces, and apply the obtained solution to tackle surface blending problems. The applications demonstrate that our proposed approach is very effective and efficient in dealing with controllable C1 continuous surface blending between time-dependent varying parametric surfaces.

Gn-blending of multiple parametric normal ringed surfaces by adding implicit closings Gn-continuous with the surfaces

The functional spline is a highly effective tool for blending multiple implicit surfaces, but the pipes to be blended are generally defined as parametric normal ringed surfaces. To bridge the gap between the equation forms, Hartmann presented a closing-based algorithm (constructing a closing for each pipe and blending the closings), where a closing is an implicit surface sealing a pipe end (inlet or outlet). So far, however, the closing construction remains imperfect: the initial normal ringed surfaces must satisfy a certain geometric requirement and the obtained closings are only G1-continuous with these surfaces (as is the resulting blending surface). This paper proposes a new algorithm that finds the desired closings without restricting its scope of application. The procedure involves the rational parametrization of the initial surfaces and seeking the closings, interpolating rational surfaces with higher-order continuity through constrained optimization. Three practical examples illustrate the strong performance of this algorithm. In particular, the last example addresses the normal elliptic surfaces.

The Geometric Modeling of Industrial Products Blending Surface Design

Abstract In this paper, the geometric modeling for blending surface as algebraic surface is proposed. The blending surface has a simple exclusive G1 continuity expression; its shape can be adjusted by changing λ values. Therefore, their shape can get the advantages of tube – modeling, it is important and necessary in this field for industrial products design.

Blending surfaces generated using the Bernstein operator

In this paper we construct blending surfaces using the univariate Bernstein operator. The surfaces have the properties that they stay on a curve (the border of the surfaces domain) and have a fixed height in a point from the domain. The surfaces are generated using a curve network, instead of the control points from the case of classical Bezier surfaces. We study the monotonicity and we give conditions to obtain concave surfaces.

基于周期B 样条曲面的环状<italic>N</italic> 边洞<italic>G<sup>n</sup></italic> 连续性过渡

The orbicular <italic>N</italic>-sided hole filling problem is usually introduced by filleting an end-point of a part with large radius. The existing methods based on quadrilateral partition or constrained-optimization can rarely generate high-order continuous blending surfaces under these circumstances. This paper first reparameterizes the boundary of the specified orbicular <italic>N</italic>-sided hole to ensure the compatibility of neighboring cross-boundary derivatives on the connecting points, preserving their <italic>Gⁿ</italic> continuity. Then we compute the control points of the periodic B-spline surface using the sufficient <italic>Gⁿ</italic> continuity condition on the pole and the algorithm of extending parametric surfaces. This method generates single blending surface, which can be converted into standard B-spline surface by adding knots without introducing errors. It only elevates the degree of the boundary by <italic>n</italic>. The construction method is simple and efficient, without iteration nor large-scale matrix solving. It achieves <italic>Gⁿ</italic> continuity under compatible conditions. The blending examples underline its feasibility and practicability.

Blends of canal surfaces from polyhedral medial transform representations

We present a new method for constructing G1 blending surfaces between an arbitrary number of canal surfaces. The topological relation of the canal surfaces is specified via a convex polyhedron and the design technique is based on a generalization of the medial surface transform. The resulting blend surface consists of trimmed envelopes of one- and two-parameter families of spheres. Blending the medial surface transform instead of the surface itself is shown to be a powerful and elegant approach for blend surface generation. The performance of our approach is demonstrated by several examples.

G n filling orbicular N-sided holes using periodic B-spline surfaces

The orbicular N-sided hole filling problem is usually introduced by filleting an end-point of a part with large radius. The existing methods based on quadrilateral partition or constrained-optimization can rarely generate high-order continuous blending surfaces under these circumstances. This paper first reparameterizes the boundary of the specified orbicular N-sided hole to ensure the compatibility of neighboring cross-boundary derivatives on the connecting points, preserving their Gn continuity. Then we compute the control points of the periodic B-spline surface using the sufficient Gn continuity condition on the pole and the algorithm of extending parametric surfaces. This method generates single blending surface, which can be converted into standard Bspline surface by adding knots without introducing errors. It only elevates the degree of the boundary by n. The construction method is simple and efficient, without iteration nor large-scale matrix solving. It achieves Gn continuity under compatible conditions. The blending examples underline its feasibility and practicability.

Construction and Rendering of Trimmed Blending Surfaces with Sharp Features on a GPU

We construct surfaces with darts, creases, and corners by blending different types of local geometries. We also render these surfaces efficiently using programmable graphics hardware. Points on the blending surface are evaluated using simplified computation which can easily be performed on a graphics processing unit. Results show an eighteen-fold to twenty-fold increase in rendering speed over a CPU version. We also demonstrate how these surfaces can be trimmed using textures.

Geometric Information and Rational Parametrization of Nonsingular Cubic Blending Surfaces

The techniques for parametrizing nonsingular cubic surfaces have shown to be of great interest in recent years. This paper is devoted to the rational parametrization of nonsingular cubic blending surfaces. We claim that these nonsingular cubic blending surfaces can be parametrized using the symbolic computation due to their excellent geometric properties. Especially for the specific forms of these surfaces, we conclude that they must be F3, F4, or F5 surfaces, and a criterion is given for deciding their surface types. Besides, using the algorithm proposed by Berry and Patterson in 2001, we obtain the uniform rational parametric representation of these specific forms. It should be emphasized that our results in this paper are invariant under any nonsingular real projective transform. Two explicit examples are presented at the end of this paper.

N-way blending problem of circular quadrics

A method of blending circular quadrics with parametric patches is proposed in this paper. It needs n rational bi-cubic Bezier patches and two S-patches to blend n (n > 2) quadrics. Bi-cubic rational Bezier patches are used to G1-continuously blend each pair of adjacent base surfaces. These patches enclose two n-sided holes. And an S-patch is used to fill each hole, which keeps nearly G1-continuity. Explicit formulas of control points for both Bezier patches and S-patches are derived from their constraint conditions, which make modeling by blending quadrics very simple and convenient. In addition, the shapes of the blending surfaces can be intuitively adjusted by free parameters.

Formula omitted] blending multiple surfaces in polar coordinates

This paper proposes a method of G n blending multiple parametric surfaces in polar coordinates. It models the geometric continuity conditions of parametric surfaces in polar coordinates and presents a mechanism of converting a Cartesian parametric surface into its polar coordinate form. The basic idea is first to reparameterize the parametric blendees into the form of polar coordinates. Then they are blended simultaneously by a basis function in the complex domain. To extend its compatibility, we also propose a method of converting polar coordinate blending surface into N NURBS patches. One application of this technique is to fill N -sided holes. Examples are presented to show its feasibility and practicability.

Polyhedral vertex blending with setbacks using rational S-patches

Setbacks not only beautify the shape of polyhedral vertex blending, but also overcome the difficulties inherited from non-setback blending. However, stitching several tensor product Bézier surfaces, the existing approaches to setback vertex blending may cause twist compatibility problems and approximate solutions, especially when the vertex gathers more edges. This paper proposes a new algorithm, yielding an exact solution using a single rational S-patch. Although S-patch developers Loop and DeRose (1989) also investigated vertex blending ( 1990), they did not configure setback. In addition, the new algorithm reduces the depth of the S-patch and extends the edge blending surfaces to use rational Bézier representation, so that circular rounding can be made. Miscellaneous examples demonstrate that the method produces smooth and natural connecting shapes.

Determination Of Euclidean Distances For Symmetry Molecules

This paper represents the geometric analysis of molecular surfaces of the molecules, indicates the blending operation of an atoms constitute to the small molecules. The decision which indicates advantages of Euclidean Voronoi diagram of an atom includes the blending surface among the atoms to make a fundamental study of docking, interactions with macromolecules. The algorithm which proposes the topological part of surfaces discussed through the Euclidean Voronoi Diagram of various accessibility procedures.