Görbék és felületek a geometriai modellezésben = Curves and surfaces in geometric modelling

B-spline görbék/felületek pontjai által, az alakzat két csomóértékének szimmetrikus változtatásakor leírt pályagörbéket vizsgáltuk, és olyan alakmódosítási eljárást adtunk, amivel a felület adott pontját/paramétervonalát előre megadott helyre mozgathatjuk a csomóértékek változtatásával. A C-Bézier, C-B-spline és F-B-spline görbék pályagörbéinek geometriai tulajdonságait írtuk le, és erre alapozva geometriai kényszereket kielégítő alakmódosításokat vizsgáltuk. Olyan általános leírási módot (linear blending) adtunk, mely egységesen kezeli az alakparaméterekkel rendelkező görbék széles osztályát, továbbá konkrét esetekben e paraméterek geometriai hatását írtuk le és kényszeres alakmódosításokra adtunk megoldást. A csomóértékeknek az interpoláló görbére gyakorolt hatását vizsgáltuk, mely alapján a harmadfokú interpoláció esetére interaktív alakmódosító eljárást dolgoztunk ki. Kontrollpontokkal adott görbék szingularitásainak detektálására a kontrollpontok helyzetén alapuló megoldást adtunk. Kontrollpont alapú szükséges és elégséges feltételt adtunk arra, hogy a Bézier-felület paramétervonalai egyenesek legyenek. Olyan Monte Carlo módszert dolgoztunk ki, amely rendezetlen ponthalmaz felülettel való interpolálásához négyszöghálót hoz létre a pontfelhő (mely elágazásokat és hurkokat is tartalmazhat) és annak topológikus gráfja ismeretében. A csonkolt Fourier-sorok terében olyan ciklikus bázist adtunk meg, amellyel végtelen simaságú zárt görbéket és felületeket írhatunk le. | We studied paths of points of B-spline curves/surfaces obtained by the symmetric alteration of two knot values and provided a constrained shape modification method that is capable of moving a point/isoparametric line of the surface to a user specified position. We described the geometric properties of paths of C-Bézier, C-B-spline and F-B-spline curves and on this basis we studied shape modifications subject to geometric constraints. We developed the general linear blending method that treats a wide class of curves with shape parameters in a uniform way; in special cases we described the geometric effects of shape parameters and provided constrained shape modification methods. We examined the impact of knots on the shape of interpolating curves, based on which we developed an interactive shape modification method for cubic interpolation. We proposed a control point based solution to the problem of singularity detection of curves described by control points. We provided control point based necessary and sufficient conditions for Bézier surfaces to have linear isoparametric lines. We developed a Monte Carlo method to generate a quadrilateral mesh (for surface interpolation) from point clouds (with possible junctions and loops) and their topological graph. We specified a cyclic basis in the space of truncated Fourier series by means of which we can describe closed curves and surfaces with C^infinity

Constrained modification of the cubic trigonometric Bézier curve with two shape parameters

A new type of cubic trigonometric Bézier curve has been introduced in [1]. This trigonometric curve has two global shape parameters λ and µ. We give a lower boundary to the shape parameters where the curve has lost the variation diminishing property. In this paper the relationship of the two shape parameters and their geometric effect on the curve is discussed. These shape parameters are independent and we prove that their geometric effect on the curve is linear. Because of the independence constrained modification is not unequivocal and it raises a number of problems which are also studied. These issues are generalized for surfaces with four shape parameters. We show that the geometric effect of the shape parameters on the surface is parabolic. Keywords: trigonometric curve, spline curve, constrained modificatio

Constrained Interpolation And Shape Preserving Approximation By Space Curves [QA297.6. K82 2006 f rb].

Dua jenis masalah rekabentuk lengkung telah ipertimbangkan. Terlebih dahulu kami mempertimbangkan interpolasi satu set titik data ruang yang bertertib dengan satu lengkung licin tertakluk kepada satu set satah kekangan yang berbentuk terhingga atau tak terhingga di mana garis cebis demi cebis yang menyambung titik data secara berturutan tidak bersilang dengan satah kekangan. Two types of curve designing problem have been considered. We first consider the interpolation of a given set of ordered spatial data points by a smooth curve in the presence of a set of finite or infinite constraint planes, where the polyline joining consecutive data points does not intersect with the constraint planes

Methods for constraint-based conceptual free-form surface design

Zusammenfassung Der constraint-basierte Entwurf von Freiformfl„chen ist eine m„chtige Methode im Computer gest�tzten Entwurf. Bekannte Realisierungen beschr„nken sich jedoch meist auf Interpolation von Rand- und isoparametrischen Kurven. In diesem Zusammenhang sind die sog. "Multi-patch" Methoden die am weitesten verbreitete Vorgehensweise. Hier versucht man Fl„chenverb„nde aus einem Netz von dreidimensionalen Kurven (oft gemischt mit unstrukturierten Punktewolken) derart zu generieren, dass die Kurven und Punkte von den Fl„chen interpoliert werden. Die Kurven werden als R„nder von rechteckigen oder dreieckigen bi-polynomialen oder polynomialen Fl„chen betrachtet. Unter dieser Einschr„nkung leidet die Flexibilit„t des Verfahrens. In dieser Dissertation schlagen wir vor, beliebige, d.h. auch nicht iso-parametrische, Kurven zu verwenden. Dadurch ergeben sich folgende Vorteile: Erstens kann so beispielsweise eine B-spline Fl„che entlang einer benutzerdefinierten Kurve verformt werden w„hrend andere Kurven oder Punkte fixiert sind. Zweitens, kann eine B-spline Fl„che Kurven interpolieren, die sich nicht auf iso-parametrische Linien der Fl„che abbilden lassen. Wir behandeln drei Arten von Constraints: Inzidenz einer beliebigen Kurve auf einer B-spline Fl„che, Fixieren von Fl„chennormalen entlang einer beliebigen Kurve (dieser Constraint dient zur Herstellung von tangentialen šberg„ngen zwischen zwei Fl„chen) und die sog. Variational Constrains. Letztere dienen unter anderem zur Optimierung der physikalischen und optischen Eigenschaften der Fl„chen. Es handelt sich hierbei um die Gausschen Normalgleichungen der Fl„chenfunktionale zweiter Ordnung, wie sie in der Literatur bekannt sind. Die Dissertation gliedert sich in zwei Teile. Der erste Teil befasst sich mit der Aufstellung der linearen Gleichungssysteme, welche die oben erw„hnten Constraints repr„sentieren. Der zweite Teil behandelt Methoden zum L”sen dieser Gleichungssysteme. Der Kern des ersten Teiles ist die Erweiterung und Generalisierung des auf Polarformen (Blossoms) basierenden Algorithmus f�r Verkettung von Polynomen auf Bezier und B-spline Basis: Gegeben sei eine B-spline Fl„che und eine B-spline Kurve im Parameterraum der Fl„che. Wir zeigen, dass die Kontrollpunkte der dreidimensionalen Fl„chenkurve, welche als polynomiale Verkettung der beiden definiert ist, durch eine im Voraus berechenbare lineare Tranformation (eine Matrix) der Fl„chenkontrollpunkte ausgedr�ckt werden k”nnen. Dadurch k”nnen Inzidenzbeziehungen zwischen Kurven und Fl„chen exakt und auf eine sehr elegante und kompakte Art definiert werden. Im Vergleich zu den bekannten Methoden ist diese Vorgehensweise effizienter, numerisch stabiler und erh”ht nicht die Konditionszahl der zu l”senden linearen Gleichungen. Die Effizienz wird erreicht durch Verwendung von eigens daf�r entwickelten Datenstrukturen und sorgf„ltige Analyse von kombinatorischen Eigenschaften von Polarformen. Die Gleichungen zur Definition von Tangentialit„ts- und Variational Constraints werden als Anwendung und Erweiterung dieses Algorithmus implementiert. Beschrieben werden auch symbolische und numerische Operationen auf B-spline Polynomen (Multiplikation, Differenzierung, Integration). Dabei wird konsistent die Matrixdarstellung von B-spline Polynomen verwendet. Das L”sen dieser Art von Constraintproblemen bedeutet das Finden der Kontrollpunkte einer B-spline Fl„che derart, dass die definierten Bedingungen erf�llt werden. Dies wird durch L”sen von, im Allgemeinen, unterbestimmten und schlecht konditionierten linearen Gleichungssystemen bewerkstelligt. Da in solchen F„llen keine eindeutige, numerisch stabile L”sung existiert, f�hren die �blichen Methoden zum L”sen von linearen Gleichungssystemen nicht zum Erfolg. Wir greifen auf die Anwendung von sog. Regularisierungsmethoden zur�ck, die auf der Singul„rwertzerlegung (SVD) der Systemmatrix beruhen. Insbesondere wird die L-curve eingesetzt, ein "numerischer Hochfrequenzfilter", der uns in die Lage versetzt eine stabile L”sung zu berechnen. Allerdings reichen auch diese Methoden im Allgemeinen nicht aus, eine Fl„che zu generieren, welche die erw�nschten „sthetischen und physikalischen Eigenschaften besitzt. Verformt man eine Tensorproduktfl„che entlang einer nicht isoparametrischen Kurve, entstehen unerw�nschte Oszillationen und Verformungen. Dieser Effekt wird "Surface-Aliasing" genannt. Wir stellen zwei Methoden vor um diese Aliasing-Effekte zu beseitigen: Die erste Methode wird vorzugsweise f�r Deformationen einer existierenden B-spline Fl„che entlang einer nicht isoparametrischen Kurve angewendet. Es erfogt eine Umparametrisierung der zu verformenden Fl„che derart, dass die Kurve in der neuen Fl„che auf eine isoparametrische Linie abgebildet wird. Die Umparametrisierung einer B- spline Fl„che ist keine abgeschlossene Operation; die resultierende Fl„che besitzt i.A. keine B-spline Darstellung. Wir berechnen eine beliebig genaue Approximation der resultierenden Fl„che mittels Interpolation von Kurvennetzen, die von der umzuparametrisierenden Fl„che gewonnen werden. Die zweite Methode ist rein algebraisch: Es werden zus„tzliche Bedingungen an die L”sung des Gleichungssystems gestellt, die die Aliasing-Effekte unterdr�cken oder ganz beseitigen. Es wird ein restriktionsgebundenes Minimum einer Zielfunktion gesucht, deren globales Minimum bei "optimaler" Form der Fl„che eingenommen wird. Als Zielfunktionen werden Gl„ttungsfunktionale zweiter Ordnung eingesetzt. Die stabile L”sung eines solchen Optimierungsproblems kann aufgrund der nahezu linearen Abh„ngigkeit des Gleichungen nur mit Hilfe von Regularisierungsmethoden gewonnen werden, welche die vorgegebene Zielfunktion ber�cksichtigen. Wir wenden die sog. Modifizierte Singul„rwertzerlegung in Verbindung mit dem L-curve Filter an. Dieser Algorithmus minimiert den Fehler f�r die geometrischen Constraints so, dass die L”sung gleichzeitig m”glichst nah dem Optimum der Zielfunktion ist.The constrained-based design of free-form surfaces is currently limited to tensor-product interpolation of orthogonal curve networks or equally spaced grids of points. The, so- called, multi-patch methods applied mainly in the context of scattered data interpolation construct surfaces from given boundary curves and derivatives along them. The limitation to boundary curves or iso-parametric curves considerably lowers the flexibility of this approach. In this thesis, we propose to compute surfaces from arbitrary (that is, not only iso-parametric) curves. This allows us to deform a B-spline surface along an arbitrary user-defined curve, or, to interpolate a B-spline surface through a set of curves which cannot be mapped to iso-parametric lines of the surface. We consider three kinds of constraints: the incidence of a curve on a B-spline surface, prescribed surface normals along an arbitrary curve incident on a surface and the, so-called, variational constraints which enforce a physically and optically advantageous shape of the computed surfaces. The thesis is divided into two parts: in the first part, we describe efficient methods to set up the equations for above mentioned linear constraints between curves and surfaces. In the second part, we discuss methods for solving such constraints. The core of the first part is the extension and generalization of the blossom-based polynomial composition algorithm for B-splines: let be given a B-spline surface and a B-spline curve in the domain of that surface. We compute a matrix which represents a linear transformation of the surface control points such that after the transformation we obtain the control points of the curve representing the polynomial composition of the domain curve and the surface. The result is a 3D B-spline curve always exactly incident on the surface. This, so-called, composition matrix represents a set of linear curve-surface incidence constraints. Compared to methods used previously our approach is more efficient, numerically more stable and does not unnecessarily increase the condition number of the matrix. The thesis includes a careful analysis of the complexity and combinatorial properties of the algorithm. We also discuss topics regarding algebraic operations on B-spline polynomials (multiplication, differentiation, integration). The matrix representation of B-spline polynomials is used throughout the thesis. We show that the equations for tangency and variational constraints are easily obtained re-using the methods elaborated for incidence constraints. The solving of generalized curve-surface constraints means to find the control points of the unknown surface given one or several curves incident on that surface. This is accomplished by solving of large and, generally, under-determined and badly conditioned linear systems of equations. In such cases, no unique and numerically stable solution exists. Hence, the usual methods such as Gaussian elimination or QR-decomposition cannot be applied in straightforward manner. We propose to use regularization methods based on Singular Value Decomposition (SVD). We apply the so-called L-curve, which can be seen as an numerical high-frequency filter. The filter automatically singles out a stable solution such that best possible satisfaction of defined constraints is achieved. However, even the SVD along with the L-curve filter cannot be applied blindly: it turns out that it is not sufficient to require only algebraic stability of the solution. Tensor-product surfaces deformed along arbitrary incident curves exhibit unwanted deformations due to the rectangular structure of the model space. We discuss a geometric and an algebraic method to remove this, so-called, Surface aliasing effect. The first method reparametrizes the surface such that a general curve constraint is converted to iso-parametric curve constraint which can be easily solved by standard linear algebra methods without aliasing. The reparametrized surface is computed by means of the approximated surface-surface composition algorithm, which is also introduced in this thesis. While this is not possible symbolically, an arbitrary accurate approximation of the resulting surface is obtained using constrained curve network interpolation. The second method states additional constraints which suppress or completely remove the aliasing. Formally we solve a constrained least square approximation problem: we minimize an surface objective function subject to defined curve constraints. The objective function is chosen such that it takes in the minimal value if the surface has optimal shape; we use a linear combination of second order surface smoothing functionals. When solving such problems we have to deal with nearly linearly dependent equations. Problems of this type are called ill-posed. Therefore sophisticated numerical methods have to be applied in order to obtain a set of degrees of freedom (control points of the surface) which are sufficient to satisfy given constraints. The remaining unused degrees of freedom are used to enforce an optically pleasing shape of the surface. We apply the Modified Truncated SVD (MTSVD) algorithm in connection with the L-curve filter which determines a compromise between an optically pleasant shape of the surface and constraint satisfaction in a particularly efficient manner

Constrained fitting of B-Spline curves based on the Force Density Method

This paper presents a novelapproach for constrained B-Spline curve approximation based on the Force Density Method (FDM). This approach aims to define a flexible technique tool for curve fitting, which allows approximating a set of points taking into account shape constraints that may be related to the production process, to the material or to other technological re- quirements. After a brief introduction on the property of the FDM and the definition of the network usedfor the formulation of the fitting problem, the paper explains in detail the mathematical approach, the methods and the techniques adopted for the definition of the proposed constrained B- Splinecurve approximation. The results suggest that the adoption of a mechanical model of bar networks allows develop- ing a more flexible tool than the traditional least squared methods (LSM) usually adopted for fitting problems. Numerical examples show that the new approach is effective in fitting prob- lems when the satisfaction of shape constraints, such as those related to production orto technological processes, are required

Synthesis of Spatially and Intrinsically Constrained Curves Using Simulated Annealing

A general technique is presented for automatic generation of B-spline curves in a spatially constrained environment, subject to specified intrinsic shape properties. Spatial constraints are characterized by a distance metric relating points on the curve to polyhedral models of obstacles which the curve should avoid. The shape of the curve is governed by constraints based on intrinsic curve properties such as parametric variation and curvature. To simultaneously address the independent goals of global obstacle avoidance and local control of intrinsic shape properties, curve synthesis is formulated as a combinatorial optimization problem and solved via simulated annealing. Several example applications are presented which demonstrate the robustness of the technique. The synthesis of both uniform and nonuniform B-spline curves is also demonstrated. An extension of the technique to general sculptured surface model synthesis is briefly described, and a preliminary example of simple surface synthesis presented

Automatic constraint-based synthesis of non-uniform rational B-spline surfaces

In this dissertation a technique for the synthesis of sculptured surface models subject to several constraints based on design and manufacturability requirements is presented. A design environment is specified as a collection of polyhedral models which represent components in the vicinity of the surface to be designed, or regions which the surface should avoid. Non-uniform rational B-splines (NURBS) are used for surface representation, and the control point locations are the design variables. For some problems the NURBS surface knots and/or weights are included as additional design variables. The primary functional constraint is a proximity metric which induces the surface to avoid a tolerance envelope around each component. Other functional constraints include: an area/arc-length constraint to counteract the expansion effect of the proximity constraint, orthogonality and parametric flow constraints (to maintain consistent surface topology and improve machinability of the surface), and local constraints on surface derivatives to exploit part symmetry. In addition, constraints based on surface curvatures may be incorporated to enhance machinability and induce the synthesis of developable surfaces;The surface synthesis problem is formulated as an optimization problem. Traditional optimization techniques such as quasi-Newton, Nelder-Mead simplex and conjugate gradient, yield only locally good surface models. Consequently, simulated annealing (SA), a global optimization technique is implemented. SA successfully synthesizes several highly multimodal surface models where the traditional optimization methods failed. Results indicate that this technique has potential applications as a conceptual design tool supporting concurrent product and process development methods