Abstract—A compound graph is a frequently encountered type of data set. Relations are given between items, and a hierarchy is defined on the items as well. We present a new method for visualizing such compound graphs. Our approach is based on visually bundling the adjacency edges, i.e., non-hierarchical edges, together. We realize this as follows. We assume that the hierarchy is shown via a standard tree visualization method. Next, we bend each adjacency edge, modeled as a B-spline curve, toward the polyline defined by the path via the inclusion edges from one node to another. This hierarchical bundling reduces visual clutter and also visualizes implicit adjacency edges between parent nodes that are the result of explicit adjacency edges between their respective child nodes. Furthermore, hierarchical edge bundling is a generic method which can be used in conjunction with existing tree visualization techniques. We illustrate our technique by providing example visualizations and discuss the results based on an informal evaluation provided by potential users of such visualizations.

This article provides a historical account of the major developments in the area of curves and surfaces as they entered the area of CAGD – Computer Aided Geometric Design – until the middle 1980s. We adopt the definition that CAGD deals with the construction and representation of free-form curves, surfaces, or volumes. 1.

3D Shape modeling has received enormous attention in computer graphics and computer vision over the past decade. Several shape modeling techniques have been proposed in literature, some are local (distributed parameter) while others are global (lumped parameter) in terms of the parameters required to describe the shape. Hybrid models that combine both ends of this parameter spectrum have been in vogue only recently. However, they do not allow a smooth transition between the two extremes of this parameter spectrum. In this paper, we introduce a new shape modeling scheme that can transform smoothly from local to global models or vice-versa. The modeling scheme utilizes a hybrid primitive called the deformable superquadric constructed in an orthonormal wavelet basis. The multiresolution wavelet basis provides the power to continuously transform from local to global shape deformations and thereby allow for a continuum of shape models -- from those with local to those with global shape desc...

This paper studies geometrically continuous spline curves of arbitrary degree. Based on the concept of universal splines we obtain geometric constructions for both the spline control points and for the B'ezier points and give algorithms for computing locally supported basis functions and for knot insertion. The geometric constructions are based on the intersection of osculating flats. The concept of universal splines is defined in such a way that these intersections are guaranteed to exist. As a result of this development we obtain a generalization of polar forms to geometrically continuous spline curves by intersecting osculating flats. The presented algorithms have been coded in Maple, and concrete examples illustrate the approach. Categories and Subject Descriptors: I.3.5 [Computer Graphics]: Computational Geometry and Object Modelling - curve, surface, solid, and object representations General Terms: Algorithms, Design Additional Key Words and Phrases: B'ezier point, blossom, de...

This chapter covers geometric continuity with emphasis on a constructive definition for piecewise parametrized surfaces. The examples in Section 1 show the need for a notion of continuity different from the direct matching of Taylor expansions used to define the continuity of piecewise functions. Section 2 defines geometric continuity for parametric curves, and for surfaces, first along edges, then around points, and finally for a whole complex of patches which is called a free-form surface spline. Here characterizes a relation between specific maps while continuity is a property of the resulting surface. The composition constraint on reparametrizations and the vertex-enclosure constraints are highlighted. Section 3 covers alternative definitions based on geometric invariants, global and regional reparametrization and briefly discusses geometric continuity in the context of implicit representations and generalized subdivision. Section 4 explains the generic construction of free-form surface splines and points to some low degree constructions. The chapter closes with a listing of additional literature.

This paper presents a new model of spline curves and surfaces. The main characteristic of this model is that it has been created from scratch by using a kind of mathematical engineering process. In a first step, a list of specifications was established. This list groups all the properties that a spline model should contain in order to appear intuitive to a non-mathematician end-user. In a second step, a new family of blending functions was derived, trying to fulfill as many items as possible of the previous list. Finally, the degrees of freedom offered by the model have been reduced to provide only shape parameters that have a visual interpretation on the screen. The resulting model includes many classical properties such as affine and perspective invariance, convex hull, variation diminution, local control andC 2 =G 2 orC 2 =G 0 continuity. But it also includesoriginal features such as a continuum between B-splines and Catmull-Rom splines, or the ability to define approximatio...

This paper presents a new, accurate, efficient and unified method for dynamic animation of one, two or three-dimensional deformable objects. The objects are modelled as d-dimensional juxtapositions of ddimensional patches defined as parametric blending of a common d-dimensional mesh of 3D control points. Animation of the object is achieved by dynamic animation of its control points. This ensures that at each time step the object shape conforms to its patches definitions, and, thus, that every property implied by the nature of the blending functions is verified. Dynamic animation of these continuous models implies no matter discretising as the control points are not considered as material points but moreover as the degrees of freedom of the continuous object. A generic (both for blending functions nature and object intrinsic dimension d) mechanical model reflecting this idea is proposed. Then, according to this modelling idea, a convenient generic dynamic animation engine is built from Lagrangian Equations. This engine relies upon an accurate and very efficient linear system. Forces and constraints handling as well as numerical resolution process are then briefly discussed in this scheme. Keywords: Dynamic animation , Lagrangian equations, spline, parametric surfaces, parametric volumes, deformable objects.

This survey presents an overview to various types of continuity of curves and surfaces, in particular parametric (C n ), visual or geometric (V n , G n ), Frenet frame (F n ), and tangent surface continuity (T n ), and discusses the relation with curve and surface modeling, visibility of (dis)continuities, and graphics rendering algorithms. It is the purpose of this paper to provide an overview of types of continuity, and to put many terms and definitions on a common footing in order to give an understanding of the subject. 1991 Mathematics Subject Classification: 41A15, 65D07, 68U05 1991 Computing Reviews Classification: I.3.5 [Computer Graphics] Computational geometry and object modeling. I.3.7 [Computer Graphics] Three-Dimensional Graphics and Realism. Key Words and Phrases: Curves, Surfaces, Continuity, Shading, Modeling. 1

This survey paper discusses the state-of-the-art in solid modeling, subdivision modeling, and physics-based modeling. Although related to each other, these research areas have yet to be integrated into a single framework. We present a historical review of each area separately, discuss research that has combined two of the above domains, and suggest directions for future work that combine concepts from all three. Solid modeling includes the study of different ways of representing and analyzing virtual solid objects. We examine several approaches to modeling solids and analyze their advantages and shortcomings. Subdivision modeling uses procedural algorithms to recursively define smooth curves, surfaces, and solids. We show the connections between subdivision modeling and spline-based modeling and demonstrate that the former is actually a generalization of the latter. Physics-based modeling augments geometric objects with physical attributes for the purposes of dynamic sculpting, physica...

We propose an interaction technique for editing splines that is aimed at professional graphic designers. These users do not take full advantage of existing spline editing software because their mental representations of drawings do not match the underlying conceptual model of the software. Although editing splines by specifying control points and tangents may be appropriate for engineers, graphic designers think more in terms of strokes, shapes, and gestures appropriate for editing drawings. Our interaction technique matches the latter model: curves can be edited by means of marks, similar to the way strokes are naturally overloaded when drawing on paper. We describe this interaction technique and the algorithms used for its implementation. KEYWORDS: Mark-based interaction, Gestures, Spline editing, Interaction models, Graphic design, CAD.