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24
Hierarchical edge bundles: Visualization of adjacency relations in hierarchical data
 IEEE Transactions on Visualization and Computer Graphics
, 2006
"... 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., nonhierarch ..."
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Cited by 258 (12 self)
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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., nonhierarchical 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 Bspline 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.
Curves and Surfaces for CAGD
, 1993
"... 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 freeform curves, s ..."
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Cited by 74 (1 self)
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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 freeform curves, surfaces, or volumes. 1.
Multiresolution Stochastic Hybrid Shape Models with Fractal Priors
 ACM Transactions on Graphics
, 1994
"... 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 t ..."
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Cited by 30 (8 self)
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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 viceversa. 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...
Polar Forms for Geometrically Continuous Spline Curves of Arbitrary Degree
 ACM Trans. Graph
, 1993
"... 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 kn ..."
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Cited by 18 (2 self)
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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...
Geometric Continuity
, 2001
"... 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. Se ..."
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Cited by 15 (4 self)
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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 freeform 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 vertexenclosure 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 freeform surface splines and points to some low degree constructions. The chapter closes with a listing of additional literature.
Geometric continuity, shape parameters, and geometric constructions for catmullrom splines
 ACM Trans. Graph
, 1988
"... CatmullRom splines have local control, can be either approximating or interpolating, and are efficiently computable. Experience with Betasplines has shown that it is useful to endow a spline with shape parameters, used to modify the shape of the curve or surface independently of the defining contr ..."
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Cited by 14 (1 self)
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CatmullRom splines have local control, can be either approximating or interpolating, and are efficiently computable. Experience with Betasplines has shown that it is useful to endow a spline with shape parameters, used to modify the shape of the curve or surface independently of the defining control vertices. Thus it is desirable to construct a subclass of the CatmullRom splines that has shape parameters. We present such a class, some members of which are interpolating and others approximating. As was done for the Betaspline, shape parameters are introduced by requiring geometric rather than parametric continuity. Splines in this class are defined by a set of control vertices and a set of shape parameter values. The shape parameters may be applied globally, affecting the entire curve, or they may be modified locally, affecting only a portion of the curve near the corresponding joint. We show that this class results from combining geometrically continuous (Betaspline) blending functions with a new set of geometrically continuous interpolating functions related to the classical Lagrange curves. We demonstrate the practicality of several members of the class by developing efficient computational algorithms. These algorithms are based on geometric constructions that take as input a control polygon and a set of shape parameter values and produce as output a sequence of Bizier control
XSplines : A Spline Model Designed for the EndUser
 Proc. SIGGRAPH'95
, 1995
"... 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 spl ..."
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Cited by 12 (3 self)
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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 nonmathematician enduser. 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 Bsplines and CatmullRom splines, or the ability to define approximatio...
Smooth parametric surfaces and Nsided patches
 Computation of Curves and Surfaces
, 1990
"... CAGD is reviewed. In particular, we are concerned with how parametric surface patches for CAGD can be pieced together to form a smooth Ck surface. The theory is applied to the problem of filling an nsided hole occurring within a smooth rectangular patch complex. A number of solutions to this proble ..."
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Cited by 11 (0 self)
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CAGD is reviewed. In particular, we are concerned with how parametric surface patches for CAGD can be pieced together to form a smooth Ck surface. The theory is applied to the problem of filling an nsided hole occurring within a smooth rectangular patch complex. A number of solutions to this problem are surveyed. 1.
Rational Continuity: Parametric, Geometric, and Frenet Frame Continuity of Rational Curves
 ACM Trans. on Graphics
, 1989
"... The parametric, geometric, or Frenet frame continuity of a rational curve has often been ensured by requiring the homogeneous polynomial curve associated with the rational curve to possess either parametric, geometric, or Frenet frame continuity, respectively. In this paper, we show that this approa ..."
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Cited by 9 (1 self)
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The parametric, geometric, or Frenet frame continuity of a rational curve has often been ensured by requiring the homogeneous polynomial curve associated with the rational curve to possess either parametric, geometric, or Frenet frame continuity, respectively. In this paper, we show that this approach is overly restrictive and derive the constraints on the associated homogeneous curve that are both necessary and sufficient to ensure that the rational curve is either parametrically, geometrically, or Frenet frame continuous.