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27
Interval Analysis For Computer Graphics
 Computer Graphics
, 1992
"... This paper discusses how interval analysis can be used to solve a wide variety of problems in computer graphics. These problems include ray tracing, interference detection, polygonal decomposition of parametric surfaces, and CSG on solids bounded by parametric surfaces. Only two basic algorithms are ..."
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Cited by 132 (2 self)
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This paper discusses how interval analysis can be used to solve a wide variety of problems in computer graphics. These problems include ray tracing, interference detection, polygonal decomposition of parametric surfaces, and CSG on solids bounded by parametric surfaces. Only two basic algorithms are required: SOLVE, which computes solutions to a system of constraints, and MINIMIZE, which computes the global minimum of a function, subject to a system of constraints. We present algorithms for SOLVE and MINIMIZE using interval analysis as the conceptual framework. Crucial to the technique is the creation of "inclusion functions" for each constraint and function to be minimized. Inclusion functions compute a bound on the range of a function, given a similar bound on its domain, allowing a branch and bound approach to constraint solution and constrained minimization. Inclusion functions also allow the MINIMIZE algorithm to compute global rather than local minima, unlike many other numerica...
Haptic Rendering Of SurfaceToSurface Sculpted Model Interaction
, 1999
"... Previous work in haptics surface tracing for virtual prototyping and surface design applications has used a point model for virtual fingersurface interaction. We extend this tracing method for surfacetosurface interactions. A straightforward extension of the pointsurface formulation to surfaces ..."
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Cited by 30 (9 self)
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Previous work in haptics surface tracing for virtual prototyping and surface design applications has used a point model for virtual fingersurface interaction. We extend this tracing method for surfacetosurface interactions. A straightforward extension of the pointsurface formulation to surfacesurface can yield extraneous, undesirable solutions, although we rework the formulation to yield more satisfactory solutions. Additionally, we derive an alternative novel velocity formulation for use in a surfacesurface tracing paradigm that exhibits additional stability beyond the Newton methods. Both methods require evaluating the surface point and first and second surface partial derivatives for both surfaces, an efficient kilohertz rate computation. These methods are integrated into a three step tracking process that uses a global minimum distance method, the local Newton formulation, and the new velocity formulation. Figure 1: Wellbehaved finger penetration into a surface shown by the...
Distance Approximations for Rasterizing Implicit Curves
 ACM Transactions on Graphics
, 1994
"... In this article we present new algorithms for rasterizing implicit curves, i.e., curves represented as level sets of functions of two variables. Considering tbe pixels as square regions of tbe plane, a “correct ” algorithm should paint those pixels whose centers lie at less than half the desired lin ..."
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Cited by 27 (0 self)
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In this article we present new algorithms for rasterizing implicit curves, i.e., curves represented as level sets of functions of two variables. Considering tbe pixels as square regions of tbe plane, a “correct ” algorithm should paint those pixels whose centers lie at less than half the desired line width from the curve, A straightforward implementation, scanning the display array evaluating the Euclidean distance from the center of each pixel to the curve, is impractical, and a standard quadtree)ikc rccursivc subdivision scheme is used instead. Then we attack the problem of testing whether or not the Euclidean distance from a point to an implicit curve is less than a given threshold For the most gcnernl case, when the implicit function is only required to have continuous firstorder derivatives, we show how tn reformulate tbe test as an unconstrained global rootfinding problem in a circular domain. For implicit functions with continuous derivativesup to order k we introduce an approximate distance of order k. The approximate distance of order k from a po]nt to an implicit curve is asymptotically equivalent to the Euclidean distance and provides a suff]cicnt tw+tfor a polynomial of degree h not to have roots inside a circle. This is the main contribution of the article, By replacing the Euclidean distance test with one of these approximate distance tests, we obtain a practical rendering algorithm, proven to be correct for algebraic curves. To speed up the computation we also introduce heuristics, which used in conjunction with loworder approximate distances almost always produce equivalent results. Tbe behavior of the algorithms is an[ilyzcd, both near regular and singular points, and several possible extensions and applications are discussed.
Sweeping of Threedimensional Objects
 ComputerAided Design
, 1989
"... this paper addresses is the following: given an arbitrary threedimensional object, which moves along an arbitrary path (possibly rotating as it does so), to compute the volume swept out by the solid object as it moves, or in other words, to find the new solid volume which represents all of the poin ..."
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Cited by 23 (0 self)
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this paper addresses is the following: given an arbitrary threedimensional object, which moves along an arbitrary path (possibly rotating as it does so), to compute the volume swept out by the solid object as it moves, or in other words, to find the new solid volume which represents all of the points in space which the object has occupied at some time during the motion
Efficient representations and techniques for computing Brep's of CSG models with NURBS primitives
, 1996
"... ..."
Interval Constraint Plotting for Interactive Visual Exploration of Implicitly Defined Relations
 Reliable Computing
, 1999
"... . Conventional plotting programs adopt techniques such as adaptive sampling to approximate, but not to guarantee, correctness and completeness in graphing functions. Moreover, implicitly defined mathematical relations can impose an even greater challenge as they either cannot be plotted directly, or ..."
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Cited by 10 (6 self)
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. Conventional plotting programs adopt techniques such as adaptive sampling to approximate, but not to guarantee, correctness and completeness in graphing functions. Moreover, implicitly defined mathematical relations can impose an even greater challenge as they either cannot be plotted directly, or otherwise are likely to be misrepresented. In this paper, we address these problems by investigating interval constraint plotting as an alternative approach that plots a hull of the specified curve. We present some empirical evidence that this hull property can be achieved by a O(n) algorithm. Practical experience shows that the hull obtained is the narrowest possible whenever the precision of the underlying floatingpoint arithmetic is adequate. We describe IASolver, a Java applet [9], that serves as testbed for this idea. Keywords: interval constraints, constraint propagation, interval arithmetic, implicitly defined relations, honest plotting, interactive plotting 1. Introduction Mathe...
Extremal Distance Maintenance for Parametric Curves and Surfaces
 In IEEE International Conference on Robotics and Automation
, 2002
"... A new extremal distance tracking algorithm is presented for parametric curves and surfaces undergoing rigid body motion. The essentially geometric extremization problem is transformed into a dynamical control problem by differentiating with respect to time. Extremization is then solved with the desi ..."
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Cited by 7 (0 self)
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A new extremal distance tracking algorithm is presented for parametric curves and surfaces undergoing rigid body motion. The essentially geometric extremization problem is transformed into a dynamical control problem by differentiating with respect to time. Extremization is then solved with the design of a stabilizing controller. We use a feedback linearizing controller. The controller simultaneously accounts for the surface shape and motion while asymptotically achieving (and maintaining)the extremal pair. Thus collision detection takes place in a framework fully analogous to the framework used for the simulation of dynamical response. 1
Primitive Geometric Operations on Planar Algebraic Curves with Gaussian Approximations
 Visual Computing
, 1992
"... We present a curve approximation method which approximates each planar algebraic curve segment by discrete curve points at each of which the curve has its gradient from a set of uniformly distributed normals. This method, called Gaussian Approximation (GAP), provides efficient algorithms for various ..."
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Cited by 7 (6 self)
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We present a curve approximation method which approximates each planar algebraic curve segment by discrete curve points at each of which the curve has its gradient from a set of uniformly distributed normals. This method, called Gaussian Approximation (GAP), provides efficient algorithms for various primitive geometric operations, especially for those related with gradients such as common tangent and convolution computations, on planar algebraic curve segments. The hierarchy of unit gradients gives the corresponding hierarchy of CAP. The approximation error at each level of the hierarchy can be modeled in the representation of GAP itself, and we can use this structure to dynamically control the precision and efficiency of geometric computation with CAP. We implemented various primitive geometric operations on planar algebraic curve segments with GAP representations on SUN4/Sparc station using C.
Torus/Sphere Intersection Based on a Configuration Space Approach
, 1998
"... This paper presents an efficient and robust geometric algorithm that classifies and detects all possible types of torus/sphere intersections, including all degenerate conic sections (circles) and singular intersections. Given a torus and a sphere, we treat one surface as an obstacle and the other ..."
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Cited by 6 (5 self)
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This paper presents an efficient and robust geometric algorithm that classifies and detects all possible types of torus/sphere intersections, including all degenerate conic sections (circles) and singular intersections. Given a torus and a sphere, we treat one surface as an obstacle and the other surface as the envelope surface of a moving ball. In this case, the Configuration space (Cspace) obstacle is the same as the constant radius offset of the original obstacle, where the radius of the moving ball is taken as the offset distance [2]. Based on the intersection between the Cspace obstacle and the trajectory of the center of the moving ball, we detect all the intersection loops and singular contact point/circle of the original torus and sphere. Moreover, we generate exactly one starting point (for numerical curve tracing) on each connected component of the intersection curve. All required computations involve vector/distance computations and circle/circle intersections, w...
An Accurate Algorithm for Rasterizing Algebraic Curves
 In Second Symposium on Solid Modeling and Applications, ACM/IEEE
, 1993
"... In this paper we introduce a new algorithm for rasterizing algebraic curves, and we discuss applications to surface and surfacesurface intersection rendering and visualization. By rustm”zing an algebraic curve we mean to determine which cells, or pixels, from a square mesh of cells in the plane, ar ..."
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Cited by 6 (0 self)
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In this paper we introduce a new algorithm for rasterizing algebraic curves, and we discuss applications to surface and surfacesurface intersection rendering and visualization. By rustm”zing an algebraic curve we mean to determine which cells, or pixels, from a square mesh of cells in the plane, are cut by a curve represented as the set of zeros of a polynomial in two variables. By using a recursive space subdivision scheme, the problem is be reduced to testing whether the curve cuts a square or not. Other mwearchers have followed this approach, but their tests are either computationally expensive, or apply just to special cases. Curves with singularities are particularly difficult to deal with, and most know algorithms fail to nmder these curves correctly.