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Offset curves/surfaces, also called parallel curves/surfaces, are defined as the locus of the points which are at constant distant d along the normal from the generator curves/surfaces. A literature survey on offset curves and surfaces was carried out by Pham [313] and more recently by Maekawa [250]. Offsets are widely used in various applications, such as tool path generation for **** pocket machining...
We present the high level architecture of a real-time interdisciplinary ocean forecasting system that employs adaptive elements in both modeling and sampling. We also discuss an important issue that arises in creating an integrated, web-accessible framework for such a system out of existing stand-alone components: transparent support for handling legacy binaries. Such binaries, that are most common...
Let us consider a curve u = u(t), v = v(t) in the parametric domain of a parametric surface r = r(u, v) as shown in Fig. 3.1. Then r = r(t) = r (u(t), v(t)) is a parametric curve lying on the surface r = r(u,v).The tangent vector to the curve on the surface is evaluated by differentiating r(t) with respect to the parameter t using the chain rule and is given by $$\mathop {\rm r}\limits^ \bullet...
An umbilic is a point on a surface where all normal curvatures are equal in all directions, and hence principal directions are indeterminate. Thus the orthogonal net of lines of curvature, which is described in Sect. 3.4, becomes singular at an umbilic. An obvious example of a surface consisting entirely of umbilical points is the sphere. Actually, spheres and planes are the only surfaces all of whose...
Interrogation is the process of extraction of information from a geometric model. In this chapter we focus on free-form curve and surface interrogation. Free form surface, also called sculptured surface, are widely used in scientific and engineering applications. For example, the hydrodynamic shape of propeller blades has an important role in marine applications, and the aerodynamic shape of turbine...
The computation of maximal and minimal distance between points sets in Euclidean space is a basic problem in computational geometry and geometric modeling. It is useful in surface intersections [210], numerical control machining, tolerance region and access space representation in solid modeling, robotics, inspection of manufactured objects [353, 297, 186, 3], and in feature recognition through the...
The differential geometry of curves and surfaces is fundamental in Computer Aided Geometric Design (CAGD). The curves and surfaces treated in differential geomentry are defined by functions which can be differentiated a certain number of times. Books by Hilbert and Cohn-Vossen [165], Koenderink [205] provide intuitive introductions to the extensive mathematical literature on three-dimensional shape...
Intersections are fundamental in computational geometry, geometric modeling and design, analysis and manufacturing applications [276,19, 175, 295, 300]. Examples of intersection problems include: Contouring of surfaces through intersection with a series of parallel planes or coaxial circular cylinders or cones for visualization (see Fig. 5.1). Numerical control machining (milling) ivolving intersection...
We first introduce three forms to represent geometric objects mathematically. They are the parametric, implicit and explicit forms. Implicit and explicit forms are often referred to as nonparametric forms. Then we briefly review the representation of curves and surfaces in Bézier and B-spline form and treat the special properties associated with each.
In Chap. 5 we have studied the classification, detection, and solution of intersection problems. In this chapter we focus on the differential geometry properties of intersection curves of two surfaces. To compute the intersection curves more accurately and efficiently, higher order approximation of intersection curves may be needed. This requires the computation of not only the tangents of the intersection...
Computation of shortest paths on free-form surfaces is an important problem in geometric design of ship hulls, robot motion planning, computation of medial axis transforms of trimmed surface patches, terrain navigation, NC machining, and cable installation on the sea floor. The history of geodesic lines begins with a study by Johann Bernoulli, who solved the problem of the shortest distance between...
We have seen in Chap. 1 that curves and surfaces in CAD/CAM systems are usually represented by piecewise polynomial equation of various type. As we will see in the remaining chapters of this book, the governing equations for general interrogation problems on such curve and surface representations (intersections, distance functions, curvature extrema, etc.
Shape interrogation is the process of extraction of information from a geometric model. It is a fundamental component of Computer Aided Design and Manufacturing (CAD/CAM) systems. The authors focus on shape interrogation of geometric models bounded by free-form surfaces. Free-form surfaces, also called sculptured surfaces, are widely used in the bodies of ships, automobiles and aircraft, which have...
Over the past 10 years, significant scientific effort has been dedicated to the problem of three‐dimensional (3‐D) surface reconstruction for structural systems. However, the critical area of marine structures remains insufficiently studied. The research presented here focuses on the problem of 3‐D surface reconstruction in the marine environment. This paper summarizes our hardware, software, and...
This paper proposes a new inspection planning algorithm, called Random Inspection Tree Algorithm (RITA). Given a perfect model of a structure, sensor specifications, robot's dynamics, and an initial configuration of a robot, RITA computes the optimal inspection trajectory that observes all points on the structure. Many inspection planning algorithms have been proposed, most of them consist of two...
The increase in frequencies and the noxious effects of HABs in global coastal waters have led to enhanced interest in monitoring and detecting of such blooms. In December 2009, a toxic bloom hit Singapore waters along East Johor Strait causing massive fish kills and great economic losses. Interdisciplinary approach involving robotic network adaptation, multi-scale-sensing using autonomous vehicles...
Uncertainty in motion planning is often caused by three main sources: motion error, sensing error, and imperfect environment map. Despite the significant effect of all three sources of uncertainty to motion planning problems, most planners take into account only one or at most two of them. We propose a new motion planner, called Guided Cluster Sampling (GCS), that takes into account all three sources...
Over the last eight years, significant scientific effort has been dedicated on the problem of 3-D surface reconstruction for structural systems. However, the critical area of marine structures remains insufficiently studied. The research presented here focuses on the problem of 3-D surface reconstruction in the marine environment. This work is an extension of our previous approach, in which a surface...
Over the last eight years, significant scientific effort has been dedicated on the problem of 3-D surface reconstruction for structural systems. However, the critical area of marine structures remains insufficiently studied. The research presented here focuses on the problem of 3-D surface reconstruction in the marine environment. This work is an extension of our previous approach, in which a surface...
This paper presents an alternative formulation for the Factorised Solution to the Simultaneous Localization and Mapping (FastSLAM) algorithm using an Adaptive Extended Kalman Filter based approach. The FastSLAM algorithm jointly estimates the pose of the robot and the location of landmarks on the feature based map by factorising the SLAM posterior into a localisation component which is implemented...
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