BookPDF Available
Nicholas M. Patrikalakis
Takashi Maekawa
Shape Interrogation
Aided Design
Figures, 8 in color
Representation of Curves and Surfaces 1
1.1 Analytic representation of curves 1
1.1.1 Plane curves 1
1.1.2 Space curves 3
1.2 Analytic representation of surfaces 4
1.3 Bezier curves and surfaces 6
1.3.1 Bernstein polynomials 6
1.3.2 Arithmetic operations of polynomials in Bernstein form 7
1.3.3 Numerical condition of polynomials in Bernstein form . 9
1.3.4 Definition of Bezier curve and its properties 12
1.3.5 Algorithms for Bezier curves 13
1.3.6 Bezier surfaces 18
1.4 B-spline curves and surfaces 20
1.4.1 B-splines 20
1.4.2 B-spline curve 21
1.4.3 Algorithms for B-spline curves 24
1.4.4 B-spline surface 29
1.5 Generalization of B-spline to NURBS 30
Differential Geometry of Curves 35
2.1 Arc length and tangent vector 35
2.2 Principal normal and curvature 39
2.3 Binormal vector and torsion 43
2.4 Frenet-Serret formulae 47
Differential Geometry of Surfaces 49
3.1 Tangent plane and surface normal 49
3.2 First fundamental form / (metric) 52
3.3 Second fundamental form // (curvature) 55
3.4 Principal curvatures 59
3.5 Gaussian and mean curvatures 64
3.5.1 Explicit surfaces 64
3.5.2 Implicit surfaces 65
3.6 Euler's theorem and Dupin's indicatrix 68
XII Contents
Nonlinear Polynomial Solvers and Robustness Issues 73
4.1 Introduction 73
4.2 Local solution methods 74
4.3 Classification of global solution methods 76
4.3.1 Algebraic and Hybrid Techniques 76
4.3.2 Homotopy (Continuation) Methods 78
4.3.3 Subdivision Methods 78
4.4 Projected Polyhedron algorithm 78
4.5 Auxiliary variable method for nonlinear systems with square
roots of polynomials 88
4.6 Robustness issues 90
4.7 Interval arithmetic 92
4.8 Rounded interval arithmetic and its implementation 95
4.8.1 Double precision floating point arithmetic 95
4.8.2 Extracting the exponent from the binary representation 98
4.8.3 Comparison of
different unit—in—the—last—place
implementations 101
4.8.4 Hardware rounding for rounded interval arithmetic . . . 102
4.8.5 Implementation of rounded interval arithmetic 103
4.9 Interval Projected Polyhedron algorithm 105
4.9.1 Formulation of the governing polynomial equations ... 105
4.9.2 Comparison of software and hardware rounding 106
5. Intersection Problems 109
5.1 Overview of intersection problems 109
5.2 Intersection problem classification Ill
5.2.1 Classification by dimension 112
5.2.2 Classification by type of geometry 112
5.2.3 Classification by number system 114
5.3 Point/point intersection 114
5.4 Point/curve intersection 114
5.4.1 Point/implicit algebraic curve intersection 114
5.4.2 Point/rational polynomial parametric curve intersection 117
5.4.3 Point/procedural parametric curve intersection 121
5.5 Point/surface intersection 121
5.5.1 Point/implicit algebraic surface intersection 121
5.5.2 Point/rational polynomial parametric surface intersec-
tion 122
5.5.3 Point/procedural parametric surface intersection 125
5.6 Curve/curve intersection 126
5.6.1 Rational polynomial parametric/implicit algebraic curve
intersection (Case D3) 126
5.6.2 Rational polynomial parametric/rational polynomial
parametric curve intersection (Case Dl) 130
Contents XIII
5.6.3 Rational polynomial parametric/procedural parame-
tric and procedural parametric/procedural parametric
curve intersections (Cases D2 and D5) 131
5.6.4 Procedural parametric/implicit algebraic curve inter-
section (Case D6) 133
5.6.5 Implicit algebraic/implicit algebraic curve intersection
(Case D8) 133
5.7 Curve/surface intersection 134
5.7.1 Rational polynomial parametric curve/implicit alge-
braic surface intersection (Case E3) 135
5.7.2 Rational polynomial parametric curve/rational poly-
nomial parametric surface intersection (Case El) 135
5.7.3 Rational polynomial parametric/procedural parame-
tric and procedural parametric/procedural parametric
curve/surface intersections (Cases E2/E6) 136
5.7.4 Procedural parametric curve/implicit algebraic sur-
face intersection (Case E7) 136
5.7.5 Implicit algebraic curve/implicit algebraic surface in-
tersection (Case Ell) 137
5.7.6 Implicit algebraic curve/rational polynomial parame-
tric surface intersection (Case E9) 137
5.8 Surface/surface intersections 137
5.8.1 Rational polynomial parametric/implicit algebraic sur-
face intersection (Case F3) 138
5.8.2 Rational polynomial parametric/rational polynomial
parametric surface intersection (Case Fl) 147
5.8.3 Implicit algebraic/implicit algebraic surface intersec-
tion (Case F8) 151
5.9 Overlapping of curves and surfaces 155
5.10 Self-intersection of curves and surfaces 157
5.11 Summary 159
6. Differential Geometry of Intersection Curves 161
6.1 Introduction 161
6.2 More differential geometry of curves 162
6.3 Transversal intersection curve 164
6.3.1 Tangential direction 164
6.3.2 Curvature and curvature vector 165
6.3.3 Torsion and third order derivative vector 167
6.3.4 Higher order derivative vector 168
6.4 Intersection curve at tangential intersection points 170
6.4.1 Tangential direction 171
6.4.2 Curvature and curvature vector 173
6.4.3 Third and higher order derivative vector 176
6.5 Examples 177
XIV Contents
6.5.1 Transversal intersection of parametric-implicit surfaces 177
6.5.2 Tangential intersection of implicit-implicit surfaces . . . 179
7. Distance Functions 181
7.1 Introduction 181
7.2 Problem formulation 182
7.2.1 Definition of the distances between two point sets .... 182
7.2.2 Geometric interpretation of stationarity of distance
function 184
7.3 More about stationary points 185
7.3.1 Classification of stationary points 185
7.3.2 Nonisolated stationary points 190
7.4 Examples 192
8. Curve and Surface Interrogation 195
8.1 Classification of interrogation methods 195
8.1.1 Zeroth-order interrogation methods 196
8.1.2 First-order interrogation methods 197
8.1.3 Second-order interrogation methods 200
8.1.4 Third-order interrogation methods 205
8.1.5 Fourth-order interrogation methods 208
8.2 Stationary points of curvature of free-form parametric surfaces210
8.2.1 Gaussian curvature 210
8.2.2 Mean curvature 213
8.2.3 Principal curvatures 214
8.3 Stationary points of curvature of explicit surfaces 215
8.4 Stationary points of curvature of implicit surfaces 221
8.5 Contouring constant curvature 223
8.5.1 Contouring levels 223
8.5.2 Finding starting points 223
8.5.3 Mathematical formulation of contouring 225
8.5.4 Examples 227
9. Umbilics and Lines of Curvature 231
9.1 Introduction 231
9.2 Lines of curvature near umbilics 232
9.3 Conversion to Monge form 237
9.4 Integration of lines of curvature 242
9.5 Local extrema of principal curvatures at umbilics 244
9.6 Perturbation of generic umbilics 250
9.7 Inflection lines of developable surfaces 256
9.7.1 Differential geometry of developable surfaces 256
9.7.2 Lines of curvature near inflection lines 262
Contents XV
Geodesies 265
10.1 Introduction 265
10.2 Geodesic equation 266
10.2.1 Parametric surfaces 266
10.2.2 Implicit surfaces 270
10.3 Two point boundary value problem 272
10.3.1 Introduction 272
10.3.2 Shooting method 273
10.3.3 Relaxation method 274
10.4 Initial approximation 275
10.4.1 Linear approximation 275
10.4.2 Circular arc approximation 277
10.5 Shortest path between a point and a curve 278
10.6 Numerical applications 281
10.6.1 Geodesic path between two points 281
10.6.2 Geodesic path between a point and a curve 282
10.7 Geodesic offsets 284
10.8 Geodesies on developable surfaces 287
Offset Curves and Surfaces 293
11.1 Introduction 293
11.1.1 Background and motivation 293
11.1.2 NC machining 293
11.1.3 Medial axis 299
11.1.4 Tolerance region 306
11.2 Planar offset curves 307
11.2.1 Differential geometry 307
11.2.2 Classification of singularities 308
11.2.3 Computation of singularities 311
11.2.4 Approximations 312
11.3 Offset surfaces 316
11.3.1 Differential geometry 316
11.3.2 Singularities of offset surfaces 318
11.3.3 Self-intersection of offsets of implicit quadratic surfaces 319
11.3.4 Self-intersection of offsets of explicit quadratic surfaces 328
11.3.5 Self-intersection of offsets of polynomial parametric
surface patches 335
11.3.6 Tracing of self-intersection curves 343
11.3.7 Approximations 345
11.4 Pythagorean hodograph 349
11.4.1 Curves 349
11.4.2 Surfaces 351
11.5 General offsets 352
11.6 Pipe surfaces 353
11.6.1 Introduction 353
XVI Contents
11.6.2 Local self-intersection
pipe surfaces
11.6.3 Global self-intersection
pipe surfaces
Problems 367
A. Color Plates 377
References 381
Index 405
... We first take the median of data points in each season (shown as solid green points in Fig. 3), and then use these seasonal-median points as control points to constrain a smooth Bézier curve. A Bézier curve is a parametric curve that uses the Bernstein polynomials as a basis (Patrikalakis & Maekawa 2010;Mortenson 1999): ...
... We use the de Casteljau algorithm (Patrikalakis & Maekawa 2010) to evaluate the Bézier curve, which defines the curve recursively ...
We study the optical light curves - primarily probing the variable emission from the accretion disk - of ~ 900 extreme variability quasars (EVQs, with maximum flux variations more than 1 mag) over an observed-frame baseline of ~ 16 years using public data from the SDSS Stripe 82, PanSTARRS-1 and the Dark Energy Survey. We classify the multi-year long-term light curves of EVQs into three categories roughly in the order of decreasing smoothness: monotonic decreasing or increasing (3.7%), single broad peak and dip (56.8%), and more complex patterns (39.5%). The rareness of monotonic cases suggests that the major mechanisms driving the extreme optical variability do not operate over timescales much longer than a few years. Simulated light curves with a damped random walk model generally under-predict the first two categories with smoother long-term trends. Despite the different long-term behaviors of these EVQs, there is little dependence of the long-term trend on the physical properties of quasars, such as their luminosity, BH mass, and Eddington ratio. The large dynamic range of optical flux variability over multi-year timescales of these EVQs allows us to explore the ensemble correlation between the short-term (< 6 months) variability and the seasonal-average flux across the decade-long baseline (the rms-mean flux relation). We find that unlike the results for X-ray variability studies, the linear short-term flux variations do not scale with the seasonal-average flux, indicating different mechanisms that drive the short-term flickering and long-term extreme variability of accretion disk emission. Finally, we present a sample of 16 EVQs, where the approximately bell-shaped large amplitude variation in the light curve can be reasonably well fit by a simple microlensing model.
... their intersection will be empty if, and only if, [25]: ...
This paper presents the complete inverse kinematic analysis of a novel redundant truss-climbing robot with 10 degrees of freedom. The robot is bipedal and has a hybrid serial-parallel architecture, where each leg consists of two parallel mechanisms connected in series. By separating the equation for inverse kinematics into two parts - with each part associated with a different leg - an analytic solution to the inverse kinematics is derived. In the obtained solution, all the joint coordinates are calculated in terms of four or five decision variables (depending on the desired orientation) whose values can be freely decided due to the redundancy of the robot. Next, the constrained inverse kinematic problem is also solved, which consists of finding the values of the decision variables that yield a desired position and orientation satisfying the joint limits. Taking the joint limits into consideration, it is shown that all the feasible solutions that yield a given desired position and orientation can be represented as 2D and 3D sets in the space of the decision variables. These sets provide a compact and complete solution to the inverse kinematics, with applications for motion planning.
A shape optimization procedure is presented. It is dedicated to the noise generated by obstacle flows. The cost function is the acoustic power efficiency, which is derived directly from the fluctuations of the aerodynamic force by a single formula from the hypothesis of tonal noise. The force is estimated from the direct solution of the 2D incompressible, unsteady flow in laminar regime over a convex symmetrical obstacle without incidence. The no-slip condition at the boundary is assured by an Immersed Boundary Method, that allows the use of the same mesh for all the geometries. The shape of the obstacle is defined by 4 Bézier curves, constrained by second-order continuity leading to 4 degrees of freedom: the aspect ratio, the position of the maximum height and two curvature parameters (up and downstream). The optimization is performed via a Particle Swarm Optimization (PSO) routine. Several tests are performed increasing complexity so that coefficients of the PSO be adjusted to the present response surface. There is up to 16 dB of difference between the power efficiency of the extrema configurations for a fixed aspect ratio (AR) and 8 dB for constrained surface or perimeter. For an AR of 1.5, the optimal shape leads to 3 dB less acoustic power than the ellipse of same AR. The shapes that minimize acoustic power are relatively different from those that minimize the mean drag.
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According to the technological development and that of the New Information and Communication Technologies (NICT), and the evolution of organizational management systems, as well, in front of the new socio-economic factors, and the successive deep international changes, resulting from the globalization, the organization services and structures as well as their process activities (agribusiness industry, automotive industry, infrastructure industry, mining industry…) became more flexible, interactive and virtual. Nowadays, the worldwide mining sector, especially in morocco, marks a radical evolution as productivity and investment regards, which allows it the upgrading of a country characterized by mining vacation to a mining production country. This vision forces a sustainable strategy for improvement in order to position in a market, which is marked by a strong competition. Such a situation encourages the mining companies to cooperate deep changes of their systems of production, as regards energy optimization, quality improvement (products, works…), of safety, of knowledge, of competence, and of industrial performance, while takingaccount thus the environmental protection. In this paper, we present the actual position of the mining sector in morocco, as regards production, investment, and processing, then we purpose our contribution relation to design a global Management system entitled SDMS (Sustainable Development Management System), ensuring a transition, on which allows the Moroccan companies to get a position into the international mining field.
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Contributions by F.- E. Wolter to Theory and Computations on Distance Function, Cut Loci, Medial Axes, Shortest Paths, Geodesics in bordered and unbordered Riemannian manifolds during the period 1979-2013.
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A geometric model of an object—in most cases being a subset of the 3D space—can be used to better understand the object's structure or behavior. Therefore data such as the geometry, the topology and other application specific data have to be represented by the model. With the help of a computer it is possible to manipulate, process or display these data. We will discuss different approaches for representing such an object: Volume-based representations describe the object in a direct way, whereas boundary representations describe the object indirectly by specifying its boundary. A variety of different surface patches can be used to model the object's boundary. For many applications it is sufficient to know only the boundary of an object. For special objects explicit or implicit mathematical representations can easily be given. An explicit representation is a map from a known parameter space for instance the unit cube to 3D-space. Implicit representations are equations or relations such as the set of zeros of a functional with three unknowns. These can be very efficient in special cases. As an example of volume-based representations we will give a brief overview of the voxel representation. We also show how the boundary of complex objects can be assembled by simpler parts such as surface patches. These come in a variety of forms: planar polygons, parametric surfaces defined by a map from 2D space to 3D space, especially spline surfaces and trimmed surfaces, multiresolutionally represented surfaces (for example wavelet-based) and surfaces obtained by subdivision schemes. In a boundary representation only the boundary of a solid is described. This is usually done by describing the boundary as a collection of surface patches attached to each other at outer edges. One of the (topologically) most complete schemes is the half-edge data structure as described by Mäntylä. Simple objects constructed via any of the methods above can be joined to build more complex objects via Boolean operators (constructive solid geometry, CSG). Constructing an object one has to assure that the object is in agreement with the topological requirements of the modeling system. Notoriously difficult problems are caused by the fact that most modeling systems can compute surface intersections only with a limited precision. This yields numerical results that may finally cause major errors such as topologically contradictory conclusions. The rather new method of ‘medial modeling’ is also presented. Here an object is described by its medial axis and an associated radius function. The medial axis itself is a collection of lower dimensional objects, that is, for a 3D solid a set of points, curves and surface patches. This medial modeling concept developed at the Welfenlab yields a very intuitive user interface (UI) useful for solid modeling, and also gives as a by-product a natural meshing of the solid for FEM computations. Additional attributes can be attached to an object, like attributes of physical origin or logical attributes. Physical attributes include photometric, haptical and other material properties, such as elasticity or roughness. Physical attributes are often specified by textures. These texture are mapped to the surface to relate surface points to certain quantities of the attribute. The most common use for these are photometric textures, although they can also be used for roughness etc. Logical attributes relate the object to its (data)environment. They can for example group objects that are somehow related, or they can associate scripts to the object, such as callbacks for user interactions.
This paper presents an overview of surface intersection problems and focuses on the rational polynomial parametric/rational polynomial parametric surface intersection case including transversal and tangential intersections. Emphasis is placed on marching methods with a discussion of the problems with conventional tracing algorithms. An approach using a validated interval ordinary differential equation system solver is outlined and illustrated with examples, which offers significant advantages in robustness over conventional marching schemes.
This paper presents a panel generation framework for seakeeping analysis of multiple bodies and offshore structures. The configurations of multiple bodies and offshore structures are different from those of a single ship. In particular, the topological structure of the free surfaces becomes complicated due to the multiple floating bodies, resulting in multiple classifications for the free surfaces based on their genus. The multi-body configuration consists of two floating bodies placed in two configurations, i.e., side by side and tandem, which would generate two holes in the free surface. For the offshore structure case, multiple holes are generated in the free-surface domain due to the legs of the offshore structure. In this work, strategies for generating body and free-surface panels are provided, and the results are analyzed. A software prototype that implements the proposed methods is developed to provide efficient panel generation for multiple bodies and offshore structures. Examples demonstrate that the proposed framework can be successfully used for seakeeping analysis of multiple bodies and offshore structures.
An analytical curvature-continuous path-smoothing algorithm is developed for the high speed machining of a linear tool path. The algorithm can be used in a post-processing stage or NC unit. Every segment junction of the linear tool path, which is the point of tangent discontinuity, is blended by inserting two cubic Bézier spiral curves. A tool path, which is composed of cubic Bézier curves and lines, is then obtained to replace the linear tool path. The new tool path is everywhere curvature-continuous, and both the tangent and curvature discontinuities are avoided. The feed motion will be more stable since the discontinuities are the most important sources of feed fluctuation. In the blending algorithm, the approximation error at the segment junction is accurately guaranteed and the control points of the two cubic Bézier transition curves are all analytically computed. The maximal curvature in every Bézier transition pair, which is critical for velocity planning, is also analytically computed. The analytical expressions provide a way to optimize the curvature radii of the transition curves to pursue the high feed speed. The path-smoothing methods for the post-processing stage and NC unit are both developed. The computational examples confirm the validity of the algorithm. The transition algorithm has been integrated into an open NC system. Cutting experiments show that the curvature-continuous tool path generates smoother feed and consumes shorter machining time than the original linear tool path.
Conference Paper
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Gives an overview of some recent methods useful for local and global shape analysis and for the design of solids. These methods include, as new tools for global and local shape analysis, the spectra of the Laplace and Laplace-Beltrami operators and the concept of stable umbilical points, i.e. stable singularities of the “principal curvature line” wire-frame model of the solid's boundary surface. Most of the material in this paper deals with the medial axis transform as a tool for shape interrogation, reconstruction, modification and design. We show that it appears to be possible to construct an intuitive user interface that allows one to mould shapes by employing the medial axis transform. We also explain that the medial axis and Voronoi diagram can also be defined and computed on free surfaces in a setting where the geodesic distance between two points p and q on a surface S is defined by the shortest surface path on S joining the two points p, q. This leads to the natural and computable generalized concepts of the geodesic medial axis and geodesic Voronoi diagram on free-form surfaces. Both can be computed with a reasonable speed and with high accuracy (of about 12 digits when double floating-point arithmetic is used for the computations)
We present an accurate and efficient method to generate a CNC tool path for a smooth free-form surface in terms of planar cubic B-spline curves which will be fed into a free-form curve interpolator. We assume the use of a three-axis CNC machine tool with a ball end-mill cutter. We first interpolate break points, which are generated by computing the offset surface–driving plane intersection curve reflecting the curvature, by a planar cubic B-spline curve. We then evaluate the maximum scallop height along a scallop curve by computing the stationary points of the distance function between the scallop curve and the design surface. Furthermore, we compute the maximum pick feed such that the maximum scallop height along a scallop curve coincides with the prescribed tolerance. Illustrative examples show the substantial improvements this method achieves over conventional methods where the tool path consists of linear or circular paths.
A solid is a connected orientable compact subset of R3 which is a 3-manifold with boundary. Moreover, its boundary consists of finitely many components, each of which is a subset of the union of finitely many almost smooth surfaces. Motivated by numerical robustness issues, we consider a finite collection of boxes, with faces parallel to the coordinate planes, which covers the boundary of the solid itself. An interval solid is the union of this collection and the solid. In this paper we develop sufficient conditions on the collection of the boxes and a 3-manifold, so that the union of the collection and the manifold is homeomorphic to the manifold itself. Finally, we outline an approach for constructing an interval solid, using interval arithmetic, homeomorphic to the solid.
: Wepresent MAPC, a library for exact representation of geometric objects -- specifically points and algebraic curves in the plane. Our library makes use of several new algorithms, whichwe present here, including methods for finding the sign of a determinant, finding intersections between two curves, and breaking a curveinto monotonic segments. These algorithms are used to speed-up the underlying computations. The library provides C++ classes that can be used to easily instantiate, manipulate, and perform queries on points and curves in the plane. The pointclassescanbeusedtorepresentpoints known in a varietyofways (e.g. as exact rational coordinates or algebraic numbers) in a unified manner. The curve class can be used to represent a portion of an algebraic curve. Wehaveused MAPC for applications dealing with algebraic points and curves, including sorting points along a curve, computing arrangementofcurves, medial axis computations and boundary evaluation of spline primitives. As com...
The medial axis transform (MAT) has potential as a powerful representation for a conceptual design tool for objects with inherent symmetry or near-symmetry. The medial axis of two-dimensional objects or medial surface of three-dimensional objects provides a conceptual design base, with transition to a detailed design occurring when the radius function is added to the medial axis or surface, since this additional information completely specifies a particular object. To make such a design tool practicable, however, it is essential to be able to convert from an MAT format to a boundary representation of an object. In this thesis, we provide the details for the conversion of the MAT of a set of two- and three-dimensional objects to a boundary representation. We demonstrate certain smoothness properties of the MAT and show the relationship between the tangent to the MAT at a point and the boundary points related to that MAT point. We classify the MAT points based on the tangency conditions at the point, and for each type of point, we detail the method for obtaining the boundary points related to it. We discuss requirements for an MAT to be locally valid in the sense that the given curves could actually be the MAT of an allowable object. We also provide a theoretical error bound on the computation for the two-dimensional case. Finally, we discuss an implementation of our algorithm both for piecewise linear two-dimensional MATs and for piecewise planar and linear three-dimensional MATs, and demonstrate some results we have obtained.