Mortenson, M.E.: Geometric Modeling, John Wiley, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Computing Gaussian Curvature Features Gaussian curvature feature points are local extrema of the image intensity surface with a high Gaussian curvature. Gaussian curvature is a parametrization invariant measure of surface curvature defined as the product of the principal normal curvatures [109]. Thus, computing these features is a two step process. First, the local intensity extrema are located and then the Gaussian curvature of the surface is evaluated. The local intensity extrema are estimated by the extrema of a parametric surface fit to the image intensity data in a local ....

Michael E. Mortenson. Geometric Modeling. John Wiley and Sons, 1985.

....is computationally intense relative to (i) ii) or (iii) It is for this combination of factors that (i) ii) or (iii) would be a better choice. 3. Defining Cubic Curve Segments by Parabolic Blending As the name suggests, parabolic blending involves the concept of using blending functions [5,6] to blend two parametrically defined second degree polynomials into a cubic polynomial over an interval. In particular, for four given points p P2 P3 and P4 a second degree polynomial, p , will be defined by the points p, P2 and P3 and another second degree polynomial, q , will be defined by ....

Mortenson, M. E., Geometric Modeling, John Wiley and Sons, Inc., Somerset, New Jersey, 1985.

....could support uniform and factorized numerical representations of solids underlying inhomogeneous structured pointsets. 6 CW complexes and object representations in solid modeling Topology has ever been a leading part in the foundations of solid modeling for more than thirty years ( 61] [56], 53] 29] See for example the applications of the pointset topology and the leading concepts of regularized r sets and regularized boolean operations for the Constructive Solid Geometry. Boundary representations and surface based models have been developed on the basis of the algebraic ....

M.E. Mortenson. Geometric Modeling. John Wiley & Sons, 1985.

....curvature for a region and not just at a single point. The other methods such as SFMs may use a region in their algorithms but their output is for a specific point on the mesh. Since a triangle mesh is a piecewise flat surface, the local curvature of such a surface is seemingly paradoxical [30]. The curvature is singular at each point on the surface infinite at vertices and edges and zero on triangle faces. We can however refer to the total curvature for regions on these surfaces. Lin and Perry [23] use the angle excess around each vertex to estimate the total Gaussian curvature. ....

....point on the surface infinite at vertices and edges and zero on triangle faces. We can however refer to the total curvature for regions on these surfaces. Lin and Perry [23] use the angle excess around each vertex to estimate the total Gaussian curvature. Angle excess itself is well known with [30] providing a nice discussion in the context of computer graphics and the Gauss Bonnett theorem. We find another application of angle excess in a series of papers [5, 6] by Delingette. He lays out a framework for a surface representation that he calls a simplex mesh that is a dual to a triangle ....

M. E. Mortenson, Geometric Modeling, 2nd ed., Wiley, New York, 1997.

....polygons like the one in Figure 1, it should be robust regarding finite arithmetic, and the implementation should not be too complicated. Such an algorithm is described in the paper. Before we proceed, a few words are dedicated to the used vocabulary adopted from the theory of geometric modelling [MORT85]: A loop represents a boundary of a polygon and consists of a closed sequence of oriented polygon edges. Each polygon must have exactly one loop. A ring represents a hole in the polygon. There can be zero or finite number of rings in a considered polygon. The rings can be nested to any level ....

Mortenson, M. E., Geometric Modeling, John Wiley & Sohns, 1985.

....NURBS definition by explicitly incorporating time. The kinematic curve is a function of both the parametric variable u and time t: c(u; t) P n i=0 p i (t)w i (t)B i;k (u) P n i=0 w i (t)B i;k (u) 1) where the B i;k (u) are the usual recursively defined piecewise rational basis functions [20, 21], p i (t) are the n 1 control points, and w i (t) are associated non negative weights. Assuming basis functions of degree k Gamma 1, the curve has n k 1 knots t i in non decreasing sequence: t 0 t 1 : t n k . In many applications, the end knots are repeated with multiplicity k in ....

M.E. Mortenson. Geometric Modeling. John Wiley and Sons, 1985.

....the capability to create new models and accept di#erent input formats. Some of the terms used in the context of geometric modeling are described in the next section. 7. 1 Terminology The following definitions have been culled from various geometric modeling and solid modeling texts [17] 27] 41] [46]. B rep stands for Boundary representation. Here the object is represented in terms of its surface boundaries: vertices, edges and faces. It is the widely used representation of a solid model. Traditionally B reps were restricted to planar, polygonal boundaries. In the past decade, there have ....

M. E. Mortenson. Geometric Modeling. John Wiley and Sons, 1985.

....complexes, cavities and holes of various types suggested by Masuda et al. [9] and others. Application of Euler characteristics and topology in design is also discussed by Lear [10] and Lee [4] A more detailed taxonomy of geometric and topological models is provided by Takala [11] Mortenson [12] and Mantyla [5] While these works tend to provide increasingly general formulations, none has concentrated on specific modeling tasks, such as modeling of sheet metal parts. These parts are usually non manifold and thus comply with general formulae such as those discussed by Gursoz et al. [8] and ....

....ELEMENTARY OPERATORS In the original Euler Poincar equation for manifold solids, the basic topological manipulations complying with the equation are termed Euler operators. They were originally introduced by Baumgart [13] and are discussed in detail by Mantyla [5] Braid et al. [14] and Morenson [12]. The same notion can be carried over to analyze sheet metal parts using Eq. 7) By historical convention, the operators are denoted by mnemonic names. The key to the (new and old) names used here is as follows: M = Make V = Vertex G = Genus (Non Manifold) K = Kill E = free Edge W = Weld (11) ....

Mortenson, M. E., Geometric Modeling, John Wiley &Sons, 2nd Ed., 1997

....to be unambiguous but not necessary unique and the geometry is typically described in an object centered coordinate frame. Constructive Solid Geometry (CSG) and Boundary Representation (B rep) are widely used in CAGD. In CSG, a part is defined by applying Boolean operations on primitive solids [14] whereas B rep defines a solid by its bounding surfaces. The Design by Features techniques [21] define the part geometry using design primitives related to actual manufacturing operations. Currently, there is no single geometric representation that would be best for every design task. Therefore, ....

....z) 1 . The superellipsoid fit is used to detect primitive solids and global shape properties such as symmetry. Rotationally symmetric shapes are constructed using surface of revolution design primitive whereas shapes with translational symmetry can be generated by extrusion. Primitive solids [14] and the approximating superellipsoid shape parameters are depicted in Table 1. Table 1: Primitive solids, and the approximating superellipsoid shape parameters. means that in general the primitive can not be recovered using the superellipsoid model we employ. Primitive solid Shape ....

Mortenson, M., "Geometric Modeling", John Wiley & Sons. 1985.

....be split into calculation of initial points and implicit curve following. Depending on the geometric model, initial point calculation can be complicated, too. We use the Constructive Solid Geometry (CSG) model, defining complicated solids by the Boolean operations applied to simple primitives [11]. For this model we have to solve many nonlinear equations in three variables to calculate initial points for edge detection. The whole mesh generation problem is sketched in Figure 1. This work is supported by the Austrian Science Fund Fonds zur Forderung der wissenschaftlichen Forschung ....

....surface and volume mesh generation is explained in x5. The used mesh optimization strategies are discussed in x6. In x7 examples are given and in x8 the current work is summarized. 2 Geometric Modeling The wide field of computational geometry provides several possibilities for geometric modeling [11]. The different models have complementary properties with respect to ease of solid description and ease of mesh generation algorithms. The Constructive Solid Geometry (CSG) model uses smooth primitives like cylinders and spheres to build more complex solids by the Boolean operations. In the ....

M. E. Mortenson. Geometric Modeling. John Wiley & Sons, New York, 1985.

.... Schemes Several representations are commonly used to define solid geometry, including parameterized sweeps and generalized cylinders, spatial partitioning schemes such as voxels, octrees, or binary space partitioning trees, constructive solid geometry (CSG) and boundary representations (b reps) [20, 29, 50]. For manufacturing applications, CSG and b reps are most common. 2.1.1 CSG With CSG, solid primitives such as cones, spheres, cubes, and half spaces are scaled, translated, and or rotated using geometric transforms and then combined via Boolean set operations (union, intersection, and ....

Michael E. Mortenson. Geometric Modeling. Wiley, New York, 1985.

....4 patches meet at each internal vertex, avoiding the problem. BMVC 1994: A J Stoddart, Slime. 2 S Patches In this section we present a very brief introduction to Bezier curves, Bezier surfaces and a generalization of Bezier surfaces called S patches. The reader is referred to standard texts [1] for more details on Bezier curves and surfaces, and to Loop and De Rose [2] for more details on S patches. When discussing Bezier curves it is useful to replace the usual single parameter u with two parameters u 1 and u 2 and a constraint that u 1 u 2 = 1. This is a notational convenience and ....

M. E. Mortenson, Geometric Modeling, John Wiley & Sons, New York, (1985).

....of a simple model, and clearly shows the role of the LOOP and COEDGE mechanisms. By the consistent usage of these topology entities, ACIS is able to model quite complex geometries very concisely. Details on the mathematical definitions of manifold cellular topology geometry can be found in [34], but a succinct and practical definition is as follows: a manifold EDGE can bound or be lengthwise attached to exactly two FACEs or surfaces simultaneously. For example, if an interior vertical web surface were added to Figure 2 5 such that it attached to the top, bottom, and side surfaces, ....

.... through curve 2, locating the end points of each segment at one third and two thirds of the arc length of curve 2) This is to be expected, and is exactly the same as what occurs when a typical analytical curve is discretized, yet because facetted data by definition will not be C 1 continuous [34], the difference is simply more pronounced. Succinctly, nodes always lie on the curve while edges may not. The command used to accomplish this virtual geometry construction is simply the importing of the decomposed subdomain mesh files. In CUBITP, use the import subdomains command (see Appendix ....

Mortenson, M. E., Geometric Modeling, John Wiley & Sons, Inc., New York, 1985, p. 94.

....addition, the shape of the LV at any time instant can be obtained by setting t to any desired value. 4) Change of strain over time can be obtained at all myocardial points. 2 4D B Spline Representation The simplest and most direct geometric element to model a time varying solid is a hyperpatch [5]. A hyperpatch is a patch bounded collection of points whose coordinates are given by continuous, four parameter, single valued mathematical functions of the form x(t) x(u; v; w; t) y(t) y(u; v; w; t) z(t) z(u; v; w; t) 1) where t is the time variable. The parametric variables u, v, and w ....

M. E. Mortenson, Geometric Modeling. John Wiley & Sons, New York, 1985.

....internal data structures from the database report and the data assemblers go the other direction they generate database reports from the internal data structures. 2.2. GEOMETRY DATA MODEL In the geometry data model, the basic geometry data of a part are organized in a boundary representation [MORT85]. In this representation, a part is described by its topological entities and its geometrical entities, as shown in Figure 2. The topological entities are structured hierarchically in the order: solid, shell, face, loop, edge, and vertex. The geometrical entities are surface, curve, point, and ....

Mortenson, Michael E., Geometric Modeling, John Wiley & Sons, New York, 1985.

....is referred to as a constructive solid geometry representation, or CSG representation, for short. Each style of representation has its advantages and disadvantages, depending on the operations we wish to perform on the objects. The reader is referred to one of the standard texts in solid modeling [13, 16], or the review article [23] for further details on these representations and their relative merits. If one looks at modelers in either camp, for example the romulus [16] geomod [25] and medusa [17] modelers of the boundary persuasion, or the padl 1 [27] padl 2 [2] and gmsolid [1] modelers of ....

....on the operations we wish to perform on the objects. The reader is referred to one of the standard texts in solid modeling [13, 16] or the review article [23] for further details on these representations and their relative merits. If one looks at modelers in either camp, for example the romulus [16], geomod [25] and medusa [17] modelers of the boundary persuasion, or the padl 1 [27] padl 2 [2] and gmsolid [1] modelers of the CSG persuasion, one nearly always finds provisions for converting to the other representation. This is an important and indispensable step that poses some challenging ....

M. Mortenson. Geometric Modeling. John Wiley & Sons, 1985. 21

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Mortenson, M.E.: Geometric Modeling, John Wiley, 1985.

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M. E. Mortenson, Geometric Modeling, John Wiley & Sons, 1997.

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Mortenson, Micheal, Geometric Modeling, New York, John Wiley & Sons, 1985.

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M.E. Mortenson, Geometric modeling, John Wiley & Sons, Inc., ISBN 0-471-12957-7, New York, 1997

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M. E. Mortenson. Geometric Modeling. Wiley, New York, NY, 1985.

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Mortenson, M.E.: Geometric Modeling, John Wiley, 1985.

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M.E. Mortenson. Geometric Modeling. John Wiley & Sons, 1985.

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Mortenson, Michael E. Geometric Modeling. John Wiley & Sons, New York, New York, 1985.

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M Mortenson. Geometric Modeling. John Wiley & Sons, Inc., 1985.

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