The Geometry of Ships

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THE GEOMETRY OF SHIPS 27 Section 5 Polygon Meshes and Subdivision Surfaces A mesh of polygons (usually triangles and/or quadrilaterals or quads) can serve as a useful approximation of a surface for some purposes of design and analysis. The concept of successive refinement of polygon meshes has led to a new alternative for mathematical surface definition known as “subdivision surfaces,” which is under rapid development at the time of this writing (Warren & Weimer 2002; Peters & Reif 2008). 5.1 Polygon Mesh. A suitable representation for a polygon mesh consists of: (a) Example triangle mesh. (b) After one cycle of subdivision and smoothing. (c) After two cycles of subdivision and smoothing. (d) After moving one vertex. Fig. 24 A triangle mesh and three subdivision surfaces based on it.
28 THE PRINCIPLES OF NAVAL ARCHITECTURE SERIES a list of 3-D points, called vertices or nodes, and a list of faces, each face being an ordered list of vertices which form a closed polygon. The lines connecting adjacent vertices in a face are called edges or links. An edge can be shared by two adjacent faces, or it can belong to only one face, when it is part of the mesh boundary. If no edge is shared by more than two faces, the mesh is said to have manifold topology. Figure 24 (a) is a small example of a triangle mesh. It has five vertices: 1: 1.0, 1.0, 0.0 2: 1.0, 1.0, 0.0 3: 0.0, 0.0, 2.0 4: 1.0, 1.0, 0.0 5: 1.0, 1.0, 0.0 four faces: 1: 1, 2, 3 2: 2, 5, 3 3: 5, 4, 3 4: 4, 1, 3 and eight edges. The four edges connecting to vertex 3 are each shared by two faces. The four edges at the plane Z 0 each belong to only one face, so they form the boundary of the mesh. A polygon mesh, and especially a triangle mesh, is easy to render for display as either a surface or a solid. It is also a commonly accepted representation for many kinds of 3-D analysis, e.g., aerodynamic and hydrodynamic flows, wave diffraction, radar cross-section, and finite element methods. 5.2 Subdivision Surfaces. Given a polygon mesh consisting of triangle and/or quad polygons, it is easy to generate a finer polygon mesh by the following linear subdivision rule: • insert a new vertex at the center of each original edge, and at the center of any quad polygon; then • connect the new vertices with new edges, so each original face is split into four new faces. This subdivision can be repeated any number of times, generating successive meshes of smaller and smaller polygons. However, subdivision alone does not improve the smoothness of the mesh; each new face constructed this way would be exactly coincident with a portion of the original face that it is descended from. The key idea of subdivision surfaces is to follow (or combine) such a subdivision step with a smoothing step that repositions each vertex to a weighted average of a small set of neighboring vertices. Then the successive meshes become progressively smoother, approaching C 2 continuity (comparable to cubic splines) at almost all points, and C 1 continuity everywhere, in the limit of infinite subdivision. There are several competing schemes for choosing the set of neighbors and assigning weights. As an example, Fig. 24 (b) and (c) show the original “coarse” triangle mesh of Fig. 24 (a) following one and two cycles of Loop subdivision. The vertices and edges of the coarse mesh can be interpreted as a “control point net,” similar in effect to the control net for a B-spline or NURBS parametric surface. For example, Fig. 24 (d) shows the effect of moving vertex 3 to (1.0, 0.5, 2.0) and regenerating the mesh. Smoothing rules can be modified at specified vertices or chains of vertices, to allow breakpoints and breaklines in the resulting surface. A subdivision surface has the following attractive properties, similar to B-spline and NURBS surfaces: • Local support: A given control point affects only a local portion of the surface. • Rigid body: The shape of the surface is invariant with respect to a rigid body displacement or rotation of the control net. • Affine stretching: The surface scales affinely in response to affine scaling of the net. • Convex hull: The surface does not extend outside the convex hull of the control points. Compared with parametric surfaces, subdivision surfaces are far freer in topology. The surface inherits the topology of its control net. A subdivision surface can have holes, any number of sides, or no sides at all. (A closed initial net produces a closed surface.) A major disadvantage of subdivision surfaces as of this writing is a lack of standardization. Because different CAD systems employ different subdivision and smoothing algorithms, subdivision surfaces cannot generally be exchanged between systems in a modifiable form. In the subdivision world, there is not yet any equivalent of the IGES file. (Of course, there are many file formats for exchanging the triangle meshes that result from subdivision.)
Page 2 and 3: The Principles of Naval Architectur
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Page 6 and 7: x PREFACE computations are presente
Page 8 and 9: Section 1 Geometric Modeling for Ma
Page 10 and 11: THE GEOMETRY OF SHIPS 3 Although fi
Page 12 and 13: THE GEOMETRY OF SHIPS 5 and their f
Page 14 and 15: THE GEOMETRY OF SHIPS 7 The concept
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Page 26 and 27: THE GEOMETRY OF SHIPS 17 plicit sur
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Page 30 and 31: THE GEOMETRY OF SHIPS 21 4.9 B-spli
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Page 34 and 35: THE GEOMETRY OF SHIPS 25 A typical
Page 38 and 39: THE GEOMETRY OF SHIPS 29 A curve ly
Page 46: Fig. 32 The lines plan of a cargo s
Page 49 and 50: THE GEOMETRY OF SHIPS 37 curves, ty
Page 51 and 52: THE GEOMETRY OF SHIPS 39 tration of
Page 53 and 54: THE GEOMETRY OF SHIPS 41 Alternativ
Page 55 and 56: THE GEOMETRY OF SHIPS 43 The genera
Page 57 and 58: THE GEOMETRY OF SHIPS 45 where: L a
Page 61 and 62: THE GEOMETRY OF SHIPS 47 Define the
Page 63 and 64: THE GEOMETRY OF SHIPS 49 Molded dis
Page 65 and 66: THE GEOMETRY OF SHIPS 51 11.2 Bonje
Page 68 and 69: THE GEOMETRY OF SHIPS 53 Section 12
Page 70 and 71: THE GEOMETRY OF SHIPS 55 Fig. 41 A
Page 74 and 75: THE GEOMETRY OF SHIPS 57 surface ab
Page 76 and 77: INDEX Note: Page numbers followed b
Page 78 and 79: Index Terms Links center of buoyanc
Page 80 and 81: Index Terms Links coordinates, homo
Page 82 and 83: Index Terms Links displacement 3 4
Page 84 and 85: Index Terms Links geometry definiti
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Index Terms Links isoparms 17 K kee
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Index Terms Links model (Cont.) sca
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Index Terms Links pole 18 polygon 1
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Index Terms Links seakeeping 2 31 3
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Index Terms Links surfaces (Cont.)
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Index Terms Links triangle mesh 27f
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The Geometry of Ships

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