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Many of the figures from this book may be reproduced free of charge in scholarly articles, proceedings, and presentations, provided only that the following citation is clearly indicated: “Reproduced with the permission of the publisher from Computer Graphics: Principles and Practice, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley. Copyright 2014 by Pearson Education, Inc.” Reproduction for any use other than as stated above requires the written permission of Pearson Education, Inc. Reproduction of any figure that bears a copyright notice other than that of Pearson Education, Inc., requires the permission of that copyright holder.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 8.1 A triangle mesh, consisting of vertices, edges, and triangular faces.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 8.2 A triangle mesh that has the geometry of a cube.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 8.3 A 1D mesh consists of a collection of vertices and straight-line edges between them. The ones that most often interest us (shown in (a) and (b)) have one or two edges meeting each vertex, and no two edges intersect except at a vertex. But there are also meshes with more than two edges at some vertices, like (c), and ones where edges do intersect at nonvertex points (d).

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 8.4 A wagon-wheel–shaped mesh. An arrow from vertex i to vertex j indicates that (i, j) is an edge of the mesh, and not (j, i).

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 8.5 The square at left is subdivided to become the octagon at right. Note that for each vertex of the square, there’s a vertex in the octagon, and for each edge of the square, its midpoint is a vertex of the octagon as well.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 8.6 A triangle with a dangling edge, like the one shown here, cannot be represented by our mesh structure.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 8.7 Two adjacent triangles in a mesh with consistent normal vectors (i.e., the normal vector tips are on the “same side” of the mesh). Note that the edge (i, j) is an edge of one triangle, but (j, i) is an edge of the other. In general, in a consistently oriented mesh, each edge appears twice, in opposite directions.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 8.8 The triangles of this mesh are oriented; the circular arrows indicate the cyclic ordering of the vertices. Note that the vertex triples (1, 2, 3) and (2, 3, 1) indicate the same oriented triangle (i.e., there are three equivalent descriptions of every oriented triangle).

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 8.9 (left) A manifold vertex has a cycle of triangles around it; (right) by contrast, a boundarylike vertex v has a chain of triangles around it—the starting and ending triangles of the chain each share only one of their edges with other triangles of the chain. The other edges of those triangles that contain v are boundary edges.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 8.10 The shared vertex is nonmanifold: No neighborhood looks planar.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 8.11 The pyramid at left has six faces; the bottom square is divided into two triangles that you cannot see. If we construct the pyramid so that after four triangles have been added, it appears as shown at the right, then the apex vertex is neither an interior vertex nor a boundary vertex according to the definitions, so this shape is neither a manifold nor a manifold with boundary. Once we add another face, it becomes a manifold with boundary, and when we add the last face, it becomes a manifold.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 8.12 The directed-edge data structure (following Figures 4 and 5 of [CKS98]). A directed edge stores a reference to its starting and ending vertex, to the previous and next edges, and to its neighbor. For each real edge of the mesh, there are two directed edges, in opposite directions. Each vertex stores its coordinates and a reference to one of the directed edges that leaves it.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 8.13 The winged-edge data structure records a “previous” and “next” pointer for the faces on each side of each edge.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 8.14 The edge from vertex a to vertex b is collapsed; the edge itself and the two adjacent faces are removed from the data structure; and the other two edges of the upper face, drawn in blue, become one, as do the two edges of the lower face, drawn in red. Nothing else changes. The two vertices v a and v b become a single vertex.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 8.15 Different geometric choices for an edge collapse in 2D. The edge AB is collapsed; one can (a) place the collapsed vertex at A for computational simplicity, (b) at the midpoint of the segment AB, or (c) at a point Q that minimizes the maximum (or average) distance from every old mesh point to the nearest new mesh point. Other goals are possible as well.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 8.16 The aspect ratio of a planar shape is determined by finding the bounding box for the object for which the ratio of length to width is greatest.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 8.17 The triangles adjacent to the edge from v a to v b both have bad aspect ratios. By replacing the edge v a v b with the edge v c v d, we get a new pair of triangles whose aspect ratios are better.