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.

Figure 24. 1 Two implicit curves in the plane Figure 24.1 Two implicit curves in the plane. (a) The circle is defined by x2 + y2 = 1, or by x2 + y2 − 1 = 0. (b) The lemniscate of Bernoulli is defined by (x2 + y2)2 = 2c(x2 − y2); changing c adjusts the angle at which the lines cross at the center of the figure eight. 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: 978-0-321-39952-6). Copyright © 2014 by Pearson Education, Inc. All rights reserved.

Figure 24. 2 A topographical map and side view of two islands Figure 24.2 A topographical map and side view of two islands. At high tide (two almost-circular red curves) the islands are separated and the shoreline has two parts; at low tide (large blue curve) the islands are joined by an isthmus, and the shoreline is a single curve. At mid-tide (green figure-eight curve), the shoreline has a singular point, where the gradient of the height function is zero. In both figures, the faint gray arrows are scaled versions of the gradient of the implicit function. 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: 978-0-321-39952-6). Copyright © 2014 by Pearson Education, Inc. All rights reserved.

Figure 24.3 The graph of F(x, y) = 1 + x3 − y2 and its associated level curves for c = .95, 1, and 1.05. The level curve for c = 1 has a cusp at the origin, where the gradient is zero. This example shows that the topology of the level set need not change at a place where the gradient is zero. 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: 978-0-321-39952-6). Copyright © 2014 by Pearson Education, Inc. All rights reserved.

Figure 24.4 (a) The sphere is defined as the zero set of the implicit function F(x, y, z) = x2 + y2 + z2 − 1; at a typical point P = (x, y, z) of the sphere, the gradient is parallel to the ray from the origin to P, hence parallel to the normal vector to the sphere at P. (b) The cylinder can be defined implicitly by x2 + y2 = 1. 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: 978-0-321-39952-6). Copyright © 2014 by Pearson Education, Inc. All rights reserved.

Figure 24.5 The intersection of a ray (defined by a point P and a direction d) and an implicit surface defined by F(x, y, z) = c must occur at a point Q = P + td (for some value of t) which satisfies F(Q) = 0. So to find the intersection, we can solve F(P + td) = c for the unknown t; the intersection point is then P + td. 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: 978-0-321-39952-6). Copyright © 2014 by Pearson Education, Inc. All rights reserved.

Figure 24.6 (a) The mesh of squares defined by the integer points of the plane can be converted to a mesh of triangles by drawing a diagonal in each square. (b) It can also be done in a more symmetric way by adding a vertex at the center of each square (shown as a small dot) and breaking the square into four triangles. 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: 978-0-321-39952-6). Copyright © 2014 by Pearson Education, Inc. All rights reserved.

Figure 24.7 (a) If we know the values vA, vB, vC, and vD of a function at the points A = (0, 0), B = (1, 0), C = (0, 1), and D = (1, 1), we can compute a value at the point (x, y) inside the unit square by first interpolating values linearly at the points P and Q of AB and CD, respectively, and then interpolating between these; alternatively, we could interpolate the values at the points R and S of the edges AC and BD, and then interpolate between those. In either case, the resultant value is (1 − x)(1 − y)vA + x(1 − y)vB + (1 − x)yvC + xyvD. (b) The graph of the resultant function when vA = 1, vB = 3, vC = 2, and vD = 0. Notice that constant-x and constant-y cross sections of the graph are linear. 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: 978-0-321-39952-6). Copyright © 2014 by Pearson Education, Inc. All rights reserved.

Figure 24.8 A complex shape created from simple implicit surfaces combined in a “blob tree,” which defines a complex implicit function in terms of unary and binary operations on simpler implicits (Courtesy of Erwin de Groot and Brian Wyvill). 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: 978-0-321-39952-6). Copyright © 2014 by Pearson Education, Inc. All rights reserved.

Figure 24.9 Starting from a grid of values, we first determine the topology of the isocurve for an associated function. Vertices are indicated by small circles at midpoints of edges. We then adjust the locations of the vertices to better match the input values (i.e., the small circles move to the place where a linear function on the edge would have a zero crossing). Thus, the process of isocurve extraction is divided into topological and geometric tasks. 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: 978-0-321-39952-6). Copyright © 2014 by Pearson Education, Inc. All rights reserved.

Figure 24. 10 Patterns of signs on a grid square Figure 24.10 Patterns of signs on a grid square. (a) All plusses (or minuses). (b) One plus (or one minus). (c) Two diagonally opposite plusses. (d) Two adjacent plusses. All other cases are rotations or reflections of these. In each case, we’ve marked a dot at the center of each edge through which the isosurface passes, and shown possible patterns of edges by which these can be connected. 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: 978-0-321-39952-6). Copyright © 2014 by Pearson Education, Inc. All rights reserved.

Figure 24.11 One grid vertex has value 0; we adjust the value slightly and compute the isocurve (which passes near the grid vertex), and then, at the end, we move the two nearby vertices to the grid vertex so that the edge between them shrinks to nothing. 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: 978-0-321-39952-6). Copyright © 2014 by Pearson Education, Inc. All rights reserved.

Figure 24.12 Two examples of patterns of plusses and minuses, and the associated bits of isosurface. 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: 978-0-321-39952-6). Copyright © 2014 by Pearson Education, Inc. All rights reserved.

Figure 24.13 Two adjacent cubes in the marching cubes algorithm share a single face. The same four vertices appear on the four edges of this face, but the edges that join together pairs of isosurface vertices in each cube are not consistent with one another; the surface that results will have a boundary rather than being a closed surface. 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: 978-0-321-39952-6). Copyright © 2014 by Pearson Education, Inc. All rights reserved.

Figure 24. 14 Water animated by the level-set method Figure 24.14 Water animated by the level-set method. Notice how droplets form and merge. (Courtesy of Stephen Marschner, ©2002 ACM, Inc. Reprinted by permission.) 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: 978-0-321-39952-6). Copyright © 2014 by Pearson Education, Inc. All rights reserved.