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 7.1 Three different functions: (a) is surjective but not injective; (b) is injective but not surjective; and (c) is bijective.

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 7.2 How is θ related to x and y?

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 7.3 The Cartesian plane, in which points are specified by x- and y-coordinates.

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 7.4 The Cartesian plane with multiple coordinate systems.

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 7.5 The coordinates of M in each coordinate system are the average of the coordinates of P and Q in that coordinate system; thus, the geometric operation of finding the midpoint of a segment corresponds to the algebraic operation of averaging coordinates, independent of what coordinate system we use.

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 7.6 The operation “divide the coordinates of a point by two” produces different results (P ʹ and P ʺ ) in the two coordinate systems— this simple algebraic operation is not independent of the coordinate system, so it doesn’t correspond to any geometric operation.

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 7.7 The arrows labeled v and w are often considered to be “the same,” even though they are clearly different entities. If we think of each of them as representing a displacement of the plane (i.e., a motion of all points of the plane up and to the right), then although the arrows themselves are distinct, they represent the same displacement.

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 7.8 The uvw directions form a right-handed coordinate system.

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 7.9 The dot product of unit vectors gives the cosine of the angle θ between them.

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 7.10 Decomposing a displacement w as a sum of two displacements, one parallel to a given vector v and one perpendicular to v.

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 7.11 A contour map shows the height above sea level with contour lines.

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 7.12 The vector n = (Q − P) ⊥ is perpendicular to the line through P and Q. A typical point X of this line has the property that (X − P) is also perpendicular to n. Indeed, a point X is on the line if and only if (X − P) ∙ n = 0.

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 7.13 The lines AB and CD are the images of the parametric functions γ and η; they intersect at the unknown point P.

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 7.14 The point Q = (1 − t) A + tB is on the edge between A and B (provided t ∈ [0, 1]), and (1 − s) Q + sC is on the line segment from Q to C (when s ∈ [0, 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: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 7.15 The function F : [0, 1] × [0, 1] → R 2 : (s, t) (1 − s) (1 − t)A + (1 − s)tB + stC, from the unit square to the triangle ABC, sends the entire s = 1 edge to the point C. All other lines in the square are sent to lines in the triangle as shown.

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 7.16 The point P divides triangle ABC into three smaller triangles, whose areas are fractions α, β, and γ of the whole; the barycentric coordinates of P are α, β, and γ, that is, P = αA + βB + γC.

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 7.17 The graph of a linear function on R 2 defined by F(x, y, z) = Ax + By + C is a plane shown in red, which is tilted relative to the large pale gray z = 0 plane as long as either A or B is nonzero; it intersects the z = 0 plane in a line ℓ (a portion of which is shown as a heavy black segment). The ray from the origin to the point (A, B) is perpendicular to ℓ.

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 7.18 Polygons (a), (b), and (d) are simple, while (c) and (e) are not. Polygon (e) has a reflex vertex, (i.e., one with vertex angle zero) at the upper right.

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 7.19 To test whether Q is in the interior of the polygon, we cast a ray in an arbitrary direction, and then count entering and leaving intersections with the edges. In this case, there are two leaving intersections (the first and third) and one entering intersection, so the point is inside.

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 7.20 (a) A nonsimple polygon whose regions have been labeled by Listing’s rule, (b) the interior defined by the positive winding number rule, (c) by the even-odd rule, and (d) by the nonzero winding number rule.

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 7.21 The signed area of the triangle QP 1 P 2 is positive if the path from Q to P 1 to P 2 to Q is counterclockwise, and negative if it’s clockwise. In the example shown, the signed area is positive.

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 7.22 The pale yellow triangle is negatively oriented; the blue one positively. The next three will be negative, positive, and negative, and their signed areas will sum to give the gray polygon’s area.

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 7.23 The gray triangle projects to three triangles, one in each coordinate plane. The signed areas of the orange, yellow, and blue triangles form the coordinates for the normal vector to the gray 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 7.24 A tilted triangle and its projection.

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 7.25 A triangle projected to the xz-plane.