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By John Vince
John Vince explains quite a lot of mathematical recommendations and problem-solving concepts linked to desktop video games, desktop animation, digital truth, CAD, and different parts of special effects during this up-to-date and multiplied fourth edition.
The first 4 chapters revise quantity units, algebra, trigonometry and coordinate structures, that are hired within the following chapters on vectors, transforms, interpolation, 3D curves and patches, analytic geometry, and barycentric coordinates. Following this, the reader is brought to the fairly new subject of geometric algebra, and the final chapters offer an creation to differential and indispensable calculus, with an emphasis on geometry.
Mathematics for special effects covers all the key parts of the topic, including:
Number sets
Algebra
Trigonometry
Coordinate systems
Transforms
Quaternions
Interpolation
Curves and surfaces
Analytic geometry
Barycentric coordinates
Geometric algebra
Differential calculus
Integral calculus
This fourth variation comprises over a hundred and twenty labored examples and over 270 illustrations, that are significant to the author’s descriptive writing kind. arithmetic for special effects presents a valid realizing of the math required for special effects, giving a desirable perception into the layout of special effects software program, and environment the scene for extra studying of extra complex books and technical learn papers.
Read or Download Mathematics for Computer Graphics (4th Edition) (Undergraduate Topics in Computer Science) PDF
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Additional info for Mathematics for Computer Graphics (4th Edition) (Undergraduate Topics in Computer Science)
Sample text
Using Cartesian coordinates, the line’s direction is determined by first identifying the vector’s tail and then measuring its components along the x- and y-axis. For example, in Fig. 2 the vector r has its tail defined by (x1 , y1 ) = (1, 2), and its head by (x2 , y2 ) = (3, 4). Vector s has its tail defined by (x3 , y3 ) = (5, 3), and its head by (x4 , y4 ) = (3, 1). The x- and y-components for r are computed as follows 46 6 Vectors Fig. 2 Two vectors r and s have the same magnitude but opposite directions xr = x2 − x1 = 3 − 1 = 2 yr = y2 − y1 = 4 − 2 = 2 and the components for s are computed as follows xs = x4 − x3 = 3 − 5 = −2 ys = y4 − y3 = 1 − 3 = −2.
6 Trigonometric Identities 29 Fig. 10 An arbitrary triangle Various programming languages include the atan2 function, which is an arctan function with two arguments: atan2(y, x). The signs of x and y provide sufficient information to locate the quadrant containing the angle, and gives the atan2 function a range of [0, 2π]. 6 Trigonometric Identities The sin and cos curves are identical, apart from being displaced by 90◦ , and are related by cos θ = sin(θ + π/2). Also, simple algebra and the theorem of Pythagoras can be used to derive other formulae such as sin θ = tan θ cos θ sin2 θ + cos2 θ = 1 1 + tan2 θ = sec2 θ 1 + cot2 θ = csc2 θ.
4 3D Vectors 47 Fig. 1 Values associated with the eight vectors in Fig. 4 3D Vectors The above vector examples are in 2D, but it is easy to extend this notation to embrace an extra dimension. 4 shows a 3D vector r with its head, tail, components and magnitude annotated. The vector, its components and magnitude are given by r = [∆x ∆y ∆z]T ∆x = xh − xt ∆y = yh − yt ∆z = zh − zt |r| = (∆x)2 + (∆y)2 + (∆z)2 . As 3D vectors play a very important role in computer animation, all future examples are three-dimensional.