Springer Undergraduate Mathematics Series
- ISBN: 1852338016
- DOI: 10.1097/01.BRS.0000018489.20501.10
Abstract
Focusing on the manipulation and representation of geometrical objects, this book explores the application of geometry to computer graphics and computer-aided design (CAD). An introduction to transformations of the plane and three-dimensional space describes how objects can be constructed from geometric primitives and manipulated. This leads into a treatment of projections and the method of rendering objects on a computer screen by application of the complete viewing operation. Subsequently, the emphasis is on the two principal curve and surface representations, namely, Bezier and B-spline (including NURBS). As in the first edition, applications of the geometric theory are exemplified throughout the book, but new features in this revised and updated edition include: the application of quaternions to computer graphics animation and orientation; discussions of the main geometric CAD surface operations and constructions: extruded, rotated and swept surfaces; offset surfaces; thickening and shelling; and skin and loft surfaces; an introduction to rendering methods in computer graphics and CAD: colour, illumination models, shading algorithms, silhouettes and shadows. Over 300 exercises are included, some new to this edition, and many of which encourage the reader to implement the techniques and algorithms discussed through the use of a computer package with graphing and computer algebra capabilities. A dedicated website also offers further resources and links to other useful websites. Designed for students of computer science and engineering as well as of mathematics, the book provides a foundation in the extensive applications of geometry in real world situations.
Springer Undergraduate Mathematics Series
EXERCISES
1.19. Suppose an affine transformation L(x, y) = (ax+ by + c, dx+ey +f)
is applied to a triangle T with vertices A, B, C and area A. Show
that the area of L(T ) is (ad − bc) · A.
1.20. Prove that a transformation maps the midpoint of a line segment to
the midpoint of the image.
1.21. Write a computer program or use a computer package to implement
the various types of transformation. Apply the program to the ex-
amples of the chapter.
There are three cases to consider: (i) (s21 + v1 · v1) = 0, (ii) (s22 + v2 · v2) = 0,
and (iii) s1 = s2 = 0. When s21 + v1 · v1 = 0, then s1 = 0 and v1 = 0, and
therefore p = 0. Likewise, when s22 + v2 · v2 = 0, then q = 0. Finally, when
s1 = s2 = 0, then −v1 · v2 = v1 × v2 = 0. Therefore, either v1 = 0 and hence
p = 0, or v2 = 0 and hence q = 0.
Let q = (s,v) = s + xi + yj + zk be any quaternion, then the conjugate
quaternion, denoted q, is defined to be (s,−v) = s− xi− yj− zk. Then
qq = (s2 + v · v,−sv + sv − (v × v))
= (s2 + v · v,0) = (s2 + |v|2,0)
= s2 + |v|2 .
The modulus of q, denoted |q|, is defined to be
|q| = (qq)1/2 = (s2 + |v|2)1/2 .
A quaternion q satisfying |q| = 1 is said to be a unit quaternion. Every non-zero
quaternion q has a multiplicative inverse quaternion, denoted q−1, satisfying
qq−1 = q−1q = 1 (see Exercise 3.15). The inverse is
q−1 =
q
|q|2 . (3.4)
Readers with a knowledge of algebraic structures may conclude that the alge-
braic properties described earlier, together with the existence of additive and
multiplicative inverses (Exercises 3.15 and 3.16), imply that the quaternions
are a non-commutative division ring.
Example 3.10
Let q = (2, (−1, 0, 3)). Then q = (2, (1, 0,−3)), and |q| = (22 + (−1, 0, 3) ·
(−1, 0, 3))1/2 =
√
14. Hence q−1 = q/|q|2 = 1
14
(2, (1, 0,−3)) =
(
1
7
, ( 1
14
, 0,− 3
14
)
)
.
EXERCISES
3.12. Determine the following sums and products of quaternions
(a) (7 + 3i + 5j− 3k) + (−2 + 3i + 6j− 4k),
(b) (9, (2,−1, 3)) + (−7, (1, 0,−2)),
(c) (2 + 4i− 9j + 5k)(5 + 3i− 2k),
(d) (−2, (3, 2,−5))(7, (0, 1, 4)), and
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