Computer Aided Engineering Design
Chapter 2 Transformations and Projections Geometric transformations provide soul or lifetovirtual objects created through geomet ...
24 COMPUTER AIDED ENGINEERING DESIGN be treated as an assemblage of finitely many points arranged in a non-arbitrary manner in s ...
TRANSFORMATIONS AND PROJECTIONS 25 are most convenient to represent such motions. The homogenous coordinate system, which has so ...
26 COMPUTER AIDED ENGINEERING DESIGN Example 2.2For a planar lamina ABCD with A (3, 5), B (2, 2), C (8, 2) and D (4, 5) in x-y p ...
TRANSFORMATIONS AND PROJECTIONS 27 where R = cos – sin 0 sin cos 0 001 θθ θθ ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ (2.4) Rigid body translation an ...
28 COMPUTER AIDED ENGINEERING DESIGN Example 2.3. Lamina ABCD with an inner point P with coordinates (4, 3), (3, 1), (8, 1), (7, ...
TRANSFORMATIONS AND PROJECTIONS 29 2.2.4 Rotation of a Point Q (xq,yq, 1) about a Point P (p,q, 1) Since the rotation matrix Rab ...
30 COMPUTER AIDED ENGINEERING DESIGN the origin O, shifting the line L parallel to itself to a translated position L. (b) Rotate ...
TRANSFORMATIONS AND PROJECTIONS 31 Figure 2.8 Reflection about an arbitrary line In Eq. (2.10), TPQ represents translation from ...
32 COMPUTER AIDED ENGINEERING DESIGN Example 2.5. To reflect a line with end points P (2, 4) and Q (6, 2) through the origin, fr ...
TRANSFORMATIONS AND PROJECTIONS 33 v v v v vv vv 1 * 2 * 1 2 11 12 13 21 22 23 31 32 33 11 22 = = 0 0 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ...
34 COMPUTER AIDED ENGINEERING DESIGN (aaa 112 + + ) = 1^221231 (a 11 a 13 + a 21 a 23 + a 31 a 33 ) = 0 (aaa 122 + + ) = 1^22223 ...
TRANSFORMATIONS AND PROJECTIONS 35 2.3.2 Shear Consider a matrix Shx = 10 010 001 ⎡ shx ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ which when applied to ...
36 COMPUTER AIDED ENGINEERING DESIGN 0 5 10 15 16 14 12 10 8 6 4 2 Figure 2.12 Shear along the y direction 2.4 Generic Transform ...
TRANSFORMATIONS AND PROJECTIONS 37 2.5 Transformations in Three-Dimensions Matrices developed for transformations in two-dimensi ...
38 COMPUTER AIDED ENGINEERING DESIGN L L z x y (a) (b) Figure 2.14 Rotation of an object: (a) about the line y–x = 0 and (b) rot ...
TRANSFORMATIONS AND PROJECTIONS 39 y x O U U′ Uyz z φ d ψ Figure 2.15(b) Computing angles from the direction cosines cos = ψψ, s ...
40 COMPUTER AIDED ENGINEERING DESIGN 2.5.2 Scaling in Three-Dimensions The scaling matrix can be extended from that in a two-dim ...
TRANSFORMATIONS AND PROJECTIONS 41 Eq. (2.30) uses the equivalence [xyzs]T≡ ⎡ ⎣⎢ ⎤ ⎦⎥ 1 x s y s z s T since both vectors represe ...
42 COMPUTER AIDED ENGINEERING DESIGN For reflection about a generic plane Π having the unit normal vector as n = [nxnynz 0] and ...
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