Computer Aided Engineering Design
TRANSFORMATIONS AND PROJECTIONS 63 Show that the reflection about an arbitrary line ax + by + c = 0 is given by ba ab ab a b a ...
64 COMPUTER AIDED ENGINEERING DESIGN Scaling of a point P(x,y) relative to a point P 0 (x 0 ,y 0 ) is defined as x* = x 0 + (x ...
TRANSFORMATIONS AND PROJECTIONS 65 You can calculate the coordinates of the rectangular section FGHIthus created. This frustum i ...
Chapter 3 4. Design of Curves The form of a real world object is often represented using points, curves, surfaces and solids. Al ...
DIFFERENTIAL GEOMETRY OF CURVES 67 values implies only reorienting the line. Another example is of a second-degree polynomial S≡ ...
68 COMPUTER AIDED ENGINEERING DESIGN which is a system of n linear equations in ai,i = 0, ..., n−1, and can be solved by inverti ...
DIFFERENTIAL GEOMETRY OF CURVES 69 Px LxyLn i n i n i –1 =0 –1 () = Σ –1() (3.8) Example 3.1. Construct a polynomial to interpol ...
70 COMPUTER AIDED ENGINEERING DESIGN α 0 = y 0 = 0 α α 1 10 10 = = 2 – 0 1 – 0 = 2 y xx α αα 2 20 120 2021 = ( – ) – ( – ) ...
DIFFERENTIAL GEOMETRY OF CURVES 71 p(x) = a 0 + a 1 x+a 2 x^2 (3.9) with coefficients a 0 ,a 1 and a 2 unknown. At x = xi, the o ...
72 COMPUTER AIDED ENGINEERING DESIGN ii=0ΣΣ Σxxi i i iΣyi 3 =0 3 =0 3 2 =0 3 1 = 4, = 8.5, = 39.25, = 5 ΣΣΣ Σ i i i ii i i i i i ...
DIFFERENTIAL GEOMETRY OF CURVES 73 3.3 Representing Curves Curves may be expressed mathematically using one of the three forms, ...
74 COMPUTER AIDED ENGINEERING DESIGN to represent a circle in the first quadrant. Herein, the parametric form of representation ...
DIFFERENTIAL GEOMETRY OF CURVES 75 For a generic curve seqment, the scalar functions x(u),y(u) and z(u) are preferred to be poly ...
76 COMPUTER AIDED ENGINEERING DESIGN AsQ approaches P, i.e. in the limit Δu→0, the length Δs becomes the differential arc length ...
DIFFERENTIAL GEOMETRY OF CURVES 77 Example 3.3. Find the length of a portion of the helix x=a cos u,y=a sin u,z=bu. To use Eq. ( ...
78 COMPUTER AIDED ENGINEERING DESIGN The vector QP×QW can be computed as QP×QW =[r(u + Δu)−r(u)]× [r(u + Δu)−r(u−Δu)] (3.28) Usi ...
DIFFERENTIAL GEOMETRY OF CURVES 79 From Eq. (3.23), T r = ds() ds so that d du ds du r = T using chain rule. Differentiating fur ...
80 COMPUTER AIDED ENGINEERING DESIGN ρ κκ = =^1 =^1 3 2 2 3 3 d du d du d du d du ds du r rr r × ⎛ ⎝ ⎞ ⎠ (3.30c) In other words, ...
DIFFERENTIAL GEOMETRY OF CURVES 81 The unit bi-normal vector Bmay be obtained from Eq. (3.30b) as B ̇r ̇ ̇r ̇r ̇ ̇r = | | × × ̇ ...
82 COMPUTER AIDED ENGINEERING DESIGN Exercises Find the parametric equation of an Archimedean spiral in a polar form. The large ...
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