Computer Aided Engineering Design
SPLINES 163 [Hint: For ti–1≠ti, use normalization; for ti–1 = ti, find C from the condition M2,i(ti) = N 2 ,i(ti) = 0]. For t∈ [ ...
164 COMPUTER AIDED ENGINEERING DESIGN r r r r 1 2 0 1 2 () =^12 [ 1] 1–21 –2 2 0 110 uuu ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ...
Chapter 6 Differential Geometry of Surfaces Surfaces define the boundaries of a solid. They themselves are bounded by curves (Fi ...
166 COMPUTER AIDED ENGINEERING DESIGN Example 6.1. Some commonly known analytical surfaces can be represented in the parametric ...
DIFFERENTIAL GEOMETRY OF SURFACES 167 Figure 6.4 Catenoid and pseudosphere x z y x (a) Catenoid (b) Pseudosphere of constant len ...
168 COMPUTER AIDED ENGINEERING DESIGN x(u,v) = ucosv, y(u,v) = u sin v, z(u,v) = av Equations in cartesian and parametric form f ...
DIFFERENTIAL GEOMETRY OF SURFACES 169 6.1.2 Tangent Plane and Normal Vector on a Surface Referring to Eq. (6.2) for tangents ru( ...
170 COMPUTER AIDED ENGINEERING DESIGN For a surface in implicit form, that is, f(x,y,z) = 0, the normal N and unit normal n at a ...
DIFFERENTIAL GEOMETRY OF SURFACES 171 6.2 Curves on a Surface A curve c on a parametric surface r(u,v) may be expressed in terms ...
172 COMPUTER AIDED ENGINEERING DESIGN The symmetric matrix G is termed as the first fundamental matrix of the surface. In litera ...
DIFFERENTIAL GEOMETRY OF SURFACES 173 The two curves are orthogonal to each other if ∂ ∂ ∂ ∂ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟⋅ ∂ ∂ ∂ ∂ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ rr ...
174 COMPUTER AIDED ENGINEERING DESIGN which using Taylor series expansion is d u du d u dud u + + +^1 du d 2 () +^1 2 () 22 2 2 ...
DIFFERENTIAL GEOMETRY OF SURFACES 175 HereL,M and N are the second fundamental form coefficients. The second fundamental matrix ...
176 COMPUTER AIDED ENGINEERING DESIGN Case 1. M^2 – LN < 0: For any departure dv from point P, there is no real value of du. ...
DIFFERENTIAL GEOMETRY OF SURFACES 177 du MMLN L ddu MMLN L = d – and = (^22) –– – vv here P is called a hyperbolic point o ...
178 COMPUTER AIDED ENGINEERING DESIGN 6.5 Curvature of a Surface: Gaussian and Mean Curvature The curve C in Figure 6.12 lies on ...
DIFFERENTIAL GEOMETRY OF SURFACES 179 Therefore, κn d ds d ds dd dd = – = – with ds^2 d d rn rn rr ⋅ rr ⋅ ⋅ ≈⋅ (6.28) We can sim ...
180 COMPUTER AIDED ENGINEERING DESIGN (L–G 11 κn) + (M–G 12 κn)μ = 0 ⇒ (– ) ( – ) (– ) ( – ) 1 = 0 0 12 22 11 12 MG NG LG MG nn ...
DIFFERENTIAL GEOMETRY OF SURFACES 181 G 11 = ru·ru = 1 + (3u^2 – 3v^2 )^2 , G 12 = ru·rv = – 6uv (3u^2 – 3v^2 ), G 22 = rv·rv = ...
182 COMPUTER AIDED ENGINEERING DESIGN Figure 6.13 Monkey saddle and its curvatures (a) Monkey saddle (b) Maximum normal curvatur ...
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