Computer Aided Engineering Design
DIFFERENTIAL GEOMETRY OF SURFACES 183 Example 6.4. A toothpaste tube is a ruled surface. For p(u) = acosui + asinuj + 0k with aa ...
184 COMPUTER AIDED ENGINEERING DESIGN ruu= [–(1 –v)acosu]i–asinu (1 –v)j ruv= [asinu– 2a/π]i–acosuj rvv= 0 All we need to show t ...
DIFFERENTIAL GEOMETRY OF SURFACES 185 6.7 Parallel Surfaces Creation of parallel surfaces is useful in design and manufacture. M ...
186 COMPUTER AIDED ENGINEERING DESIGN MK LN GG H LG NG GG = 0 = = , 2 = ( + ) = + 12 11 22 12 22 11 11 22 ⇒ κκ κ κ (6.45) Create ...
DIFFERENTIAL GEOMETRY OF SURFACES 187 nu · rv = a 1 ru·rv + b 1 rv·rv⇒–M = a 1 G 12 + b 1 G 22 ⇒b 1 = 0 (QM = 0, G 12 = 0) There ...
188 COMPUTER AIDED ENGINEERING DESIGN (a) (b) Example 6.5.Find the equation of surfaces parallel to the sphere x(u,v) = a cos u ...
DIFFERENTIAL GEOMETRY OF SURFACES 189 For a curve on a plane, we can form a surface of revolution by rotating it about a given l ...
190 COMPUTER AIDED ENGINEERING DESIGN rvv = (–u cos v,–u sin v, 0), ruv = (–sinv, cos v, 0), N = ru×rv = (– cos v, –sinv,u) nrrr ...
DIFFERENTIAL GEOMETRY OF SURFACES 191 T() = 1000 0100 001 0001 v v ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ Thus, the equation of the cylinde ...
192 COMPUTER AIDED ENGINEERING DESIGN r(t,θ) = (t) + rt[– cos θ(– cos t, – sin t, 0) + sin (^22) + θ ab (b sin t,–b cos t,a)] F ...
DIFFERENTIAL GEOMETRY OF SURFACES 193 6.10 Curve of Intersection between Two Surfaces In engineering design, one has to deal wit ...
194 COMPUTER AIDED ENGINEERING DESIGN Hence, λ^3 κb = p×Δp=m (say) (6.57) Therefore, | m | = λ^3 κ (6.58) Using the operator Δ o ...
DIFFERENTIAL GEOMETRY OF SURFACES 195 Then f = (xi + yj + zk) = (x,y,z) = Nf, g = (i + 0j + 0k) = (1, 0, 0) = Ng Here,Nf and N ...
196 COMPUTER AIDED ENGINEERING DESIGN Intersection of a Cylinder and a Sphere: Viviani Curve (1692) Let the equations of the sph ...
DIFFERENTIAL GEOMETRY OF SURFACES 197 Curve of Intersection of Two Perpendicular Cylinders The equation of two cylinders can be ...
198 COMPUTER AIDED ENGINEERING DESIGN –2 –1 (^01) 2 4 3 2 1 0 –2 –1 0 1 2 8 4 2 0 6 4 2 –5 0 5 (a) (b) Figure P6.1 (c) (d) (e) ( ...
DIFFERENTIAL GEOMETRY OF SURFACES 199 Develop a procedure for viewing surface geometry. Your program should display a parametri ...
200 COMPUTER AIDED ENGINEERING DESIGN Hint: One may express nu and nv as respective linear combinations of ru and rv, that is, n ...
Chapter 7 Design of Surfaces A closed, connected composite surface represents the shape of a solid. This surface, in turn, is co ...
202 COMPUTER AIDED ENGINEERING DESIGN In Chapter 6, differential properties of surface patches are discussed using analytical su ...
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