2. Machining of Curved Geometries PinionForging die for BladeIntake manifold Pelton cup Knee joint prothesis Impeller

7. PARAMETRIC SURFACES Surfaces of Known Form Plane surface Cylindrical surface Conical surface Spherical Surface Toroidal Surface

10. PARAMETRIC CURVES Parametric Representation of Curve x = f ( u ); y = g( u ); z = 0; 0  u  1.0 u = 0 u = 1

11. PARAMETRIC CURVES Parametric Representation of Curve x = f ( u ); y = g( u ); z = 0; 0  u  1.0 u = 0 u = 1

12. CNC Programming

14. OFFEST CURVE Parametric Representation of Curve x = f ( u ); y = g( u ); z = 0; 0  u  1.0 pu =pu = dx du dy du p n = - 0 dy du dx du 0

15. OFFEST CURVE Parametric Representation of Curve x = f ( u ); y = g( u ); z = 0; 0  u  1.0 Parametric Representation of Offset Curve X = f ( u ) + n x r Y = g( u ) + n y r Z = 0 nxnx nyny 0 pnpn pnpn n =n = =

16. Machining of Curved Geometries

17. CNC Programming

18. Gouging

19. CNC Programming 1 r min k max =

20. TOOL SELECTION Parametric Representation of Curve x = f ( u ); y = g( u ); z = 0; 0  u  1.0 pu =pu = dx du dy du 0 d2xd2x du 2 d2yd2y 0 p uu = (p u x p uu ). (p u x p uu ) (p u. p u ) 3 k2 =k2 =

21. PARAMETRIC SURFACES Surfaces of Known Form Plane surface Cylindrical surface Conical surface Spherical Surface Toroidal Surface

22. u w

23. PARAMETRIC SURFACES Flat End MillBall End Mill

26. FREE-FORM SURFACES Parametric Surface x = f (u, w); y = g (u, w); z = (u, w); 0  u  1.0; 0  w  1.0 Offset Surface X = f (u, w) + n x.r; Y = g(u, w) + n y.r; Z = (u, w) + n z.r; 0  u  1.0; 0  w  1.0 p u = δxδx δu δu δyδy δu δu δzδz δu δu p w = δxδx δw δw δyδy δw δw δzδz δw δw (p u x p w ) n =n =

27. FREE-FORM SURFACES Parametric Surface x = f (u, w); y = g (u, w); z = (u, w); 0  u  1.0; 0  w  1.0 (p u x p w ) n =n = E = (p u. p u ) F = (p u. p w ) G = (p w. p w ) L = (p uu. n ) M = (p uw. n ) N = (p ww. n ) (LN – M 2 ) K = (EG – F 2 ) = k 1 k 2 (EN + GL -2FM) H = 2 (EG – F 2 ) = 0.5(k 1 + k 2 )

37. RULED SURFACE p(u, 0) p(u, 1) Input Two curves p(u, 0), p(u, 1) p(u, w) = (1-w) p(u, 0) + w p(u, 1)

39. BILINEAR SURFACE p(0, 0) p(0, 1) p(1, 0) p(1, 1)

41. BILINEAR SURFACE (HYPERBOLIC PARABOLOID) Four Corner Points p(0, 0), p(1, 0), p(0, 1), p(1, 1) p(u, w) = (1-u) (1-w) p(0, 0) + u (1-w) p(1, 0) + (1-u) w p(0, 1) + u w p(1, 1)

42. BILINEAR SURFACE (HYPERBOLIC PARABOLOID)