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6 in a matrix form, we have ! 7) a0x a0y a0z 2Â3 where matrix A contains the unknown coefficients. By plugging u ¼ 0 and u ¼ 1 into Eq. 7, we have respectively P(0) ¼ P0 ¼ [0 1] A, and P(1) ¼ P1 ¼ [1 1] A. By rewriting the above equations in a matrix form, we have ! ! 8) 1 1 2Â2 2Â3 Pð1Þ 2Â3 P1 2Â3 where P0 and P1 must be known in order to define the straight line. From Eq. À1 ! ! 9) 1 1 P1 1 0 P1 Hence, from Eq. 7, we have PðuÞ ¼ UA ¼ ½ u À1 1 1 1 0 ! P0 P1 ! 10) where (1 À u) and u are so-called basis functions associated with the characteristic points (in this case, the start and end points P0 and P1) of the straight line.
16 Crosshatch pattern of a typical cut-out layer: (a) overall and (b) zoom-in area. , 1998b; McMahon and Browne, 1998) to fabricate functional parts as well as a die or mold for production. For example, the cover die of a mechanical part is machined from an 8 in. 25 in. Â 2 in. 17(a). P11 in Pro/MFG. In addition to volume milling and contour surface milling, drilling operations are conducted to create the waterlines. 17b). 17 Cover die machining: (a) virtual and (b) CNC. 5) is employed to illustrate the e-Design paradigm and tool environment.
11) where matrix A contains 9 (3Â3) unknown coefficients. 1 Spline CurvedThree Points For a quadratic spline curve, we assume the three distinct points are P0 ¼ P(0), P1 ¼ P(½), and P2 ¼ P(1). Note that P1 does not have to be located at u ¼ ½. By plugging u ¼ 0, u ¼ ½, and u ¼ 1 into Eq. 11, we have respectively P(0) ¼ P0 ¼ [0 0 1] A, P(½) ¼ P1 ¼ [¼ ½ 1] A, and P(1) ¼ P2 ¼ [1 1 1] A. 12) P1 5 4 6 2 7 4 4 2 1 5A3Â3 4 5 P2 3Â3 1 1 1 Pð1Þ 3Â3 where P0, P1, and P2 are known. 13) Hence from Eq. 6(a).