Computational geometry for design and manufacture by I. D. Faux, Michael J. Pratt

By I. D. Faux, Michael J. Pratt

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H1fl + --- + h,fa : /* G X[x]} is an ideal generated by / i , . . , / s . The set of generators / i , . . , fs is called a 6os«s of the ideal. Definition 3: A variety is the set of common zeros of a set of polynomials / ! , . . , V ( / i , . . ,on)=01l<«

By condition (2), we can assume that g2 = gi + axp\, 93=g2+a2pl, 9i = 5 3 +a3pl, (5) where a, = bo,i + &i,jX + b2,iV + b^^z. From these three equations, a conformability condition axp\ + a2p\ + azpl = 0 (6) must hold. From {fi,F2}, we can compute three ascending sets Ai (i — 1,2,3) under the order x > y > z. Setting prem(gj, ^4j) = 0 for i = 1,2,3 leads to a system of linear equations with bij as unknowns. The solution of this system has 10 free parameters. A suitable choice of free parameters gives gi = 2z2xy + 2x2y - 580x2yz - 6496yz - 4306x2/ - 4842xz + 645x 2 z + 1088xy2 + 1358xz2 + 32Q5y2z + 1708z2y + 1090x3jy + 1425x3z + 10682/3x - 538y3z - 737x2y2 - 1295xV + 5 4 6 y V 4- 1425xz3 - 528yz3 - 14146 - 15395x2 - 4Q57y2 - 9319^2 + 5710x3 - 2684y3 + 853^ 3 - 1295x4 + 534y4 + U71xy2z - 4274xyz Constructing Piecewise Algebraic Blending Surfaces 55 + 19190x + 145381/ + 16924z, m = 0i + 2(606a: 4- 606^ - 1212z + 5^ 2 - y2 - 7z2 + 9xy + hyz - llxz) • (x - yf m = g±- 2(606^ - 606y 4- 13s 2 4- y2 - z2 - llxy - 3yz + xz)(y - z)2\ The blending surface corresponding to this solution is shown in Fig.

2. Simple pipe surface blending The above example illustrates some advantages of using piecewise algebraic surfaces to* blend algebraic surfaces. First, the blending surface has a lower degree and thus is topologically simpler. Second, there are fewer free parameters in the general solutions, which makes the control of shape easier. 5 (1) According to the given initial surfaces and transversal surfaces (or planes), determine the defining region for the PAS. Some heuristic rules should be applied into the step.

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