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Extra resources for Proceedings ISSAC 2008 (ACM)
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S, t˜, u ˜ s, t˜, u P(˜ ˜) = a ˜(˜ s, t˜, u ˜), ˜b(˜ s, t˜, u ˜), c˜(˜ s, t˜, u ˜), d(˜ ˜) which has the same implicit representation as P(s, t, u). ˜ s, t˜, u Since P(˜ ˜) contains the base point (0, 1, 0), it doesn’t contain the term t2 . Therefore, without loss of generality, we may assume that, in non-homogeneous variables, the quadratic surface P(s, t) has the form P(s, t) = PT1 + PTs s + PTt t + PTs2 s2 + PTst st, PRELIMINARIES where P1 , Ps , Pt , Pst and Ps2 are all column vectors in F 4.
That C = BA is fully and uniquely determined by M C C This terminates the correctness proof. The claimed complexity derives immediately from Propositions 2 and 3 that are proved in the next subsections. 1 k! (k − i)! P ˜ m,n Proposition 2 Algorithm Eval computes M in M(mn)+ O(mn) ops. Proof. The series exp(X) mod X n+1 and the factorials 1, . . , n! are computed by recurrence relations in O(n) ops. The computation of S can be done in M(s ) for the size s P ˜ m,n of the corresponding diagonal of M .
Here s0 and t0 lie in the algebraically closed extension filed of F . A QSOB is a rational quadratic parametric surface with one simple base point. The total degree of a QSOB in s, t is 2 and its implicit degree is 22 − 1 = 3. In this paper, we will provide an efficient and rigorous method for implicitization and parametrization of a QSOB using the method of moving planes. A moving plane is a family of planes with parameter pair (s, t): A1 x + B1 y + C1 z + D1 = 0 A2 x + B2 y + C2 z + D2 = 0 be an arbitrary line.