Surface algorithms using bounds on derivatives

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304 D. Filip et al. / <strong>Surface</strong> <strong>algorithms</strong> SURF2I(0, S1, SI(0, 0), SI(1, 0), SI(0, 1), SI(1, 1), 0, 0, 1, 1, K1, rl, $2, $2(0, 0), $2(1,0), $2(0, 1), $2(1, 1), 0, 0, 1, 1, K2, r2) END SURFACE INTERSECT Procedure SURF2I(d, S1, P1, P2, P3, P4, ull, vll, u12, v12, K1, rl, $2, Q1, Q2, Q3, Q4, u21, v21, u22, v22, K2, r2) /* Input parameter d: the current subdivision depth */ /* Input Parameters for S1 ($2 are similar): */ /. P1, P2, P3, P4 are coners points */ /* (ull, vll) is lower left corner of parameter domain •/ /* (u12, v12) is upper right comer of parameter domain */ /. K1 is bounding box swell factor * ,/ /* rl is the maximum subdivision depth necessary •/ (A0, A1) = min-max bounding box of (P1, P2, P3, P4} A0 = A0 - (K1, K1, K1) A1 = A1 + (K1, K1, K1) (B0, B1) = min-max bounding box of (Q1, Q2, Q3, Q4} B0 = B0 - (K2, K2, K2) B1 = B1 + (K2, K2, K2) if (A0, A1) and (B0, B1) do not intersect then return else if kl >/rl and k2 >~ r2 then /* S1 and $2 are flat * ,/ Output intersection curve of the linear approximations to S1 and $2 else if kl >~ rl and k2 < r2 then /* S1 flat and $2 not flat */ /* only generate subpatches for $2 */ u2 = (u21 + u22)/2 v2 = (v21 + v22)/2 Q5 = $2(u2, v21) Q6 = $2(u21, v2) Q7 = $2(u2, v2) Q8 = $2(u22, v2) Q9 = $2(u2, v22) SURF2I(d + 1, $1, P1, P2, P3, P4, ull, vll, u12, v12, K1, rl, $2, Q1, Q5, Q7, Q6, u21, v21, u2, v2, K2/4, r2) SURF2I(d + 1, S1, P1, P2, P3, P4, u/l, vll, u12, v12, K1, rl, $2, Q5, Q2, Q7, Q8, u2, v21, u22, v2, K2/4, r2) SURF2I(d + 1, S1, P1, P2, P3, P4, ull, vll, u12, v12, K1, rl, $2, Q6, Q7, Q3, Q9, u21, v2, u2, v22, K2/4, r2) SURF2I(d + 1, S1, P1, P2, P3, P4, ull, vll, u12, v12, K1, rl, $2, Q7, Q8, Q9, Q4, u2, v2, u22, v22, K2/4, r2) else if kl < rl and k2 >t r2 then ... /* $2 flat and S1 not flat ,,/ else ... /* both $1 and $2 are not flat * ,/ END SURF2I We remark here on some properities of this algorithm. First, it is independent of any surface representation and only depends on being able to evaluate the surface and knowing <strong>bounds</strong> on the second order partials of the surfaces. Second, the algorithm only needs to subdivide the parameter space and does not need to subdivide the surface definition, like in traditional B6zier subdivision. Third, the algorithm is adaptive in the sense that only the portions of the surfaces whose bounding boxes intersect need to be subdivided. Fourth, it is only necessary to find the
D. Filip et at. / <strong>Surface</strong> <strong>algorithms</strong> 305 min-max box of the comer points to generate the bounding box of the surface rather than finding the min-max of all the control points (if the surface is a polynomial). Also note that K1 and K2 are independent of the tolerance and so may be computed once and then stored with the surface definition. From empirical evidence we find that even if the surface is a polynomial that this algorithm can run an order of magnitude or more faster than the traditional algorithm based on subdivision and flatness testing of the control points. One reason is that all the evaluations can be done <strong>using</strong> Homer's rule on the power basis form of a polynomial. As in almost every surface intersector, the output of the line segments must be sorted if contiguous intersection curves are desired. These intersection curves may then be refined by a Newton like method to generate more accurate curves [Houghton et al. '85]. 4. Calculating the <strong>bounds</strong> of the partials of order two In this section we discuss how to obtain the numbers M 1 = l[ a2I/on2 II, ME = II o2f/auav II, and M 3 = II a21/av2 II. Fortunately these numbers can be calculated in a straightforward manner for many of the common surface types used in CAGD. Also, it is not important that the calculation of these numbers be done very quickly since they only need be done once for a given surface as a preprocessing step and then stored with the surface definition. Because many of the formulas for <strong>bounds</strong> on the surfaces can be broken down into a sequence of curve problems, we first discuss finding <strong>bounds</strong> for curves. The <strong>bounds</strong> for curves are also needed if the <strong>algorithms</strong> described in Section 3 are applied to the curve case. 4.1. Bounds on curves To find the <strong>bounds</strong> of the partials of order two for some of the surfaces discussed in the next section, we require knowledge of the <strong>bounds</strong> of certain curves and their derivatives of order one and two. The curves involved are often polynomials or rational polynomials. In this section we only discuss finding these <strong>bounds</strong> for curves which are functions from [0, 1] to the reals since the surface case usually deals with <strong>bounds</strong> for each component function seperately and then later combines the <strong>bounds</strong>. One simple method for finding the desired <strong>bounds</strong> on a polynomial curve c(t) of degree n is first to write it in its B~zier form (Boehm et al. '84] where i=n c( t ) = ~ biB~, i=0 i 1 - t)n-lt i u~[0,1], b~R, is an element of the Bemstein basis (a basis for the vector space of polynomials of degree n). Because each basis element is positive over [0, 1] and the basis forms a partition of unity (i.e. sum to one), it is easy to show sup Ic(t) l~< max (Ibil}. tE[O,1] O
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