Curve Construction via Local Fitting
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1 Curve Construction via Local Fitting Suppose we are given points and tangents Q k, and T k (k = 0,..., n), and a fitting tolerance ε. We want to fit this data with the minimum (in some sense) number of rational quadratic Bezier segments. Using double interior knots, the result will be a G 1 -continuous quadratic NURBS curve. ME525x NURBS Curve and Surface Modeling Page 585
2 The algorithm uses repeated binary subdivision at each step to determine the maximum number of points that can be fitted with one conic arc to within ε. For example, assume (Q k, T k ) k = 0,..., 7); and suppose the data can be fit (according to the method given below) with two arcs, the first from Q 0 to Q 5 and the second from Q 5 to Q 7. The subdivision determines this as follows: ME525x NURBS Curve and Surface Modeling Page 586
3 = 3 ; the algorithm determines that Q 0 to 2 Q 3 can be fitted = 5 ; it determines that Q 0 to Q 2 5 can be fitted = 6 ; it determines Q 0 to Q 2 6 cannot be fitted. ME525x NURBS Curve and Surface Modeling Page 587
4 The points Q 0 to Q 5 would then be fitted with a conic arc, and afterward, the subdivision would continue considering the points Q 5 to Q 7. Now consider the fitting method. The general conic arc is given by: C ( u) = ( 1 u) 2 w b Q b + 2u ( 1 u) w t Q t + u 2 w e Q e ( 1 u) 2 w b + 2u ( 1 u) w t + u 2 w e where the subscripts b, t, and e, denote beginning, interior, and ending control points and weights. ME525x NURBS Curve and Surface Modeling Page 588
5 The algorithm starts by setting Q b = Q 0, T b = T 0, and Q e = Q n, T e = T n. We assume that T b and T e are not parallel. The algorithm is as follows: 1. Compute the middle control point Q t. 2. Check if all points Q b+1,..., Q e-1 are in the triangle Q b Q t Q e ; if not, then subdivide to get a new Q e, and go back to step 1. ME525x NURBS Curve and Surface Modeling Page 589
6 3. For each Q i, i = b + 1,..., e - 1, there exists a unique conic passing through Q i and determined by Q b, Q t, Q e and w t i, with w b = w e = 1. (how to get u i and w t i ) ME525x NURBS Curve and Surface Modeling Page 590
7 Q t Q S Q 1 Q b V Q e Recall that if w t i = 0, we get a line between Q b and Q e ME525x NURBS Curve and Surface Modeling Page 591
8 Thus, V ( u) = ( 1 u) 2 Q b + u 2 Q e ( 1 u) 2 + u 2 It follows that the ratio of the distances Q b V and VQ e is u 2 : (1 - u 2 ). Thus we can solve for parameter u by u = a a ME525x NURBS Curve and Surface Modeling Page 592
9 where, a = Q b V VQ e Now find the weight for the interior control point. Let Q 1 be the intersection of the w t = 1 conic and the line segment Q t V, i.e., Q 1 = C(u; w t = 1) with u computed as above. ME525x NURBS Curve and Surface Modeling Page 593
10 The points Q 1 and Q can be expressed in terms of V and Q t as follows: Q1 i = ( 1 s) V + sq t Q i = ( 1 r) V + rq t where, ME525x NURBS Curve and Surface Modeling Page 594
11 s 2u ( 1 u) = ( 1 u) 2 + 2u ( 1 u) + u 2 = 2u ( 1 u) 2u ( 1 u) w t r = ( 1 u) 2 + 2u ( 1 u) w t + u 2 are affine coordinates along the line VQ t with 0 at V and 1 at Q t. Using these coordinates we can show that: ME525x NURBS Curve and Surface Modeling Page 595
12 1 s w t = : s 1 r r Q t Q Q 1 : V Q t Q QV Since the point Q is given, r is already determined from the ratio of VQ and VQ t. ME525x NURBS Curve and Surface Modeling Page 596
13 Thus the algorithm to compute the conic segment passing through a specified point Q requires the following steps: compute V = Q b Q e Q t Q set a = Q b V VQ e set u = a / (1 + a) set s = 2u(1 - u) ME525x NURBS Curve and Surface Modeling Page 597
14 set r = VQ / VQ t 1 s 1 r set w t = : s r Note that this formulation is independent of the dimensionality of the data. Also, it can be used to check whether or not the point Q is within the convex hull. If it is outside the convex hull, then either u or r is outside the interval [0, 1]. Thus steps 2 and 3 can be combined. ME525x NURBS Curve and Surface Modeling Page 598
15 The next step in the overall algorithm is to determine whether the conics defined by each of the Q i, i = b + 1,..., e - 1, lie within a fitting tolerance ε. We do this by checking the shoulder points. Recall that for a conic, the shoulder point is defined as, S = ( 1 s) M + sq t ME525x NURBS Curve and Surface Modeling Page 599
16 where, s = w t w t Thus, we could compute d = S max S min < ε ME525x NURBS Curve and Surface Modeling Page 600
17 or, to avoid the distance calculation, s max s min ε < - e where, e = Q t M is a scaling factor. To avoid unnecessary computation, the algorithm should be implemented to check this scatter condition as each Q i is considered, rather than as a separate step, after fitting. ME525x NURBS Curve and Surface Modeling Page 601
18 ME525x NURBS Curve and Surface Modeling Page 602
19 4. Given that a subset of data meets the convex hull and scatter conditions, the goal is to choose a value for w t which somehow best fits the data. A good approach is to use the center of gravity of the individual shoulder points in the segment, thus, e 1 1 s f = b e s i i = b + 1 ME525x NURBS Curve and Surface Modeling Page 603
20 from which the weight of the best fit arc is: w t = s f s f ME525x NURBS Curve and Surface Modeling Page 604
21 Scattered data ME525x NURBS Curve and Surface Modeling Page 605
22 A segmentation example ε = 0.1. ME525x NURBS Curve and Surface Modeling Page 606
23 A segmentation example ε = ME525x NURBS Curve and Surface Modeling Page 607
24 A segmentation example ε = ME525x NURBS Curve and Surface Modeling Page 608
25 Piecewise rational curve fit ME525x NURBS Curve and Surface Modeling Page 609
Rational Bezier Surface
Rational Bezier Surface The perspective projection of a 4-dimensional polynomial Bezier surface, S w n ( u, v) B i n i 0 m j 0, u ( ) B j m, v ( ) P w ij ME525x NURBS Curve and Surface Modeling Page 97
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