Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 2 pot

... ¯a 2 = p 2 ,q 2 , [l 2 ,r 2 ], we first have to find all x-values satisfying the equality p 1 (x) q 1 (x) = p 2 (x) q 2 (x) . We extract all the roots of the univariate polynomial p 1 q 2 − p 2 q 1 ∈ ... segment traits 2 Arr non caching segment traits 2 Arr segment traits 2 Arr polyline traits 2 Arr conic traits 2 Arr rational arc traits 2 ArrBasicTraits 2 ArrX...

Ngày tải lên: 10/08/2014, 02:20

... bx 3 +6cx 2 − 4 dx+ e with a>0, we define the invariants A = ae − 4bd +3c 2 ,B= ace +2bcd − ad 2 − eb 2 − c 3 , ∆ 1 = A 3 − 27 B 2 ,∆ 2 = b 2 − ac, ∆ 3 = c 2 − bd, ∆ 4 = d 2 − ce, W 1 = ad − bc, W 2 = ... and {2, 2} means two double real roots. condition real roots ∆ 1 > 0 ∧T>0 ∧ ∆ 2 > 0 {1, 1, 1, 1} ∆ 1 > 0 ∧(T ≤ 0 ∨ ∆ 2 ≤ 0) {} ∆ 1 < 0 {1, 1} ∆ 1 =...

Ngày tải lên: 10/08/2014, 02:20

... a hyperboloid x 2 + y 2 −z 2 = a in the vicinity of the origin, for a ≈ 0. For a>0, we have a hyperboloid of one sheet, and for a<0, we have a hyperboloid of two sheets, see Fig. 5 .22 . The transition ... automatically form part of the polar variety. Figs. 5 .20 –5 .21 show an example of a sphere and a line that are defined by the equation (x 2 + y 2 + z 2 − 1)  (x...

Ngày tải lên: 10/08/2014, 02:20

... computing the x-values for which (tx + v) 2 − 4s(rx 2 + ux + w)=0, which gives the following quadratic equation: (t 2 − 4rs)x 2 +2( tv −2su)+(v 2 − 4sw)=0. (1 .2) Let x 1 ,x 2 be the real-valued roots ... York Monique Teillaud Jean-Daniel Boissonnat INRIA Sophia-Antipolis 20 04 route des Lucioles B.P. 93 069 02 Sophia-Antipolis, France E-mail: Jean-Daniel .Boissonnat@ so...

Ngày tải lên: 10/08/2014, 02:20

... polynomial 4083 324 84x 4 + 51939673y 4 − 66477 920 4x 3 y − 24 101506y 3 x +564185 724 x 2 y 2 − 25 0019406x 3 + 17767644y 3 +22 1 120 964x 2 y − 123 026 916y 2 x + 16691919x 2 + 4764152y 2 +14441004xy + 104 829 00x + 23 05740y ... look for a relocation (x  2 ,y  2 ) of the center of C 2 such that |  (x 1 − x  2 ) 2 +(y 1 − y  2 ) 2 − (r 1 + r 2 )|≥ε. T...

Ngày tải lên: 10/08/2014, 02:20

... −λ i p i ) 2 − λ 2 i 4 − λ 2 i p 2 i + λ i p 2 i − µ i ≤ (x 2 + λ j 2 ) 2 +(x −λ j p j ) 2 − λ 2 j 4 − λ 2 j p 2 j + λ j p 2 j − µ j ⇐⇒ (X −C i ) 2 − r 2 i ≤ (X −C j ) 2 − r 2 j ⇐⇒ Σ i (X) ≤ ... have for all j =1, ,n, λ i (x − p i ) 2 − µ i ≤ λ j (x − p j ) 2 − µ j ⇐⇒ λ i x 2 − 2 i p i · x + λ i p 2 i − µ i ≤ λ j x 2 − 2 j p j · x + λ j p 2...

Ngày tải lên: 10/08/2014, 02:20

... =  r 2 − 4l 2 ≤  r 2 − 4εr +4ε 2 = r − 2 . We then deduce c −c  ≤ 2 (r −ε) r −h − 2 ≤ 2 r r − √ r 2 − 4l 2 − 2 . We then get c −c   r ≤ 2 (r + √ r 2 − 4l 2 − 2 ) (r 2 ) 2 − (r 2 − 4l 2 ) ≤ εr l 2 − ... S  Noting h =  r 2 − 1 4 x −y 2 = √ r 2 − 4l 2 the distance from c to line xy, we have d(c  ,S) ≤ min(c  − x, c  − y) ≤  (c −c  ...

Ngày tải lên: 10/08/2014, 02:20

... Wein- garten endomorphism. 3 Algebraic Issues in Computational Geometry 147 We will not consider examples such as P 1 = x 2 +y 2 −1,P 2 = x 2 +y 2 +z 2 −1, where (P 1 ,P 2 )=(x 2 +y 2 −1,z 2 )andI(C)=(x 2 +y 2 −1,z), ... G(x, y) is not necessarily a square-free polynomial. Consider for instance the case P 1 = x 2 + y 2 −1,P 2 = x 2 + y 2 + z 2 2, wher...

Ngày tải lên: 10/08/2014, 02:20

... closest point on the medial axis, see Fig. 5. 12. (See p. 109 in Sect. 2. 7 for the definition of the medial axis; see also Sect. 6 .2. 2, pp. 24 4 24 7, for a more extensive discussion about the medial ... distance at most O(ε 2 lfs(x)). The theorem was first formulated for ε-samples (with the same bound of 0.1) by Amenta and Bern [22 ], see also Theorem 6 in Chap. 6 (p. 24 8) for...

Ngày tải lên: 10/08/2014, 02:20

... Exercise 1. For curve reconstruction, see [24 , 125 , 127 ]. • Exercise 2. The separation of critical points of the distance functions to an ε-sample a smooth surface is studied in [ 122 ]. • Exercise ... in nature to Boissonnat s early algorithm. 24 8 F. Cazals, J. Giesen Fig. 6.14. For a non-smooth curve, some Voronoi centers may not converge to the medial axis ε-sample. Amenta and B...

Ngày tải lên: 10/08/2014, 02:20