C. Bradford Barber, D. Dobkin, and H. Huhdanpaa. The quickhull algorithm for convex hulls. ACM Trans. Math. Softw., 22(4):469-- 483, December 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....that can be controlled by the user (by default, we use 3 of the length of the data set s diagonal) Re triangulating the Simplified Mesh. After the simplification, the new set of points (which may contain up to #b points in addition to the vertices in the simplified set) is sent to qhull [1], which returns a (Delaunay) tetrahedralization. The problem now is that any face that contains (at least) one ghost vertex must be considered to be transparent; there can be a significant number of such faces. Eliminating Transparent Cells. At first, our code treated transparent faces ....

C. Bradford Barber, D. Dobkin, and H. Huhdanpaa. The quickhull algorithm for convex hulls. ACM Trans. Math. Softw., 22(4):469-- 483, December 1996.

....improve the results of protein structure prediction. Methods Delaunay tessellation is a canonical tessellation of space based on nearest neighbors (Aurenhammer 1991, Sugihara 1995) A Delaunay tessellation of a set of points is equivalent to a convex hull of the set in one higher dimension (Barber et al. 1993). For example, to determine the Delaunay tessellation of a set of points in 3D, we lift the points to a paraboloid and compute their convex hull in 4D. In general, a (d 1) dimensional convex hull of a set of points is a simplicial complex which is represented by its vertices, d dimensional facets ....

....atomic backbone structure) The first step in this process is extracting the set of 3D coordinates of the C a atoms from the PDB entry file. Delaunay tessellation of this set of points is then performed using the program qhull which implements the Quickhull algorithm developed by Barber et al. (Barber et al. 1993) and is distributed by the University of Minnesota Geometry Center. The Quickhull algorithm is a variation of the randomized, incremental algorithm of Clarkson and Shor. The program qhull produces the Delaunay tessellation by computing the convex hull of this set of points in four dimensions and ....

Barber C. B., Dobkin D. P., Huhdanpaa H. 1993. The quickhull algorithm for convex hull, Tech. Rep. GCG53, Geometry Center, University of Minnesota, Minneapolis, MN 55454.

....de nes a family of iso performance lines, and for a given family, the optimal methods are those that lie on the most northwest iso performance line. Thus, a classi er is optimal for some conditions if and only if it lies on the northwest boundary (i.e. above the line y = x) of the convex hull (Barber, Dobkin, Huhdanpaa, 1996) of the set of points in ROC space. 2 We discuss this in detail in Section 3. We call the convex hull of the set of points in ROC space the ROC convex hull (rocch) of the corresponding set of classi ers. Figure 3 shows four ROC curves with the ROC convex hull drawn as the border between the ....

.... 1; else I is a negative example Fcount = Fcount 1; end if end while add point ( Fcount N , Tcount P ) to end of R; 2. Find the convex hull of the set of points representing the predictive behavior of all classi ers of interest, for example by using the QuickHull algorithm (Barber et al. 1996). 3. For each set of class and cost distributions of interest, nd the slope (or range of slopes) of the corresponding iso performance lines. 4. For each set of class and cost distributions, the optimal classi er will be the point on the convex hull that intersects the iso performance line with ....

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Barber, C. B., Dobkin, D. P., & Huhdanpaa, H. (1996). The quickhull algorithm for convex hulls. ACM Transactions on Mathematical Software, 22 (4), 469-483. Available from ftp://geom.umn.edu/ pub/software/qhull.tar.Z.

....We present extensive experimental results to show that it performs well on a variety of input distributions. For input points from d , we compared the performance of our program with existing ones, including LEDA [Mehlhorn and N aher 1995] and Triangle [Shewchuk 1996] for d = 2, and Qhull [Barber et al. 1996] for d 2. After considering practical issues, several older algorithms were not selected for comparison. Among them, the best algorithm may be due to Bentley and Friedman [1978] variant of Prim s using k d trees) Their experiments suggest that it runs in O(n log n) time and that it performs ....

....Shewchuk s fast and robust two dimensional Delaunay triangulator Triangle [Shewchuk 1996] version 1.3) in combination with our own implementation of Kruskal s algorithm; we used the default divide and conquer Delaunay Triangulation algorithm based on exact arithmetic. e) Qhull: we used Qhull [Barber et al. 1996] (version 2.5) to compute the Delaunay triangulation and combined it with our own implementation of Kruskal s algorithm. The Delaunay triangulation is computed by lifting the input points to a paraboloid and computing the convex hull for the lifted points. f) Kruskal: our own implementation of ....

Barber, B. C., Dobkin, D. P., and Huhdanpaa, H. 1996. The quickhull algorithm for convex hulls. ACM Trans. on Mathematical Software 22, 4 (Dec.), 469-483.

....computing convex hulls and reviews perturbation schemes as well as previous work on exact modular arithmetic. There is a variety of convex hull algorithms and programs. In the past few years, there have been some implementations of general dimension convex hull algorithms, e.g. K. Clarkson s and [BDH93, BMS94] As far as algorithms are concerned, we restrict attention to Beneath Beyond. This is an incremental method that repeats the following step: given the convex hull of a subset of the points, it adds one point and updates the convex hull. If the point lies outside the given polytope, the ....

....with respect to the existing partial convex hull. This approach requires that the input points have distinct coordinates along the chosen axis, a condition which is simulated by our perturbation. More efficient approaches exist for enumerating the visible facets, e.g. CS89, Sei91, BDS 92, BDH93] The visibility test consists of deciding whether a facet defined by d 1 points is visible by another point. If the points defining every facet of the current hull are given in positive orientation, then it is possible to implement the visibility test as a single Orientation test. The ....

C.B. Barber, D.P. Dobkin, and H. Huhdanpaa. The Quickhull algorithm for convex hull. Tech. Report GCG53, Geometry Center, Univ. of Minnesota, July 1993. RR n2551 20 Ioannis Z. Emiris

....is dual to the Voronoi diagram 3.1 in the sense that there is a natural bijection between the two complexes which reverses the face inclusions. E#cient algorithms for computing Voronoi cells and Delaunay tessellations of point sets are publicly available; as a example we mention the qhull package [31]. Apart from degenerate cases, each Delaunay cell is a tetrahedron with for points of A at its corners. This procedure therefore defines 4 edges (sets of 4 mutually adjacent vertices) in a (protein) structures in a parameterfree way. The (2 )edges of a contact graph and 3 edges can of course be ....

C. B. Barber, D. P. Dobkin, and H. Huhdanpaa. The quickhull algorithm for convex hulls. ACM Trans. Math. Software, 22:469--483, 1996.

....costly than AEoating point arithmetic, it is a lot faster. We also implemented another method for computing the Minkowski sum. It consists in computing all vertices vA vB for all mn pairs of vertices v A 2 A; vB 2 B and then calculating their convex hull [AB89] with the Quick Hull algorithm [BDH93] This algorithm also uses AEoating point arithmetic. The performance were even poorer. Instrument N p N v Nm K N s K s WALL 54 837 ESRI 1 3 32 162 30.8 44 11.0 ESRI 2 3 32 171 30.3 46 10.8 HORN 7 HST 7 188 420 51.3 146 11.9 SVS ANTENNA 10 727 670 144.5 257 22.3 VOGO 10 296 770 ....

C. B. Barber, D. P. Dobkin, and H. Huhdanpaa. The Quickhull algorithm for convex hull. Technical Report GCG53, Geometry Center, Univ. of Minnesota, July 1993.

....the convolution method, it is a lot faster. In fact we also implemented the method that consists, for computing the Minkowski sum, in computing all vertices v A v B for all mn pairs of vertices v A 2 A; v B 2 B and then calculating their convex hull [AB89, Avn89] with the Quick Hull algorithm [BDH93] The performances were even poorer. Equipment N p N v Nm K N s K s INIT 54 837 ESRI 1 3 32 162 30.8 44 11.0 ESRI 2 3 32 171 30.3 46 10.8 HORN 7 HST 7 188 420 51.3 146 11.9 SVS ANTENNA 10 727 670 144.5 257 22.3 VOGO 10 296 770 91.5 161 17.7 TVGAVO ENS 4 70 348 62.3 107 19.3 Total ....

C. B. Barber, D. P. Dobkin, and H. Huhdanpaa. The Quickhull algorithm for convex hull. Technical Report GCG53, Geometry Center, Univ. of Minnesota, July 1993.

....even more important on data set ellipse2 which is representative of real applications. RR n# 3298 18 O. Devillers 5.5.3 Comparison with other software We have compared with some Delaunay softwares available on the WWW. # qhull by Bradford Barber and Hannu Huhdanpaa, duality with 3D convex hull [BDH93] (available at http: www.geom.umn.edu locate qhull) # div conquer by Jonathan Shewchuk, divide and conquer [She96] # sweep by Jonathan Shewchuk, plane sweep # incremental by Jonathan Shewchuk, incremental with M#cke et al. localization. These three codes supports exact arithmetic on double ....

C. B. Barber, D. P. Dobkin, and H. Huhdanpaa. The Quickhull algorithm for convex hull. Technical Report GCG53, Geometry Center, Univ. of Minnesota, July 1993.

....of flow) We therefore allow users to associate an annotation with a volume of the data space, rather than a single point in the space. To specify a volume, the user positions pegs that define the region s extreme vertices. The convex hull of the pegs is computed using the quickhull algorithm [2] and is rendered in either wireframe or transparent mode. Vertices can be added, deleted and moved, and the volume redrawn repeatedly. Figure 2 shows a volume which has been defined in this way. This implementation provides a simple way to draw volumes. However, since it uses a convex hull, ....

C. Barber, D. Dobkin, and H. Huhdanpaa. The quickhull algorithm for convex hull. Technical Report GCG53, Geometry Center, U. Minnesota, July 1993.

....origin. Each face is transformed to its dual point: if the plane equation of the face is ax by cz 1 = 0, the dual point is (a, b, c) The 3 D convex hull of all dual points (belonging to both A and B)is computed. We have used an excellent available implementation of the QuickHull algorithm [Barber93]. The plane equations of the resulting convex hull are now dualized back to vertices. These are the vertices of the intersection A # B. The correctness of this algorithm is simple to prove [Preparata85] The complexity of 3 D convex hull is O(n log n) so this step dominates the theoretical ....

Barber, C.B., Dobkin, D.P., Huhdanpaa, H.T., The Quickhull algorithm for convex hull, GCG53, The Geometry Center, Minneapolis, 1993 (ftp.geom.umn.edu/pub/software/qhull.tar.Z).

....without randomization. Each test is an average and range for ten trials. Quickhull partitions points to the first visible facet. The randomized version starts with a random initial simplex. In a previous report, we compared actual time and space for Quickhull and a randomized incremental program [Barber et al. 1993]. Those figures support our results here. First consider uniform random distributions projected to a sphere. Each coordinate is selected randomly from the interval [ 0.5, 0.5] The sphere is radius 0.5 centered at the origin. All points are extreme. In R 3 to R 7 , there is little difference ....

Barber, C. B., Dobkin, D. P., and Huhdanpaa, H. 1993. The Quickhull algorithm for convex hull. Technical Report GCG53, The Geometry Center.

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