http://ergawiki.di.uoa.gr/index.php?title=Implicitization&feed=atom&action=historyImplicitization - Revision history2024-03-29T06:16:41ZRevision history for this page on the wikiMediaWiki 1.15.1http://ergawiki.di.uoa.gr/index.php?title=Implicitization&diff=121&oldid=prevEmiris: /* Implicitization */2015-12-14T22:14:00Z<p><span class="autocomment">Implicitization</span></p>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>== Implicitization ==</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>== Implicitization <ins class="diffchange diffchange-inline">experiments </ins>==</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>These are (outdated) results obtained using Maple 11 exact solving methods.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>These are (outdated) results obtained using Maple 11 exact solving methods.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>| 1.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>| 1.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>| cardioid</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>| cardioid</div></td></tr>
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</table>Emirishttp://ergawiki.di.uoa.gr/index.php?title=Implicitization&diff=120&oldid=prevEmiris: /* Implicitization */2015-12-14T22:13:11Z<p><span class="autocomment">Implicitization</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Implicitization ==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Implicitization ==</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>These results <del class="diffchange diffchange-inline">have been </del>obtained using Maple 11 exact solving methods.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>These <ins class="diffchange diffchange-inline">are (outdated) </ins>results obtained using Maple 11 exact solving methods.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Curves'''</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Curves'''</div></td></tr>
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</table>Emirishttp://ergawiki.di.uoa.gr/index.php?title=Implicitization&diff=119&oldid=prevEmiris at 22:12, 14 December 20152015-12-14T22:12:14Z<p></p>
<a href="http://ergawiki.di.uoa.gr/index.php?title=Implicitization&diff=119&oldid=118">Show changes</a>Emirishttp://ergawiki.di.uoa.gr/index.php?title=Implicitization&diff=118&oldid=prevEmiris: /* References */2015-12-14T22:07:27Z<p><span class="autocomment">References</span></p>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>==<del class="diffchange diffchange-inline">References</del>==</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>==<ins class="diffchange diffchange-inline">Footnotes</ins>==</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">1 I.Z.Emiris, V.Fisikopoulos, C.Konaxis, L.Peñaranda. An oracle-based, output sensitive algorithm for projections of resultant polytopes. International Journal of Computational Geometry and Applications vol. 23, pp. 397-423, (Special issue) [World Scientific[http://www.worldscientific.com/doi/abs/10.1142/S0218195913600108]] 2013. </del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">2 I.Z.Emiris, V.Fisikopoulos, C.Konaxis, L.Peñaranda. ResPol: A software to compute a projection of the Newton polytope of the resultant, https://sourceforge.net/projects/respol/</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">3 I. Z. Emiris, T. Kalinka, Ch. Konaxis, T. Luu Ba, [http://ergawiki.di.uoa.gr/experiments/EBKK11.pdf Implicitization of curves and surfaces using predicted support, 2011. Theor. Comp. Sci.].</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">6 </del>This is the computation time of enumeration of regular triangulations algorithm using reverse search. We would like to thank Fumihiko TAKEUCHI for running the experiments and providing the results. Experiments were done on a Blade 100, 550Mhz, 2GB memory with SunOS 5.9.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">1 </ins>This is the computation time of enumeration of regular triangulations algorithm using reverse search. We would like to thank Fumihiko TAKEUCHI for running the experiments and providing the results. Experiments were done on a Blade 100, 550Mhz, 2GB memory with SunOS 5.9.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">7 </del>This is the computation time of points2alltriangs client of TOPCOM package. Experiments were done on a Intel(R) Pentium(R) 4 CPU 3.20GHz, 1.5GB memory with x86_64 Debian GNU/Linux.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">2 </ins>This is the computation time of points2alltriangs client of TOPCOM package. Experiments were done on a Intel(R) Pentium(R) 4 CPU 3.20GHz, 1.5GB memory with x86_64 Debian GNU/Linux.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">8 </del>This is the computation time of points2triangs client of TOPCOM package. Experiments were done on a Intel(R) Pentium(R) 4 CPU 3.20GHz, 1.5GB memory with x86_64 Debian GNU/Linux.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">3 </ins>This is the computation time of points2triangs client of TOPCOM package. Experiments were done on a Intel(R) Pentium(R) 4 CPU 3.20GHz, 1.5GB memory with x86_64 Debian GNU/Linux.</div></td></tr>
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</table>Emirishttp://ergawiki.di.uoa.gr/index.php?title=Implicitization&diff=117&oldid=prevEmiris: /* Implicitization experiments on curves and surfaces */2015-12-14T22:05:59Z<p><span class="autocomment">Implicitization experiments on curves and surfaces</span></p>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">== </del>Implicitization using predicted support has been implemented in ''Maple''. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Implicitization using predicted support has been implemented in ''Maple''.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In our experiments we use two support prediction methods: one applies only for curves and can be fully implemented in ''Maple'', another is general (implemented in respol). We present our [http://ergawiki.di.uoa.gr/experiments/simpl.mpl '''''Maple'' implementation'''] of the implicitization with predicted support and some [http://ergawiki.di.uoa.gr/experiments/examples.mw examples]. Here we apply numeric (SVD) as well as exact methods for linear solving in order to obtain coefficients of the implicit equation. </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In our experiments we use two support prediction methods: one applies only for curves and can be fully implemented in ''Maple'', another is general (implemented in respol). We present our [http://ergawiki.di.uoa.gr/experiments/simpl.mpl '''''Maple'' implementation'''] of the implicitization with predicted support and some [http://ergawiki.di.uoa.gr/experiments/examples.mw examples]. Here we apply numeric (SVD) as well as exact methods for linear solving in order to obtain coefficients of the implicit equation. </div></td></tr>
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</table>Emirishttp://ergawiki.di.uoa.gr/index.php?title=Implicitization&diff=116&oldid=prevEmiris: /* Implicitization experiments on curves and surfaces */2015-12-14T22:05:30Z<p><span class="autocomment">Implicitization experiments on curves and surfaces</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The former describes the geometric object as the set of values of parametric expressions over a set of values of the parameters.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The former describes the geometric object as the set of values of parametric expressions over a set of values of the parameters.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The latter describes the same object as the zero-set of a single polynomial. Example: the unit circle is given parametrically by x=cos(t), y=sin(t); its implicit equation is x^2+y^2=1. We shall call x,y the implicit variables.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The latter describes the same object as the zero-set of a single polynomial. Example: the unit circle is given parametrically by x=cos(t), y=sin(t); its implicit equation is x^2+y^2=1. We shall call x,y the implicit variables.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>For the case of curves, there are two parametric equations in one parameter. For the case of surfaces, there are 3 parametric expressions in two parameters. Each parametric expression can be transformed to a polynomial. For example, the expression x=f(t)/g(t), is transformed to the polynomial F(t)=x*g(t)-f(t). We consider the system of all such polynomials and define its sparse resultant. </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>For the case of curves, there are two parametric equations in one parameter. For the case of surfaces, there are 3 parametric expressions in two parameters. Each parametric expression can be transformed to a polynomial. For example, the expression x=f(t)/g(t), is transformed to the polynomial F(t)=x*g(t)-f(t). We consider the system of all such polynomials and define its sparse resultant. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>At this point, our first approach was to compute the Newton polytope of the sparse resultant, or resultant polytope, and then to recover from this polytope the Newton polytope of the implicit equation, or implicit polytope. This method was very inefficient as it can be seen in the examples below. We follow a more direct approach and compute a projection of the resultant's Newton polytope, without computing the resultant polytope itself. This projection gives information on the implicit equation's support and it is defined by the coefficients of the polynomials which involve implicit variables. Then, we compute the implicit equation by interpolation, which amounts to linear algebra.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>At this point, our first approach was to compute the Newton polytope of the sparse resultant, or resultant polytope, and then to recover from this polytope the Newton polytope of the implicit equation, or implicit polytope. This method was very inefficient as it can be seen in the examples below. We follow a more direct approach and compute a projection of the resultant's Newton polytope, without computing the resultant polytope itself. This projection gives information on the implicit equation's support and it is defined by the coefficients of the polynomials which involve implicit variables. Then, we compute the implicit equation by interpolation, which amounts to linear algebra.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The first section contains experimental results on the support prediction step which is the computation of the resultant polytope, or its projection, given the system of polynomials constructed by the parametric expressions. We developed an output-sensitive algorithm for computing resultant polytopes and their projections [[EFKP]http://www.worldscientific.com/doi/abs/10.1142/S0218195913600108].</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The first section contains experimental results on the support prediction step which is the computation of the resultant polytope, or its projection, given the system of polynomials constructed by the parametric expressions. We developed an output-sensitive algorithm for computing resultant polytopes and their projections [[EFKP]http://www.worldscientific.com/doi/abs/10.1142/S0218195913600108].</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Implicitization using predicted support has been implemented in ''Maple''. </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Implicitization using predicted support has been implemented in ''Maple''. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> ==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> ==</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>In our experiments we use two support prediction methods: one applies only for curves and can be fully implemented in ''Maple'', another is general (implemented in respol). We present our [http://ergawiki.di.uoa.gr/experiments/simpl.mpl '''''Maple'' implementation'''] of the implicitization with predicted support and some [http://ergawiki.di.uoa.gr/experiments/examples.mw examples]. Here we apply numeric (SVD) as well as exact methods for linear solving <del class="diffchange diffchange-inline">inorder </del>to obtain coefficients of the implicit equation. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>In our experiments we use two support prediction methods: one applies only for curves and can be fully implemented in ''Maple'', another is general (implemented in respol). We present our [http://ergawiki.di.uoa.gr/experiments/simpl.mpl '''''Maple'' implementation'''] of the implicitization with predicted support and some [http://ergawiki.di.uoa.gr/experiments/examples.mw examples]. Here we apply numeric (SVD) as well as exact methods for linear solving <ins class="diffchange diffchange-inline">in order </ins>to obtain coefficients of the implicit equation. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>For <del class="diffchange diffchange-inline">more </del>information see [[<del class="diffchange diffchange-inline">EKK</del>]http://ergawiki.di.uoa.gr/experiments/<del class="diffchange diffchange-inline">Implicitization-snc</del>.pdf] [http://www.<del class="diffchange diffchange-inline">example</del>.com <del class="diffchange diffchange-inline">link title]and [[EKK-TCS]http:</del>//<del class="diffchange diffchange-inline">ergawiki.di.uoa.gr</del>/<del class="diffchange diffchange-inline">experiments</del>/<del class="diffchange diffchange-inline">EBKKtcs11.pdf</del>].</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>For <ins class="diffchange diffchange-inline">background </ins>information see [[<ins class="diffchange diffchange-inline">EKKL-TCS</ins>]http://ergawiki.di.uoa.gr/experiments/<ins class="diffchange diffchange-inline">EBKKtcs11</ins>.pdf]<ins class="diffchange diffchange-inline">. For recent progress see </ins>[<ins class="diffchange diffchange-inline">[EKK2015]</ins>http://www.<ins class="diffchange diffchange-inline">sciencedirect</ins>.com/<ins class="diffchange diffchange-inline">science</ins>/<ins class="diffchange diffchange-inline">article</ins>/<ins class="diffchange diffchange-inline">pii</ins>/<ins class="diffchange diffchange-inline">S1524070315000260</ins>].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Support prediction ==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Support prediction ==</div></td></tr>
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</table>Emirishttp://ergawiki.di.uoa.gr/index.php?title=Implicitization&diff=115&oldid=prevEmiris: /* References */2015-12-14T22:03:07Z<p><span class="autocomment">References</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>2 I.Z.Emiris, V.Fisikopoulos, C.Konaxis, L.Peñaranda. ResPol: A software to compute a projection of the Newton polytope of the resultant, https://sourceforge.net/projects/respol/</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>2 I.Z.Emiris, V.Fisikopoulos, C.Konaxis, L.Peñaranda. ResPol: A software to compute a projection of the Newton polytope of the resultant, https://sourceforge.net/projects/respol/</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>3 <del class="diffchange diffchange-inline">I. Z. Emiris, T. Kalinka, Ch. Konaxis, [http://ergawiki.di.uoa.gr/experiments/Impl2011.pdf Implicitization of curves and surfaces using predicted support, 2011]. </del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>3 I. Z. Emiris, T. Kalinka, Ch. Konaxis, T. Luu Ba, [http://ergawiki.di.uoa.gr/experiments/EBKK11.pdf Implicitization of curves and surfaces using predicted support, 2011. Theor. Comp. Sci.].</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">4 </del>I. Z. Emiris, T. Kalinka, Ch. Konaxis, T. Luu Ba, [http://ergawiki.di.uoa.gr/experiments/EBKK11.pdf Implicitization of curves and surfaces using predicted support, 2011. Theor. Comp. Sci.].</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>6 This is the computation time of enumeration of regular triangulations algorithm using reverse search. We would like to thank Fumihiko TAKEUCHI for running the experiments and providing the results. Experiments were done on a Blade 100, 550Mhz, 2GB memory with SunOS 5.9.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>6 This is the computation time of enumeration of regular triangulations algorithm using reverse search. We would like to thank Fumihiko TAKEUCHI for running the experiments and providing the results. Experiments were done on a Blade 100, 550Mhz, 2GB memory with SunOS 5.9.</div></td></tr>
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</table>Emirishttp://ergawiki.di.uoa.gr/index.php?title=Implicitization&diff=114&oldid=prevEmiris: /* References */2015-12-14T22:02:38Z<p><span class="autocomment">References</span></p>
<table style="background-color: white; color:black;">
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>2 I.Z.Emiris, V.Fisikopoulos, C.Konaxis, L.Peñaranda. ResPol: A software to compute a projection of the Newton polytope of the resultant, https://sourceforge.net/projects/respol/</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>2 I.Z.Emiris, V.Fisikopoulos, C.Konaxis, L.Peñaranda. ResPol: A software to compute a projection of the Newton polytope of the resultant, https://sourceforge.net/projects/respol/</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">4 </del>I. Z. Emiris, T. Kalinka, Ch. Konaxis, [http://ergawiki.di.uoa.gr/experiments/Impl2011.pdf Implicitization of curves and surfaces using predicted support, 2011]. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">3 </ins>I. Z. Emiris, T. Kalinka, Ch. Konaxis, [http://ergawiki.di.uoa.gr/experiments/Impl2011.pdf Implicitization of curves and surfaces using predicted support, 2011]. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">5 </del>I. Z. Emiris, T. Kalinka, Ch. Konaxis, T. Luu Ba, [http://ergawiki.di.uoa.gr/experiments/EBKK11.pdf Implicitization of curves and surfaces using predicted support, 2011. <del class="diffchange diffchange-inline">Submitted to journal</del>]. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">4 </ins>I. Z. Emiris, T. Kalinka, Ch. Konaxis, T. Luu Ba, [http://ergawiki.di.uoa.gr/experiments/EBKK11.pdf Implicitization of curves and surfaces using predicted support, 2011. <ins class="diffchange diffchange-inline">Theor. Comp. Sci.</ins>].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>6 This is the computation time of enumeration of regular triangulations algorithm using reverse search. We would like to thank Fumihiko TAKEUCHI for running the experiments and providing the results. Experiments were done on a Blade 100, 550Mhz, 2GB memory with SunOS 5.9.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>6 This is the computation time of enumeration of regular triangulations algorithm using reverse search. We would like to thank Fumihiko TAKEUCHI for running the experiments and providing the results. Experiments were done on a Blade 100, 550Mhz, 2GB memory with SunOS 5.9.</div></td></tr>
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</table>Emirishttp://ergawiki.di.uoa.gr/index.php?title=Implicitization&diff=113&oldid=prevEmiris: /* Implicitization experiments on curves and surfaces */2015-12-14T22:01:43Z<p><span class="autocomment">Implicitization experiments on curves and surfaces</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The first section contains experimental results on the support prediction step which is the computation of the resultant polytope, or its projection, given the system of polynomials constructed by the parametric expressions. We developed an output-sensitive algorithm for computing resultant polytopes and their projections [[EFKP]http://www.worldscientific.com/doi/abs/10.1142/S0218195913600108].</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The first section contains experimental results on the support prediction step which is the computation of the resultant polytope, or its projection, given the system of polynomials constructed by the parametric expressions. We developed an output-sensitive algorithm for computing resultant polytopes and their projections [[EFKP]http://www.worldscientific.com/doi/abs/10.1142/S0218195913600108].</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Its implementation, called ResPol can be found in [<del class="diffchange diffchange-inline">2</del>]. Let us demonstrate how to prepare the input for ResPol, given the parametric equations of the surface: </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Its implementation, called ResPol can be found in [<ins class="diffchange diffchange-inline">[Sourceforge]https://sourceforge.net/projects/respol/</ins>]. Let us demonstrate how to prepare the input for ResPol, given the parametric equations of the surface: </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>x = 1/2*s^2 - 1/2*t^2 - 1/4*s^4 + 3/2*s^2*t^2 - 1/4*t^4,<br></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>x = 1/2*s^2 - 1/2*t^2 - 1/4*s^4 + 3/2*s^2*t^2 - 1/4*t^4,<br></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[3,2,2],[9,0,4],[0,12,0],[0,0,16],[4,4,0],[0,0,6],[8,4,0],[0,8,0],[3,0,4],[0,2,4],[3,2,2].</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[3,2,2],[9,0,4],[0,12,0],[0,0,16],[4,4,0],[0,0,6],[8,4,0],[0,8,0],[3,0,4],[0,2,4],[3,2,2].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><!--</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><ref name="respol">I.Z.Emiris, V.Fisikopoulos, C.Konaxis, L.Peñaranda. ResPol: A software to compute a projection of the Newton polytope of the resultant, https://sourceforge.net/projects/respol/</ref>. For experimental results of this method see [2].</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">--></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">Implicitization using predicted support has been implemented in ''Maple''. </del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">In our experiments we use two support prediction methods: one applies only for curves and can be fully implemented in ''Maple'', another is general (implemented in respol). We present our [http://ergawiki.di.uoa.gr/experiments/simpl.mpl ''Maple'' implementation] of the implicitization with predicted support and some [http://ergawiki.di.uoa.gr/experiments/examples.mw examples]. Here we apply numeric (SVD) as well as exact methods for linear solving in order to obtain coefficients of the implicit equation. </del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">For more information see [4],</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">== Implicitization using predicted support has been implemented in ''Maple''. </ins></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"><!--<ref name</del>=<del class="diffchange diffchange-inline">"EKK"> I</del>. <del class="diffchange diffchange-inline">Z</del>. <del class="diffchange diffchange-inline">Emiris, T</del>. <del class="diffchange diffchange-inline">Kalinka, Ch</del>. <del class="diffchange diffchange-inline">Konaxis, Implicitization </del>of <del class="diffchange diffchange-inline">curves and surfaces using </del>predicted support<del class="diffchange diffchange-inline">, 2011 </del>[http://ergawiki.di.uoa.gr/experiments/<del class="diffchange diffchange-inline">Implicitization-snc</del>.<del class="diffchange diffchange-inline">pdf</del>] <del class="diffchange diffchange-inline"></ref></del>.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"> </ins>=<ins class="diffchange diffchange-inline">=</ins></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">--></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">In our experiments we use two support prediction methods: one applies only for curves and can be fully implemented in ''Maple'', another is general (implemented in respol)</ins>. <ins class="diffchange diffchange-inline">We present our [http://ergawiki</ins>.<ins class="diffchange diffchange-inline">di</ins>.<ins class="diffchange diffchange-inline">uoa</ins>.<ins class="diffchange diffchange-inline">gr/experiments/simpl.mpl '''''Maple'' implementation'''] </ins>of <ins class="diffchange diffchange-inline">the implicitization with </ins>predicted support <ins class="diffchange diffchange-inline">and some </ins>[http://ergawiki.di.uoa.gr/experiments/<ins class="diffchange diffchange-inline">examples</ins>.<ins class="diffchange diffchange-inline">mw examples</ins>]<ins class="diffchange diffchange-inline">. Here we apply numeric (SVD) as well as exact methods for linear solving inorder to obtain coefficients of the implicit equation</ins>. </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[<del class="diffchange diffchange-inline">5</del>].</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"><!--<ref name="EKK"> I</del>. <del class="diffchange diffchange-inline">Z</del>. <del class="diffchange diffchange-inline">Emiris, T. Kalinka, Ch. Konaxis, </del>Implicitization <del class="diffchange diffchange-inline">of curves and surfaces using predicted support, 2011</del>. <del class="diffchange diffchange-inline">Submitted to journal</del>. [http://ergawiki.di.uoa.gr/experiments/EBKKtcs11.pdf] <del class="diffchange diffchange-inline"></ref></del>.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">For more information see </ins>[<ins class="diffchange diffchange-inline">[EKK</ins>]<ins class="diffchange diffchange-inline">http://ergawiki</ins>.<ins class="diffchange diffchange-inline">di</ins>.<ins class="diffchange diffchange-inline">uoa</ins>.<ins class="diffchange diffchange-inline">gr/experiments/</ins>Implicitization<ins class="diffchange diffchange-inline">-snc</ins>.<ins class="diffchange diffchange-inline">pdf] [http://www</ins>.<ins class="diffchange diffchange-inline">example.com link title]and </ins>[<ins class="diffchange diffchange-inline">[EKK-TCS]</ins>http://ergawiki.di.uoa.gr/experiments/EBKKtcs11.pdf].</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">--></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Support prediction ==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Support prediction ==</div></td></tr>
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</table>Emirishttp://ergawiki.di.uoa.gr/index.php?title=Implicitization&diff=112&oldid=prevEmiris: /* Implicitization experiments on curves and surfaces */2015-12-14T21:56:06Z<p><span class="autocomment">Implicitization experiments on curves and surfaces</span></p>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The first section contains experimental results on the support prediction step which is the computation of the resultant polytope, or its projection, given the system of polynomials constructed by the parametric expressions. We developed an output-sensitive algorithm for computing resultant polytopes and their projections [<del class="diffchange diffchange-inline">1</del>]<del class="diffchange diffchange-inline">.</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The first section contains experimental results on the support prediction step which is the computation of the resultant polytope, or its projection, given the system of polynomials constructed by the parametric expressions. We developed an output-sensitive algorithm for computing resultant polytopes and their projections [<ins class="diffchange diffchange-inline">[EFKP</ins>]http://<ins class="diffchange diffchange-inline">www</ins>.<ins class="diffchange diffchange-inline">worldscientific.com/doi</ins>/abs/<ins class="diffchange diffchange-inline">10</ins>.<ins class="diffchange diffchange-inline">1142</ins>/<ins class="diffchange diffchange-inline">S0218195913600108]</ins>.</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"><!-- </del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"><ref name="socg12"> I.Z.Emiris, V.Fisikopoulos, C.Konaxis, L.Peñaranda. An output-sensitive algorithm for computing projections of resultant polytopes. To appear in 28th Annual Symposium on Computational Geometry (SoCG 2012), Chapel Hill, NC, USA. (available from arXiv:1108.5985) </del>http://<del class="diffchange diffchange-inline">arxiv</del>.<del class="diffchange diffchange-inline">org</del>/abs/<del class="diffchange diffchange-inline">1108</del>.<del class="diffchange diffchange-inline">5985<</del>/<del class="diffchange diffchange-inline">ref></del>. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">--></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Its implementation, called ResPol can be found in [2]. Let us demonstrate how to prepare the input for ResPol, given the parametric equations of the surface: </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Its implementation, called ResPol can be found in [2]. Let us demonstrate how to prepare the input for ResPol, given the parametric equations of the surface: </div></td></tr>
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</table>Emiris