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<div class="title">ImplicitPolynomial3Shape.ih</div>  </div>
</div><!--header-->
<div class="contents">
<div class="fragment"><div class="line"><a name="l00001"></a><span class="lineno">    1</span>&#160;/**</div>
<div class="line"><a name="l00002"></a><span class="lineno">    2</span>&#160; *  This program is free software: you can redistribute it and/or modify</div>
<div class="line"><a name="l00003"></a><span class="lineno">    3</span>&#160; *  it under the terms of the GNU Lesser General Public License as</div>
<div class="line"><a name="l00004"></a><span class="lineno">    4</span>&#160; *  published by the Free Software Foundation, either version 3 of the</div>
<div class="line"><a name="l00005"></a><span class="lineno">    5</span>&#160; *  License, or  (at your option) any later version.</div>
<div class="line"><a name="l00006"></a><span class="lineno">    6</span>&#160; *</div>
<div class="line"><a name="l00007"></a><span class="lineno">    7</span>&#160; *  This program is distributed in the hope that it will be useful,</div>
<div class="line"><a name="l00008"></a><span class="lineno">    8</span>&#160; *  but WITHOUT ANY WARRANTY; without even the implied warranty of</div>
<div class="line"><a name="l00009"></a><span class="lineno">    9</span>&#160; *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the</div>
<div class="line"><a name="l00010"></a><span class="lineno">   10</span>&#160; *  GNU General Public License for more details.</div>
<div class="line"><a name="l00011"></a><span class="lineno">   11</span>&#160; *</div>
<div class="line"><a name="l00012"></a><span class="lineno">   12</span>&#160; *  You should have received a copy of the GNU General Public License</div>
<div class="line"><a name="l00013"></a><span class="lineno">   13</span>&#160; *  along with this program.  If not, see &lt;http://www.gnu.org/licenses/&gt;.</div>
<div class="line"><a name="l00014"></a><span class="lineno">   14</span>&#160; *</div>
<div class="line"><a name="l00015"></a><span class="lineno">   15</span>&#160; **/</div>
<div class="line"><a name="l00016"></a><span class="lineno">   16</span>&#160; </div>
<div class="line"><a name="l00017"></a><span class="lineno">   17</span>&#160;/**</div>
<div class="line"><a name="l00018"></a><span class="lineno">   18</span>&#160; * @file ImplicitPolynomial3Shape.ih</div>
<div class="line"><a name="l00019"></a><span class="lineno">   19</span>&#160; * @author Jacques-Olivier Lachaud (\c jacques-olivier.lachaud@univ-savoie.fr )</div>
<div class="line"><a name="l00020"></a><span class="lineno">   20</span>&#160; * Laboratory of Mathematics (CNRS, UMR 5807), University of Savoie, France</div>
<div class="line"><a name="l00021"></a><span class="lineno">   21</span>&#160; *</div>
<div class="line"><a name="l00022"></a><span class="lineno">   22</span>&#160; * @date 2012/02/14</div>
<div class="line"><a name="l00023"></a><span class="lineno">   23</span>&#160; *</div>
<div class="line"><a name="l00024"></a><span class="lineno">   24</span>&#160; * Implementation of inline methods defined in ImplicitPolynomial3Shape.h</div>
<div class="line"><a name="l00025"></a><span class="lineno">   25</span>&#160; *</div>
<div class="line"><a name="l00026"></a><span class="lineno">   26</span>&#160; * This file is part of the DGtal library.</div>
<div class="line"><a name="l00027"></a><span class="lineno">   27</span>&#160; */</div>
<div class="line"><a name="l00028"></a><span class="lineno">   28</span>&#160; </div>
<div class="line"><a name="l00029"></a><span class="lineno">   29</span>&#160; </div>
<div class="line"><a name="l00030"></a><span class="lineno">   30</span>&#160;//////////////////////////////////////////////////////////////////////////////</div>
<div class="line"><a name="l00031"></a><span class="lineno">   31</span>&#160;#include &lt;cstdlib&gt;</div>
<div class="line"><a name="l00032"></a><span class="lineno">   32</span>&#160;//////////////////////////////////////////////////////////////////////////////</div>
<div class="line"><a name="l00033"></a><span class="lineno">   33</span>&#160; </div>
<div class="line"><a name="l00034"></a><span class="lineno">   34</span>&#160;///////////////////////////////////////////////////////////////////////////////</div>
<div class="line"><a name="l00035"></a><span class="lineno">   35</span>&#160;// IMPLEMENTATION of inline methods.</div>
<div class="line"><a name="l00036"></a><span class="lineno">   36</span>&#160;///////////////////////////////////////////////////////////////////////////////</div>
<div class="line"><a name="l00037"></a><span class="lineno">   37</span>&#160; </div>
<div class="line"><a name="l00038"></a><span class="lineno">   38</span>&#160;///////////////////////////////////////////////////////////////////////////////</div>
<div class="line"><a name="l00039"></a><span class="lineno">   39</span>&#160;// ----------------------- Standard services ------------------------------</div>
<div class="line"><a name="l00040"></a><span class="lineno">   40</span>&#160; </div>
<div class="line"><a name="l00041"></a><span class="lineno">   41</span>&#160;//-----------------------------------------------------------------------------</div>
<div class="line"><a name="l00042"></a><span class="lineno">   42</span>&#160;template &lt;typename TSpace&gt;</div>
<div class="line"><a name="l00043"></a><span class="lineno">   43</span>&#160;inline</div>
<div class="line"><a name="l00044"></a><span class="lineno">   44</span>&#160;DGtal::ImplicitPolynomial3Shape&lt;TSpace&gt;::~ImplicitPolynomial3Shape()</div>
<div class="line"><a name="l00045"></a><span class="lineno">   45</span>&#160;{</div>
<div class="line"><a name="l00046"></a><span class="lineno">   46</span>&#160;}</div>
<div class="line"><a name="l00047"></a><span class="lineno">   47</span>&#160;//-----------------------------------------------------------------------------</div>
<div class="line"><a name="l00048"></a><span class="lineno">   48</span>&#160;template &lt;typename TSpace&gt;</div>
<div class="line"><a name="l00049"></a><span class="lineno">   49</span>&#160;inline</div>
<div class="line"><a name="l00050"></a><span class="lineno">   50</span>&#160;DGtal::ImplicitPolynomial3Shape&lt;TSpace&gt;::</div>
<div class="line"><a name="l00051"></a><span class="lineno">   51</span>&#160;ImplicitPolynomial3Shape( const Polynomial3 &amp; poly )</div>
<div class="line"><a name="l00052"></a><span class="lineno">   52</span>&#160;{</div>
<div class="line"><a name="l00053"></a><span class="lineno">   53</span>&#160;  init( poly );</div>
<div class="line"><a name="l00054"></a><span class="lineno">   54</span>&#160;}</div>
<div class="line"><a name="l00055"></a><span class="lineno">   55</span>&#160;//-----------------------------------------------------------------------------</div>
<div class="line"><a name="l00056"></a><span class="lineno">   56</span>&#160;template &lt;typename TSpace&gt;</div>
<div class="line"><a name="l00057"></a><span class="lineno">   57</span>&#160;inline</div>
<div class="line"><a name="l00058"></a><span class="lineno">   58</span>&#160;DGtal::ImplicitPolynomial3Shape&lt;TSpace&gt; &amp;</div>
<div class="line"><a name="l00059"></a><span class="lineno">   59</span>&#160;DGtal::ImplicitPolynomial3Shape&lt;TSpace&gt;::</div>
<div class="line"><a name="l00060"></a><span class="lineno">   60</span>&#160;operator=( const ImplicitPolynomial3Shape &amp; other )</div>
<div class="line"><a name="l00061"></a><span class="lineno">   61</span>&#160;{</div>
<div class="line"><a name="l00062"></a><span class="lineno">   62</span>&#160;  if ( this != &amp;other )</div>
<div class="line"><a name="l00063"></a><span class="lineno">   63</span>&#160;  {</div>
<div class="line"><a name="l00064"></a><span class="lineno">   64</span>&#160;    myPolynomial = other.myPolynomial;</div>
<div class="line"><a name="l00065"></a><span class="lineno">   65</span>&#160; </div>
<div class="line"><a name="l00066"></a><span class="lineno">   66</span>&#160;    myFx= other.myFx;</div>
<div class="line"><a name="l00067"></a><span class="lineno">   67</span>&#160;    myFy= other.myFy;</div>
<div class="line"><a name="l00068"></a><span class="lineno">   68</span>&#160;    myFz= other.myFz;</div>
<div class="line"><a name="l00069"></a><span class="lineno">   69</span>&#160; </div>
<div class="line"><a name="l00070"></a><span class="lineno">   70</span>&#160;    myFxx= other.myFxx;</div>
<div class="line"><a name="l00071"></a><span class="lineno">   71</span>&#160;    myFxy= other.myFxy;</div>
<div class="line"><a name="l00072"></a><span class="lineno">   72</span>&#160;    myFxz= other.myFxz;</div>
<div class="line"><a name="l00073"></a><span class="lineno">   73</span>&#160; </div>
<div class="line"><a name="l00074"></a><span class="lineno">   74</span>&#160;    myFyx= other.myFyx;</div>
<div class="line"><a name="l00075"></a><span class="lineno">   75</span>&#160;    myFyy= other.myFyy;</div>
<div class="line"><a name="l00076"></a><span class="lineno">   76</span>&#160;    myFyz= other.myFyz;</div>
<div class="line"><a name="l00077"></a><span class="lineno">   77</span>&#160; </div>
<div class="line"><a name="l00078"></a><span class="lineno">   78</span>&#160;    myFzx= other.myFzx;</div>
<div class="line"><a name="l00079"></a><span class="lineno">   79</span>&#160;    myFzy= other.myFzy;</div>
<div class="line"><a name="l00080"></a><span class="lineno">   80</span>&#160;    myFzz= other.myFzz;</div>
<div class="line"><a name="l00081"></a><span class="lineno">   81</span>&#160; </div>
<div class="line"><a name="l00082"></a><span class="lineno">   82</span>&#160;    myUpPolynome = other.myUpPolynome;  </div>
<div class="line"><a name="l00083"></a><span class="lineno">   83</span>&#160;    myLowPolynome = other.myLowPolynome;</div>
<div class="line"><a name="l00084"></a><span class="lineno">   84</span>&#160;  }</div>
<div class="line"><a name="l00085"></a><span class="lineno">   85</span>&#160;  return *this;</div>
<div class="line"><a name="l00086"></a><span class="lineno">   86</span>&#160;}</div>
<div class="line"><a name="l00087"></a><span class="lineno">   87</span>&#160;//-----------------------------------------------------------------------------</div>
<div class="line"><a name="l00088"></a><span class="lineno">   88</span>&#160;template &lt;typename TSpace&gt;</div>
<div class="line"><a name="l00089"></a><span class="lineno">   89</span>&#160;inline</div>
<div class="line"><a name="l00090"></a><span class="lineno">   90</span>&#160;void</div>
<div class="line"><a name="l00091"></a><span class="lineno">   91</span>&#160;DGtal::ImplicitPolynomial3Shape&lt;TSpace&gt;::</div>
<div class="line"><a name="l00092"></a><span class="lineno">   92</span>&#160;init( const Polynomial3 &amp; poly )</div>
<div class="line"><a name="l00093"></a><span class="lineno">   93</span>&#160;{</div>
<div class="line"><a name="l00094"></a><span class="lineno">   94</span>&#160;  myPolynomial = poly;</div>
<div class="line"><a name="l00095"></a><span class="lineno">   95</span>&#160; </div>
<div class="line"><a name="l00096"></a><span class="lineno">   96</span>&#160;  myFx= derivative&lt;0&gt;( poly );</div>
<div class="line"><a name="l00097"></a><span class="lineno">   97</span>&#160;  myFy= derivative&lt;1&gt;( poly );</div>
<div class="line"><a name="l00098"></a><span class="lineno">   98</span>&#160;  myFz= derivative&lt;2&gt;( poly );</div>
<div class="line"><a name="l00099"></a><span class="lineno">   99</span>&#160; </div>
<div class="line"><a name="l00100"></a><span class="lineno">  100</span>&#160;  myFxx= derivative&lt;0&gt;( myFx );</div>
<div class="line"><a name="l00101"></a><span class="lineno">  101</span>&#160;  myFxy= derivative&lt;1&gt;( myFx );</div>
<div class="line"><a name="l00102"></a><span class="lineno">  102</span>&#160;  myFxz= derivative&lt;2&gt;( myFx);</div>
<div class="line"><a name="l00103"></a><span class="lineno">  103</span>&#160; </div>
<div class="line"><a name="l00104"></a><span class="lineno">  104</span>&#160;  myFyx= derivative&lt;0&gt;( myFy );</div>
<div class="line"><a name="l00105"></a><span class="lineno">  105</span>&#160;  myFyy= derivative&lt;1&gt;( myFy );</div>
<div class="line"><a name="l00106"></a><span class="lineno">  106</span>&#160;  myFyz= derivative&lt;2&gt;( myFy );</div>
<div class="line"><a name="l00107"></a><span class="lineno">  107</span>&#160; </div>
<div class="line"><a name="l00108"></a><span class="lineno">  108</span>&#160;  myFzx= derivative&lt;0&gt;( myFz );</div>
<div class="line"><a name="l00109"></a><span class="lineno">  109</span>&#160;  myFzy= derivative&lt;1&gt;( myFz );</div>
<div class="line"><a name="l00110"></a><span class="lineno">  110</span>&#160;  myFzz= derivative&lt;2&gt;( myFz );</div>
<div class="line"><a name="l00111"></a><span class="lineno">  111</span>&#160; </div>
<div class="line"><a name="l00112"></a><span class="lineno">  112</span>&#160;  // These two polynomials are used for mean curvature estimation.</div>
<div class="line"><a name="l00113"></a><span class="lineno">  113</span>&#160;  myUpPolynome = myFx*(myFx*myFxx+myFy*myFyx+myFz*myFzx)+</div>
<div class="line"><a name="l00114"></a><span class="lineno">  114</span>&#160;                                myFy*(myFx*myFxy+myFy*myFyy+myFz*myFzy)+</div>
<div class="line"><a name="l00115"></a><span class="lineno">  115</span>&#160;                                myFz*(myFx*myFxz+myFy*myFyz+myFz*myFzz)-</div>
<div class="line"><a name="l00116"></a><span class="lineno">  116</span>&#160;                                ( myFx*myFx +myFy*myFy+myFz*myFz )*(myFxx+myFyy+myFzz);</div>
<div class="line"><a name="l00117"></a><span class="lineno">  117</span>&#160; </div>
<div class="line"><a name="l00118"></a><span class="lineno">  118</span>&#160;  myLowPolynome = myFx*myFx +myFy*myFy+myFz*myFz;</div>
<div class="line"><a name="l00119"></a><span class="lineno">  119</span>&#160;}</div>
<div class="line"><a name="l00120"></a><span class="lineno">  120</span>&#160;//-----------------------------------------------------------------------------</div>
<div class="line"><a name="l00121"></a><span class="lineno">  121</span>&#160;template &lt;typename TSpace&gt;</div>
<div class="line"><a name="l00122"></a><span class="lineno">  122</span>&#160;inline</div>
<div class="line"><a name="l00123"></a><span class="lineno">  123</span>&#160;double</div>
<div class="line"><a name="l00124"></a><span class="lineno">  124</span>&#160;DGtal::ImplicitPolynomial3Shape&lt;TSpace&gt;::</div>
<div class="line"><a name="l00125"></a><span class="lineno">  125</span>&#160;operator()(const RealPoint &amp;aPoint) const</div>
<div class="line"><a name="l00126"></a><span class="lineno">  126</span>&#160;{</div>
<div class="line"><a name="l00127"></a><span class="lineno">  127</span>&#160;  return myPolynomial( aPoint[ 0 ] )( aPoint[ 1 ] )( aPoint[ 2 ] );</div>
<div class="line"><a name="l00128"></a><span class="lineno">  128</span>&#160;}</div>
<div class="line"><a name="l00129"></a><span class="lineno">  129</span>&#160;//-----------------------------------------------------------------------------</div>
<div class="line"><a name="l00130"></a><span class="lineno">  130</span>&#160;template &lt;typename TSpace&gt;</div>
<div class="line"><a name="l00131"></a><span class="lineno">  131</span>&#160;inline</div>
<div class="line"><a name="l00132"></a><span class="lineno">  132</span>&#160;bool</div>
<div class="line"><a name="l00133"></a><span class="lineno">  133</span>&#160;DGtal::ImplicitPolynomial3Shape&lt;TSpace&gt;::</div>
<div class="line"><a name="l00134"></a><span class="lineno">  134</span>&#160;isInside(const RealPoint &amp;aPoint) const</div>
<div class="line"><a name="l00135"></a><span class="lineno">  135</span>&#160;{</div>
<div class="line"><a name="l00136"></a><span class="lineno">  136</span>&#160;  return orientation( aPoint ) == INSIDE;</div>
<div class="line"><a name="l00137"></a><span class="lineno">  137</span>&#160;}</div>
<div class="line"><a name="l00138"></a><span class="lineno">  138</span>&#160;//-----------------------------------------------------------------------------</div>
<div class="line"><a name="l00139"></a><span class="lineno">  139</span>&#160;template &lt;typename TSpace&gt;</div>
<div class="line"><a name="l00140"></a><span class="lineno">  140</span>&#160;inline</div>
<div class="line"><a name="l00141"></a><span class="lineno">  141</span>&#160;DGtal::Orientation</div>
<div class="line"><a name="l00142"></a><span class="lineno">  142</span>&#160;DGtal::ImplicitPolynomial3Shape&lt;TSpace&gt;::</div>
<div class="line"><a name="l00143"></a><span class="lineno">  143</span>&#160;orientation(const RealPoint &amp;aPoint) const</div>
<div class="line"><a name="l00144"></a><span class="lineno">  144</span>&#160;{</div>
<div class="line"><a name="l00145"></a><span class="lineno">  145</span>&#160;  Ring v = this-&gt;operator()(aPoint);</div>
<div class="line"><a name="l00146"></a><span class="lineno">  146</span>&#160;  if ( v &lt; (Ring)0 )</div>
<div class="line"><a name="l00147"></a><span class="lineno">  147</span>&#160;    return INSIDE;</div>
<div class="line"><a name="l00148"></a><span class="lineno">  148</span>&#160;  else if ( v &gt; (Ring)0 )</div>
<div class="line"><a name="l00149"></a><span class="lineno">  149</span>&#160;    return OUTSIDE;</div>
<div class="line"><a name="l00150"></a><span class="lineno">  150</span>&#160;  else</div>
<div class="line"><a name="l00151"></a><span class="lineno">  151</span>&#160;    return ON;</div>
<div class="line"><a name="l00152"></a><span class="lineno">  152</span>&#160;}</div>
<div class="line"><a name="l00153"></a><span class="lineno">  153</span>&#160;//-----------------------------------------------------------------------------</div>
<div class="line"><a name="l00154"></a><span class="lineno">  154</span>&#160;template &lt;typename TSpace&gt;</div>
<div class="line"><a name="l00155"></a><span class="lineno">  155</span>&#160;inline</div>
<div class="line"><a name="l00156"></a><span class="lineno">  156</span>&#160;typename DGtal::ImplicitPolynomial3Shape&lt;TSpace&gt;::RealVector</div>
<div class="line"><a name="l00157"></a><span class="lineno">  157</span>&#160;DGtal::ImplicitPolynomial3Shape&lt;TSpace&gt;::</div>
<div class="line"><a name="l00158"></a><span class="lineno">  158</span>&#160;gradient( const RealPoint &amp;aPoint ) const</div>
<div class="line"><a name="l00159"></a><span class="lineno">  159</span>&#160;{</div>
<div class="line"><a name="l00160"></a><span class="lineno">  160</span>&#160;  // ISO C++ tells that an object created at return time will not be</div>
<div class="line"><a name="l00161"></a><span class="lineno">  161</span>&#160;  // copied into the caller context, but will be already defined in</div>
<div class="line"><a name="l00162"></a><span class="lineno">  162</span>&#160;  // the correct context.</div>
<div class="line"><a name="l00163"></a><span class="lineno">  163</span>&#160;  return RealVector</div>
<div class="line"><a name="l00164"></a><span class="lineno">  164</span>&#160;      ( myFx ( aPoint[ 0 ] )( aPoint[ 1 ] )( aPoint[ 2 ] ),</div>
<div class="line"><a name="l00165"></a><span class="lineno">  165</span>&#160;        myFy ( aPoint[ 0 ] )( aPoint[ 1 ] )( aPoint[ 2 ] ),</div>
<div class="line"><a name="l00166"></a><span class="lineno">  166</span>&#160;        myFz ( aPoint[ 0 ] )( aPoint[ 1 ] )( aPoint[ 2 ] ) );</div>
<div class="line"><a name="l00167"></a><span class="lineno">  167</span>&#160; </div>
<div class="line"><a name="l00168"></a><span class="lineno">  168</span>&#160;}</div>
<div class="line"><a name="l00169"></a><span class="lineno">  169</span>&#160; </div>
<div class="line"><a name="l00170"></a><span class="lineno">  170</span>&#160; </div>
<div class="line"><a name="l00171"></a><span class="lineno">  171</span>&#160;// ------------------------------------------------------------ Added by Anis Benyoub</div>
<div class="line"><a name="l00172"></a><span class="lineno">  172</span>&#160;//-----------------------------------------------------------------------------</div>
<div class="line"><a name="l00173"></a><span class="lineno">  173</span>&#160; </div>
<div class="line"><a name="l00174"></a><span class="lineno">  174</span>&#160;/**</div>
<div class="line"><a name="l00175"></a><span class="lineno">  175</span>&#160; * @param aPoint any point in the Euclidean space.</div>
<div class="line"><a name="l00176"></a><span class="lineno">  176</span>&#160; * This computation is based on the hessian formula of the mean curvature</div>
<div class="line"><a name="l00177"></a><span class="lineno">  177</span>&#160; * k=-(∇F ∗ H (F ) ∗ ∇F T − |∇F |^2 *Trace(H (F ))/2|∇F |^3</div>
<div class="line"><a name="l00178"></a><span class="lineno">  178</span>&#160; * we define it as positive for a sphere</div>
<div class="line"><a name="l00179"></a><span class="lineno">  179</span>&#160; * @return the mean curvature value of the polynomial at \a aPoint.</div>
<div class="line"><a name="l00180"></a><span class="lineno">  180</span>&#160; * </div>
<div class="line"><a name="l00181"></a><span class="lineno">  181</span>&#160;*/</div>
<div class="line"><a name="l00182"></a><span class="lineno">  182</span>&#160;template &lt;typename TSpace&gt;</div>
<div class="line"><a name="l00183"></a><span class="lineno">  183</span>&#160;inline</div>
<div class="line"><a name="l00184"></a><span class="lineno">  184</span>&#160;double</div>
<div class="line"><a name="l00185"></a><span class="lineno">  185</span>&#160;DGtal::ImplicitPolynomial3Shape&lt;TSpace&gt;::</div>
<div class="line"><a name="l00186"></a><span class="lineno">  186</span>&#160;meanCurvature( const RealPoint &amp;aPoint ) const</div>
<div class="line"><a name="l00187"></a><span class="lineno">  187</span>&#160;{</div>
<div class="line"><a name="l00188"></a><span class="lineno">  188</span>&#160;  double temp= myLowPolynome( aPoint[ 0 ] )( aPoint[ 1 ] )( aPoint[ 2 ] );</div>
<div class="line"><a name="l00189"></a><span class="lineno">  189</span>&#160;  temp = sqrt(temp);</div>
<div class="line"><a name="l00190"></a><span class="lineno">  190</span>&#160;  double downValue = 2.0*(temp*temp*temp);</div>
<div class="line"><a name="l00191"></a><span class="lineno">  191</span>&#160;  double upValue = myUpPolynome( aPoint[ 0 ] )( aPoint[ 1 ] )( aPoint[ 2 ] );</div>
<div class="line"><a name="l00192"></a><span class="lineno">  192</span>&#160; </div>
<div class="line"><a name="l00193"></a><span class="lineno">  193</span>&#160; </div>
<div class="line"><a name="l00194"></a><span class="lineno">  194</span>&#160;  return -(upValue/downValue);</div>
<div class="line"><a name="l00195"></a><span class="lineno">  195</span>&#160;}</div>
<div class="line"><a name="l00196"></a><span class="lineno">  196</span>&#160; </div>
<div class="line"><a name="l00197"></a><span class="lineno">  197</span>&#160; </div>
<div class="line"><a name="l00198"></a><span class="lineno">  198</span>&#160; </div>
<div class="line"><a name="l00199"></a><span class="lineno">  199</span>&#160;//-----------------------------------------------------------------------------</div>
<div class="line"><a name="l00200"></a><span class="lineno">  200</span>&#160;template &lt;typename TSpace&gt;</div>
<div class="line"><a name="l00201"></a><span class="lineno">  201</span>&#160;inline</div>
<div class="line"><a name="l00202"></a><span class="lineno">  202</span>&#160;double</div>
<div class="line"><a name="l00203"></a><span class="lineno">  203</span>&#160;DGtal::ImplicitPolynomial3Shape&lt;TSpace&gt;::</div>
<div class="line"><a name="l00204"></a><span class="lineno">  204</span>&#160;gaussianCurvature( const RealPoint &amp;aPoint ) const</div>
<div class="line"><a name="l00205"></a><span class="lineno">  205</span>&#160;{</div>
<div class="line"><a name="l00206"></a><span class="lineno">  206</span>&#160;  /*</div>
<div class="line"><a name="l00207"></a><span class="lineno">  207</span>&#160;    JOL: new Gaussian curvature formula (in sage)</div>
<div class="line"><a name="l00208"></a><span class="lineno">  208</span>&#160;    var(&#39;Fx&#39;,&#39;Fy&#39;,&#39;Fz&#39;,&#39;Fxx&#39;,&#39;Fxy&#39;,&#39;Fxz&#39;,&#39;Fyy&#39;,&#39;Fyz&#39;,&#39;Fzz&#39;)</div>
<div class="line"><a name="l00209"></a><span class="lineno">  209</span>&#160;    M=Matrix(4,4,[[Fxx,Fxy,Fxz,Fx],[Fxy,Fyy,Fyz,Fy],[Fxz,Fyz,Fzz,Fz],[Fx,Fy,Fz,0]])</div>
<div class="line"><a name="l00210"></a><span class="lineno">  210</span>&#160;    det(M)</div>
<div class="line"><a name="l00211"></a><span class="lineno">  211</span>&#160;# Fxz^2*Fy^2 - 2*Fx*Fxz*Fy*Fyz + Fx^2*Fyz^2 - 2*Fxy*Fxz*Fy*Fz + 2*Fx*Fxz*Fyy*Fz - 2*Fx*Fxy*Fyz*Fz + 2*Fxx*Fy*Fyz*Fz + Fxy^2*Fz^2 - Fxx*Fyy*Fz^2 + 2*Fx*Fxy*Fy*Fzz - Fxx*Fy^2*Fzz - Fx^2*Fyy*Fzz</div>
<div class="line"><a name="l00212"></a><span class="lineno">  212</span>&#160;    G = -det(M) / ( Fx^2 + Fy^2 + Fz^2 )^2</div>
<div class="line"><a name="l00213"></a><span class="lineno">  213</span>&#160;   */</div>
<div class="line"><a name="l00214"></a><span class="lineno">  214</span>&#160;  const double   x = aPoint[ 0 ];</div>
<div class="line"><a name="l00215"></a><span class="lineno">  215</span>&#160;  const double   y = aPoint[ 1 ];</div>
<div class="line"><a name="l00216"></a><span class="lineno">  216</span>&#160;  const double   z = aPoint[ 2 ];</div>
<div class="line"><a name="l00217"></a><span class="lineno">  217</span>&#160;  const double  Fx = myFx( x )( y )( z );</div>
<div class="line"><a name="l00218"></a><span class="lineno">  218</span>&#160;  const double  Fy = myFy( x )( y )( z );</div>
<div class="line"><a name="l00219"></a><span class="lineno">  219</span>&#160;  const double  Fz = myFz( x )( y )( z );</div>
<div class="line"><a name="l00220"></a><span class="lineno">  220</span>&#160;  const double Fx2 = Fx * Fx;</div>
<div class="line"><a name="l00221"></a><span class="lineno">  221</span>&#160;  const double Fy2 = Fy * Fy;</div>
<div class="line"><a name="l00222"></a><span class="lineno">  222</span>&#160;  const double Fz2 = Fz * Fz;</div>
<div class="line"><a name="l00223"></a><span class="lineno">  223</span>&#160;  const double  G2 = Fx2 + Fy2 + Fz2;</div>
<div class="line"><a name="l00224"></a><span class="lineno">  224</span>&#160;  const double Fxx = myFxx( x )( y )( z );</div>
<div class="line"><a name="l00225"></a><span class="lineno">  225</span>&#160;  const double Fxy = myFxy( x )( y )( z );</div>
<div class="line"><a name="l00226"></a><span class="lineno">  226</span>&#160;  const double Fxz = myFxz( x )( y )( z );</div>
<div class="line"><a name="l00227"></a><span class="lineno">  227</span>&#160;  const double Fyy = myFyy( x )( y )( z );</div>
<div class="line"><a name="l00228"></a><span class="lineno">  228</span>&#160;  const double Fyz = myFyz( x )( y )( z );</div>
<div class="line"><a name="l00229"></a><span class="lineno">  229</span>&#160;  const double Fzz = myFzz( x )( y )( z );</div>
<div class="line"><a name="l00230"></a><span class="lineno">  230</span>&#160;  const double Ax2 = ( Fyz * Fyz - Fyy * Fzz ) * Fx2;</div>
<div class="line"><a name="l00231"></a><span class="lineno">  231</span>&#160;  const double Ay2 = ( Fxz * Fxz - Fxx * Fzz ) * Fy2; </div>
<div class="line"><a name="l00232"></a><span class="lineno">  232</span>&#160;  const double Az2 = ( Fxy * Fxy - Fxx * Fyy ) * Fz2;</div>
<div class="line"><a name="l00233"></a><span class="lineno">  233</span>&#160;  const double Axy = ( Fxy * Fzz - Fxz * Fyz ) * Fx * Fy;</div>
<div class="line"><a name="l00234"></a><span class="lineno">  234</span>&#160;  const double Axz = ( Fxz * Fyy - Fxy * Fyz ) * Fx * Fz;</div>
<div class="line"><a name="l00235"></a><span class="lineno">  235</span>&#160;  const double Ayz = ( Fxx * Fyz - Fxy * Fxz ) * Fy * Fz;</div>
<div class="line"><a name="l00236"></a><span class="lineno">  236</span>&#160;  const double det = Ax2 + Ay2 + Az2 + 2 * ( Axy + Axz + Ayz );</div>
<div class="line"><a name="l00237"></a><span class="lineno">  237</span>&#160;  return - det / ( G2*G2 );</div>
<div class="line"><a name="l00238"></a><span class="lineno">  238</span>&#160;}</div>
<div class="line"><a name="l00239"></a><span class="lineno">  239</span>&#160; </div>
<div class="line"><a name="l00240"></a><span class="lineno">  240</span>&#160;template&lt; typename TSpace &gt;</div>
<div class="line"><a name="l00241"></a><span class="lineno">  241</span>&#160;inline</div>
<div class="line"><a name="l00242"></a><span class="lineno">  242</span>&#160;void</div>
<div class="line"><a name="l00243"></a><span class="lineno">  243</span>&#160;DGtal::ImplicitPolynomial3Shape&lt;TSpace&gt;::principalCurvatures</div>
<div class="line"><a name="l00244"></a><span class="lineno">  244</span>&#160;( const RealPoint &amp; aPoint,</div>
<div class="line"><a name="l00245"></a><span class="lineno">  245</span>&#160;  double &amp; k1,</div>
<div class="line"><a name="l00246"></a><span class="lineno">  246</span>&#160;  double &amp; k2 ) const</div>
<div class="line"><a name="l00247"></a><span class="lineno">  247</span>&#160;{</div>
<div class="line"><a name="l00248"></a><span class="lineno">  248</span>&#160;    double H = meanCurvature( aPoint );</div>
<div class="line"><a name="l00249"></a><span class="lineno">  249</span>&#160;    double G = gaussianCurvature( aPoint );</div>
<div class="line"><a name="l00250"></a><span class="lineno">  250</span>&#160;    double tmp = std::sqrt( fabs( H * H - G ));</div>
<div class="line"><a name="l00251"></a><span class="lineno">  251</span>&#160;    k2 = H + tmp;</div>
<div class="line"><a name="l00252"></a><span class="lineno">  252</span>&#160;    k1 = H - tmp;</div>
<div class="line"><a name="l00253"></a><span class="lineno">  253</span>&#160;}</div>
<div class="line"><a name="l00254"></a><span class="lineno">  254</span>&#160; </div>
<div class="line"><a name="l00255"></a><span class="lineno">  255</span>&#160;template&lt; typename TSpace &gt;</div>
<div class="line"><a name="l00256"></a><span class="lineno">  256</span>&#160;inline</div>
<div class="line"><a name="l00257"></a><span class="lineno">  257</span>&#160;void</div>
<div class="line"><a name="l00258"></a><span class="lineno">  258</span>&#160;DGtal::ImplicitPolynomial3Shape&lt;TSpace&gt;::principalDirections</div>
<div class="line"><a name="l00259"></a><span class="lineno">  259</span>&#160;( const RealPoint &amp; aPoint,</div>
<div class="line"><a name="l00260"></a><span class="lineno">  260</span>&#160;  RealVector &amp; d1,</div>
<div class="line"><a name="l00261"></a><span class="lineno">  261</span>&#160;  RealVector &amp; d2 ) const</div>
<div class="line"><a name="l00262"></a><span class="lineno">  262</span>&#160;{</div>
<div class="line"><a name="l00263"></a><span class="lineno">  263</span>&#160;  const RealVector grad_F = gradient( aPoint );</div>
<div class="line"><a name="l00264"></a><span class="lineno">  264</span>&#160;  const auto           Fn = grad_F.norm();</div>
<div class="line"><a name="l00265"></a><span class="lineno">  265</span>&#160;  if ( Fn &lt; 1e-8 )</div>
<div class="line"><a name="l00266"></a><span class="lineno">  266</span>&#160;    {</div>
<div class="line"><a name="l00267"></a><span class="lineno">  267</span>&#160;      d1 = d2 = RealVector();</div>
<div class="line"><a name="l00268"></a><span class="lineno">  268</span>&#160;      return;</div>
<div class="line"><a name="l00269"></a><span class="lineno">  269</span>&#160;    }</div>
<div class="line"><a name="l00270"></a><span class="lineno">  270</span>&#160;  RealVector u, v;</div>
<div class="line"><a name="l00271"></a><span class="lineno">  271</span>&#160;  const RealVector n = grad_F / Fn;</div>
<div class="line"><a name="l00272"></a><span class="lineno">  272</span>&#160;  u  = RealVector( 1.0, 0.0, 0.0 ).crossProduct( n );</div>
<div class="line"><a name="l00273"></a><span class="lineno">  273</span>&#160;  auto u_norm = u.norm();</div>
<div class="line"><a name="l00274"></a><span class="lineno">  274</span>&#160;  if ( u_norm &lt; 1e-8 )</div>
<div class="line"><a name="l00275"></a><span class="lineno">  275</span>&#160;    {</div>
<div class="line"><a name="l00276"></a><span class="lineno">  276</span>&#160;      u  = RealVector( 0.0, 1.0, 0.0 ).crossProduct( n );</div>
<div class="line"><a name="l00277"></a><span class="lineno">  277</span>&#160;      u_norm = u.norm();</div>
<div class="line"><a name="l00278"></a><span class="lineno">  278</span>&#160;    }</div>
<div class="line"><a name="l00279"></a><span class="lineno">  279</span>&#160;  u /= u_norm;</div>
<div class="line"><a name="l00280"></a><span class="lineno">  280</span>&#160;  v = n.crossProduct( u );</div>
<div class="line"><a name="l00281"></a><span class="lineno">  281</span>&#160;  double k_min, k_max;</div>
<div class="line"><a name="l00282"></a><span class="lineno">  282</span>&#160;  principalCurvatures( aPoint, k_min, k_max );</div>
<div class="line"><a name="l00283"></a><span class="lineno">  283</span>&#160;  const double   x = aPoint[ 0 ];</div>
<div class="line"><a name="l00284"></a><span class="lineno">  284</span>&#160;  const double   y = aPoint[ 1 ];</div>
<div class="line"><a name="l00285"></a><span class="lineno">  285</span>&#160;  const double   z = aPoint[ 2 ];</div>
<div class="line"><a name="l00286"></a><span class="lineno">  286</span>&#160;  // Computing Hessian matrix</div>
<div class="line"><a name="l00287"></a><span class="lineno">  287</span>&#160;  const double Fxx = myFxx( x )( y )( z );</div>
<div class="line"><a name="l00288"></a><span class="lineno">  288</span>&#160;  const double Fxy = myFxy( x )( y )( z );</div>
<div class="line"><a name="l00289"></a><span class="lineno">  289</span>&#160;  const double Fxz = myFxz( x )( y )( z );</div>
<div class="line"><a name="l00290"></a><span class="lineno">  290</span>&#160;  const double Fyy = myFyy( x )( y )( z );</div>
<div class="line"><a name="l00291"></a><span class="lineno">  291</span>&#160;  const double Fyz = myFyz( x )( y )( z );</div>
<div class="line"><a name="l00292"></a><span class="lineno">  292</span>&#160;  const double Fzz = myFzz( x )( y )( z );</div>
<div class="line"><a name="l00293"></a><span class="lineno">  293</span>&#160;  const RealVector HessF_u = { Fxx * u[ 0 ] + Fxy * u[ 1 ] + Fxz * u[ 2 ],</div>
<div class="line"><a name="l00294"></a><span class="lineno">  294</span>&#160;                               Fxy * u[ 0 ] + Fyy * u[ 1 ] + Fyz * u[ 2 ],</div>
<div class="line"><a name="l00295"></a><span class="lineno">  295</span>&#160;                               Fxz * u[ 0 ] + Fyz * u[ 1 ] + Fzz * u[ 2 ] };</div>
<div class="line"><a name="l00296"></a><span class="lineno">  296</span>&#160;  const RealVector HessF_v = { Fxx * v[ 0 ] + Fxy * v[ 1 ] + Fxz * v[ 2 ],</div>
<div class="line"><a name="l00297"></a><span class="lineno">  297</span>&#160;                               Fxy * v[ 0 ] + Fyy * v[ 1 ] + Fyz * v[ 2 ],</div>
<div class="line"><a name="l00298"></a><span class="lineno">  298</span>&#160;                               Fxz * v[ 0 ] + Fyz * v[ 1 ] + Fzz * v[ 2 ] };</div>
<div class="line"><a name="l00299"></a><span class="lineno">  299</span>&#160;  const double Fuu = u.dot( HessF_u );</div>
<div class="line"><a name="l00300"></a><span class="lineno">  300</span>&#160;  const double Fuv = u.dot( HessF_v );</div>
<div class="line"><a name="l00301"></a><span class="lineno">  301</span>&#160;  const double Fvv = v.dot( HessF_v );</div>
<div class="line"><a name="l00302"></a><span class="lineno">  302</span>&#160;  if ( fabs( k_min * Fn - Fuu ) &gt;= fabs( k_min * Fn - Fvv ) )</div>
<div class="line"><a name="l00303"></a><span class="lineno">  303</span>&#160;    {</div>
<div class="line"><a name="l00304"></a><span class="lineno">  304</span>&#160;      // Choose k1 = k_min and k2 = k_max,</div>
<div class="line"><a name="l00305"></a><span class="lineno">  305</span>&#160;      // to avoid null k1*Fn - Fuu = -(k2*Fn - Fvv) = 0</div>
<div class="line"><a name="l00306"></a><span class="lineno">  306</span>&#160;      double k1 = k_min;</div>
<div class="line"><a name="l00307"></a><span class="lineno">  307</span>&#160;      double k2 = k_max;</div>
<div class="line"><a name="l00308"></a><span class="lineno">  308</span>&#160;      d1 = RealVector( ( k1 * Fn - Fuu ) * v[ 0 ] + Fuv * u[ 0 ],</div>
<div class="line"><a name="l00309"></a><span class="lineno">  309</span>&#160;                       ( k1 * Fn - Fuu ) * v[ 1 ] + Fuv * u[ 1 ],</div>
<div class="line"><a name="l00310"></a><span class="lineno">  310</span>&#160;                       ( k1 * Fn - Fuu ) * v[ 2 ] + Fuv * u[ 2 ] );</div>
<div class="line"><a name="l00311"></a><span class="lineno">  311</span>&#160;      d2 = -1.0 * RealVector( ( k2 * Fn - Fvv ) * u[ 0 ] + Fuv * v[ 0 ],</div>
<div class="line"><a name="l00312"></a><span class="lineno">  312</span>&#160;                              ( k2 * Fn - Fvv ) * u[ 1 ] + Fuv * v[ 1 ],</div>
<div class="line"><a name="l00313"></a><span class="lineno">  313</span>&#160;                              ( k2 * Fn - Fvv ) * u[ 2 ] + Fuv * v[ 2 ] );</div>
<div class="line"><a name="l00314"></a><span class="lineno">  314</span>&#160;    }</div>
<div class="line"><a name="l00315"></a><span class="lineno">  315</span>&#160;  else</div>
<div class="line"><a name="l00316"></a><span class="lineno">  316</span>&#160;    {</div>
<div class="line"><a name="l00317"></a><span class="lineno">  317</span>&#160;      // Choose k2 = k_min and k1 = k_max,</div>
<div class="line"><a name="l00318"></a><span class="lineno">  318</span>&#160;      // then | k_max*Fn - Fuu | &gt;= | k_max*Fn - Fvv | &gt;= 0</div>
<div class="line"><a name="l00319"></a><span class="lineno">  319</span>&#160;      double k1 = k_max;</div>
<div class="line"><a name="l00320"></a><span class="lineno">  320</span>&#160;      double k2 = k_min;</div>
<div class="line"><a name="l00321"></a><span class="lineno">  321</span>&#160;      d2 = RealVector( ( k1 * Fn - Fuu ) * v[ 0 ] + Fuv * u[ 0 ],</div>
<div class="line"><a name="l00322"></a><span class="lineno">  322</span>&#160;                       ( k1 * Fn - Fuu ) * v[ 1 ] + Fuv * u[ 1 ],</div>
<div class="line"><a name="l00323"></a><span class="lineno">  323</span>&#160;                       ( k1 * Fn - Fuu ) * v[ 2 ] + Fuv * u[ 2 ] );</div>
<div class="line"><a name="l00324"></a><span class="lineno">  324</span>&#160;      d1 = -1.0 * RealVector( ( k2 * Fn - Fvv ) * u[ 0 ] + Fuv * v[ 0 ],</div>
<div class="line"><a name="l00325"></a><span class="lineno">  325</span>&#160;                              ( k2 * Fn - Fvv ) * u[ 1 ] + Fuv * v[ 1 ],</div>
<div class="line"><a name="l00326"></a><span class="lineno">  326</span>&#160;                              ( k2 * Fn - Fvv ) * u[ 2 ] + Fuv * v[ 2 ] );</div>
<div class="line"><a name="l00327"></a><span class="lineno">  327</span>&#160;    }</div>
<div class="line"><a name="l00328"></a><span class="lineno">  328</span>&#160;  d1 /= d1.norm();</div>
<div class="line"><a name="l00329"></a><span class="lineno">  329</span>&#160;  d2 /= d2.norm();</div>
<div class="line"><a name="l00330"></a><span class="lineno">  330</span>&#160;}</div>
<div class="line"><a name="l00331"></a><span class="lineno">  331</span>&#160; </div>
<div class="line"><a name="l00332"></a><span class="lineno">  332</span>&#160;/**</div>
<div class="line"><a name="l00333"></a><span class="lineno">  333</span>&#160; *@param aPoint any point in the Euclidean space.</div>
<div class="line"><a name="l00334"></a><span class="lineno">  334</span>&#160; *@param accuracy refers to the precision </div>
<div class="line"><a name="l00335"></a><span class="lineno">  335</span>&#160; *@param maxIter refers to the maximum iterations the fonction user authorises</div>
<div class="line"><a name="l00336"></a><span class="lineno">  336</span>&#160; *@param gamma refers to the step</div>
<div class="line"><a name="l00337"></a><span class="lineno">  337</span>&#160; *@return the nearest point on the surface to the one given in parameter.</div>
<div class="line"><a name="l00338"></a><span class="lineno">  338</span>&#160; */</div>
<div class="line"><a name="l00339"></a><span class="lineno">  339</span>&#160;template &lt;typename TSpace&gt;</div>
<div class="line"><a name="l00340"></a><span class="lineno">  340</span>&#160;inline</div>
<div class="line"><a name="l00341"></a><span class="lineno">  341</span>&#160;typename DGtal::ImplicitPolynomial3Shape&lt;TSpace&gt;::RealPoint </div>
<div class="line"><a name="l00342"></a><span class="lineno">  342</span>&#160;DGtal::ImplicitPolynomial3Shape&lt;TSpace&gt;::nearestPoint</div>
<div class="line"><a name="l00343"></a><span class="lineno">  343</span>&#160;( const RealPoint &amp;aPoint, const double accuracy, </div>
<div class="line"><a name="l00344"></a><span class="lineno">  344</span>&#160;  const int maxIter, const double gamma ) const</div>
<div class="line"><a name="l00345"></a><span class="lineno">  345</span>&#160;{</div>
<div class="line"><a name="l00346"></a><span class="lineno">  346</span>&#160;   RealPoint X = aPoint;</div>
<div class="line"><a name="l00347"></a><span class="lineno">  347</span>&#160;   for ( int numberIter = 0; numberIter &lt; maxIter; numberIter++ )</div>
<div class="line"><a name="l00348"></a><span class="lineno">  348</span>&#160;     {</div>
<div class="line"><a name="l00349"></a><span class="lineno">  349</span>&#160;       double val_X = (*this)( X );</div>
<div class="line"><a name="l00350"></a><span class="lineno">  350</span>&#160;       if ( fabs( val_X ) &lt; accuracy ) break;</div>
<div class="line"><a name="l00351"></a><span class="lineno">  351</span>&#160;       RealVector grad_X = (*this).gradient( X );</div>
<div class="line"><a name="l00352"></a><span class="lineno">  352</span>&#160;       double  n2_grad_X = grad_X.dot( grad_X );</div>
<div class="line"><a name="l00353"></a><span class="lineno">  353</span>&#160;       if ( n2_grad_X &gt; 0.000001 ) grad_X /= n2_grad_X;</div>
<div class="line"><a name="l00354"></a><span class="lineno">  354</span>&#160;       X -= val_X * gamma * grad_X ;</div>
<div class="line"><a name="l00355"></a><span class="lineno">  355</span>&#160;     }</div>
<div class="line"><a name="l00356"></a><span class="lineno">  356</span>&#160;   return X;</div>
<div class="line"><a name="l00357"></a><span class="lineno">  357</span>&#160;}</div>
<div class="line"><a name="l00358"></a><span class="lineno">  358</span>&#160; </div>
<div class="line"><a name="l00359"></a><span class="lineno">  359</span>&#160;///////////////////////////////////////////////////////////////////////////////</div>
<div class="line"><a name="l00360"></a><span class="lineno">  360</span>&#160;// Interface - public :</div>
<div class="line"><a name="l00361"></a><span class="lineno">  361</span>&#160; </div>
<div class="line"><a name="l00362"></a><span class="lineno">  362</span>&#160;/**</div>
<div class="line"><a name="l00363"></a><span class="lineno">  363</span>&#160; * Writes/Displays the object on an output stream.</div>
<div class="line"><a name="l00364"></a><span class="lineno">  364</span>&#160; * @param out the output stream where the object is written.</div>
<div class="line"><a name="l00365"></a><span class="lineno">  365</span>&#160; */</div>
<div class="line"><a name="l00366"></a><span class="lineno">  366</span>&#160;template &lt;typename TSpace&gt;</div>
<div class="line"><a name="l00367"></a><span class="lineno">  367</span>&#160;inline</div>
<div class="line"><a name="l00368"></a><span class="lineno">  368</span>&#160;void</div>
<div class="line"><a name="l00369"></a><span class="lineno">  369</span>&#160;DGtal::ImplicitPolynomial3Shape&lt;TSpace&gt;::selfDisplay ( std::ostream &amp; out ) const</div>
<div class="line"><a name="l00370"></a><span class="lineno">  370</span>&#160;{</div>
<div class="line"><a name="l00371"></a><span class="lineno">  371</span>&#160;  out &lt;&lt; &quot;[ImplicitPolynomial3Shape] P(x,y,z) = &quot; &lt;&lt; myPolynomial;</div>
<div class="line"><a name="l00372"></a><span class="lineno">  372</span>&#160;}</div>
<div class="line"><a name="l00373"></a><span class="lineno">  373</span>&#160; </div>
<div class="line"><a name="l00374"></a><span class="lineno">  374</span>&#160;/**</div>
<div class="line"><a name="l00375"></a><span class="lineno">  375</span>&#160; * Checks the validity/consistency of the object.</div>
<div class="line"><a name="l00376"></a><span class="lineno">  376</span>&#160; * @return &#39;true&#39; if the object is valid, &#39;false&#39; otherwise.</div>
<div class="line"><a name="l00377"></a><span class="lineno">  377</span>&#160; */</div>
<div class="line"><a name="l00378"></a><span class="lineno">  378</span>&#160;template &lt;typename TSpace&gt;</div>
<div class="line"><a name="l00379"></a><span class="lineno">  379</span>&#160;inline</div>
<div class="line"><a name="l00380"></a><span class="lineno">  380</span>&#160;bool</div>
<div class="line"><a name="l00381"></a><span class="lineno">  381</span>&#160;DGtal::ImplicitPolynomial3Shape&lt;TSpace&gt;::isValid() const</div>
<div class="line"><a name="l00382"></a><span class="lineno">  382</span>&#160;{</div>
<div class="line"><a name="l00383"></a><span class="lineno">  383</span>&#160;  return true;</div>
<div class="line"><a name="l00384"></a><span class="lineno">  384</span>&#160;}</div>
<div class="line"><a name="l00385"></a><span class="lineno">  385</span>&#160; </div>
<div class="line"><a name="l00386"></a><span class="lineno">  386</span>&#160; </div>
<div class="line"><a name="l00387"></a><span class="lineno">  387</span>&#160; </div>
<div class="line"><a name="l00388"></a><span class="lineno">  388</span>&#160;///////////////////////////////////////////////////////////////////////////////</div>
<div class="line"><a name="l00389"></a><span class="lineno">  389</span>&#160;// Implementation of inline functions                                        //</div>
<div class="line"><a name="l00390"></a><span class="lineno">  390</span>&#160; </div>
<div class="line"><a name="l00391"></a><span class="lineno">  391</span>&#160;template &lt;typename TSpace&gt;</div>
<div class="line"><a name="l00392"></a><span class="lineno">  392</span>&#160;inline</div>
<div class="line"><a name="l00393"></a><span class="lineno">  393</span>&#160;std::ostream&amp;</div>
<div class="line"><a name="l00394"></a><span class="lineno">  394</span>&#160;DGtal::operator&lt;&lt; ( std::ostream &amp; out,</div>
<div class="line"><a name="l00395"></a><span class="lineno">  395</span>&#160;                    const ImplicitPolynomial3Shape&lt;TSpace&gt; &amp; object )</div>
<div class="line"><a name="l00396"></a><span class="lineno">  396</span>&#160;{</div>
<div class="line"><a name="l00397"></a><span class="lineno">  397</span>&#160;  object.selfDisplay( out );</div>
<div class="line"><a name="l00398"></a><span class="lineno">  398</span>&#160;  return out;</div>
<div class="line"><a name="l00399"></a><span class="lineno">  399</span>&#160;}</div>
<div class="line"><a name="l00400"></a><span class="lineno">  400</span>&#160; </div>
<div class="line"><a name="l00401"></a><span class="lineno">  401</span>&#160;//                                                                           //</div>
<div class="line"><a name="l00402"></a><span class="lineno">  402</span>&#160;///////////////////////////////////////////////////////////////////////////////</div>
<div class="line"><a name="l00403"></a><span class="lineno">  403</span>&#160; </div>
<div class="line"><a name="l00404"></a><span class="lineno">  404</span>&#160; </div>
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