茂之钳
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493 lines
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<a href="#pub-methods">Public 成员函数</a> |
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<a href="classvde_1_1curves_1_1BSplineCurve-members.html">所有成员列表</a> </div>
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<div class="headertitle">
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<div class="title">vde::curves::BSplineCurve类 参考<div class="ingroups"><a class="el" href="group__curves.html">曲线曲面模块 (vde::curves)</a></div></div> </div>
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</div><!--header-->
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<div class="contents">
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<p>B 样条曲线（非有理、均匀/非均匀节点）
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<a href="classvde_1_1curves_1_1BSplineCurve.html#details">更多...</a></p>
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<p><code>#include <<a class="el" href="bspline__curve_8h_source.html">bspline_curve.h</a>></code></p>
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<table class="memberdecls">
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<tr class="heading"><td colspan="2"><h2 class="groupheader"><a name="pub-methods"></a>
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Public 成员函数</h2></td></tr>
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<tr class="memitem:ac122dfe6b9515f29d0415ae8c6918984"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classvde_1_1curves_1_1BSplineCurve.html#ac122dfe6b9515f29d0415ae8c6918984">BSplineCurve</a> (std::vector< Point3D > <a class="el" href="classvde_1_1curves_1_1BSplineCurve.html#a04741a3297bbd26c5ab8ac712fc86906">control_points</a>, std::vector< double > <a class="el" href="classvde_1_1curves_1_1BSplineCurve.html#ae2f04689aac05a5cc631daa2d4d2661b">knots</a>, int <a class="el" href="classvde_1_1curves_1_1BSplineCurve.html#a4139f8b3b8bd315c5506136681e60273">degree</a>)</td></tr>
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<tr class="memdesc:ac122dfe6b9515f29d0415ae8c6918984"><td class="mdescLeft"> </td><td class="mdescRight">构造 B 样条曲线 <a href="classvde_1_1curves_1_1BSplineCurve.html#ac122dfe6b9515f29d0415ae8c6918984">更多...</a><br /></td></tr>
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<tr class="separator:ac122dfe6b9515f29d0415ae8c6918984"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:aa16013651a9b35a2f755c56b94436278"><td class="memItemLeft" align="right" valign="top">Point3D </td><td class="memItemRight" valign="bottom"><a class="el" href="classvde_1_1curves_1_1BSplineCurve.html#aa16013651a9b35a2f755c56b94436278">evaluate</a> (double t) const</td></tr>
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<tr class="memdesc:aa16013651a9b35a2f755c56b94436278"><td class="mdescLeft"> </td><td class="mdescRight">在参数 t 处求值曲线点 <a href="classvde_1_1curves_1_1BSplineCurve.html#aa16013651a9b35a2f755c56b94436278">更多...</a><br /></td></tr>
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<tr class="separator:aa16013651a9b35a2f755c56b94436278"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:a2fd32824beffc6ed33e731d1aba6288f"><td class="memItemLeft" align="right" valign="top">Vector3D </td><td class="memItemRight" valign="bottom"><a class="el" href="classvde_1_1curves_1_1BSplineCurve.html#a2fd32824beffc6ed33e731d1aba6288f">derivative</a> (double t, int order=1) const</td></tr>
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<tr class="memdesc:a2fd32824beffc6ed33e731d1aba6288f"><td class="mdescLeft"> </td><td class="mdescRight">在参数 t 处求导数 <a href="classvde_1_1curves_1_1BSplineCurve.html#a2fd32824beffc6ed33e731d1aba6288f">更多...</a><br /></td></tr>
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<tr class="separator:a2fd32824beffc6ed33e731d1aba6288f"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:a4139f8b3b8bd315c5506136681e60273"><td class="memItemLeft" align="right" valign="top">int </td><td class="memItemRight" valign="bottom"><a class="el" href="classvde_1_1curves_1_1BSplineCurve.html#a4139f8b3b8bd315c5506136681e60273">degree</a> () const</td></tr>
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<tr class="memdesc:a4139f8b3b8bd315c5506136681e60273"><td class="mdescLeft"> </td><td class="mdescRight">曲线阶次 <a href="classvde_1_1curves_1_1BSplineCurve.html#a4139f8b3b8bd315c5506136681e60273">更多...</a><br /></td></tr>
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<tr class="separator:a4139f8b3b8bd315c5506136681e60273"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:a020663a039886ef2b17009405e5db855"><td class="memItemLeft" align="right" valign="top">std::pair< double, double > </td><td class="memItemRight" valign="bottom"><a class="el" href="classvde_1_1curves_1_1BSplineCurve.html#a020663a039886ef2b17009405e5db855">domain</a> () const</td></tr>
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<tr class="memdesc:a020663a039886ef2b17009405e5db855"><td class="mdescLeft"> </td><td class="mdescRight">参数定义域 <a href="classvde_1_1curves_1_1BSplineCurve.html#a020663a039886ef2b17009405e5db855">更多...</a><br /></td></tr>
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<tr class="separator:a020663a039886ef2b17009405e5db855"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:a04741a3297bbd26c5ab8ac712fc86906"><td class="memItemLeft" align="right" valign="top">const std::vector< Point3D > & </td><td class="memItemRight" valign="bottom"><a class="el" href="classvde_1_1curves_1_1BSplineCurve.html#a04741a3297bbd26c5ab8ac712fc86906">control_points</a> () const</td></tr>
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<tr class="memdesc:a04741a3297bbd26c5ab8ac712fc86906"><td class="mdescLeft"> </td><td class="mdescRight">控制点访问 <a href="classvde_1_1curves_1_1BSplineCurve.html#a04741a3297bbd26c5ab8ac712fc86906">更多...</a><br /></td></tr>
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<tr class="separator:a04741a3297bbd26c5ab8ac712fc86906"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:ae2f04689aac05a5cc631daa2d4d2661b"><td class="memItemLeft" align="right" valign="top">const std::vector< double > & </td><td class="memItemRight" valign="bottom"><a class="el" href="classvde_1_1curves_1_1BSplineCurve.html#ae2f04689aac05a5cc631daa2d4d2661b">knots</a> () const</td></tr>
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<tr class="memdesc:ae2f04689aac05a5cc631daa2d4d2661b"><td class="mdescLeft"> </td><td class="mdescRight">节点向量访问 <a href="classvde_1_1curves_1_1BSplineCurve.html#ae2f04689aac05a5cc631daa2d4d2661b">更多...</a><br /></td></tr>
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<tr class="separator:ae2f04689aac05a5cc631daa2d4d2661b"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:af4e982a7104d5a7e1df0ec94539518f1"><td class="memItemLeft" align="right" valign="top">int </td><td class="memItemRight" valign="bottom"><a class="el" href="classvde_1_1curves_1_1BSplineCurve.html#af4e982a7104d5a7e1df0ec94539518f1">find_span</a> (double t) const</td></tr>
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<tr class="memdesc:af4e982a7104d5a7e1df0ec94539518f1"><td class="mdescLeft"> </td><td class="mdescRight">查找 t 所在的节点区间 <a href="classvde_1_1curves_1_1BSplineCurve.html#af4e982a7104d5a7e1df0ec94539518f1">更多...</a><br /></td></tr>
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<tr class="separator:af4e982a7104d5a7e1df0ec94539518f1"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:a233f6553350307715708fff3961e535d"><td class="memItemLeft" align="right" valign="top">std::vector< double > </td><td class="memItemRight" valign="bottom"><a class="el" href="classvde_1_1curves_1_1BSplineCurve.html#a233f6553350307715708fff3961e535d">basis_functions</a> (double t, int span=-1) const</td></tr>
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<tr class="memdesc:a233f6553350307715708fff3961e535d"><td class="mdescLeft"> </td><td class="mdescRight">在 t 处求非零基函数值 <a href="classvde_1_1curves_1_1BSplineCurve.html#a233f6553350307715708fff3961e535d">更多...</a><br /></td></tr>
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<tr class="separator:a233f6553350307715708fff3961e535d"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:aacd9293d70a48686ffe89542d2343051"><td class="memItemLeft" align="right" valign="top">std::vector< double > </td><td class="memItemRight" valign="bottom"><a class="el" href="classvde_1_1curves_1_1BSplineCurve.html#aacd9293d70a48686ffe89542d2343051">basis_derivatives</a> (double t, int span=-1) const</td></tr>
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<tr class="memdesc:aacd9293d70a48686ffe89542d2343051"><td class="mdescLeft"> </td><td class="mdescRight">在 t 处求非零基函数的一阶导数 <a href="classvde_1_1curves_1_1BSplineCurve.html#aacd9293d70a48686ffe89542d2343051">更多...</a><br /></td></tr>
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<tr class="separator:aacd9293d70a48686ffe89542d2343051"><td class="memSeparator" colspan="2"> </td></tr>
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</table>
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<a name="details" id="details"></a><h2 class="groupheader">详细描述</h2>
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<div class="textblock"><p>B 样条曲线（非有理、均匀/非均匀节点） </p>
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<p>由 n+1 个控制点、m+1 个节点向量和阶次 p 定义。 节点向量必须满足 m = n + p + 1。 参数域为 [knots[p], knots[n+1]]，由前 p 个和后 p 个节点修剪。</p>
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<p>与 Bézier 的主要区别：</p><ul>
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<li>局部控制：修改单个控制点仅影响 [t_i, t_{i+p+1}] 区间</li>
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<li>节点插入可引入更多控制点而不改变形状</li>
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<li>通过多重节点可达 C^{p-k} 连续性 </li>
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</ul>
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<p class="definition">在文件 <a class="el" href="bspline__curve_8h_source.html">bspline_curve.h</a> 第 <a class="el" href="bspline__curve_8h_source.html#l00023">23</a> 行定义.</p>
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</div><h2 class="groupheader">构造及析构函数说明</h2>
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<a id="ac122dfe6b9515f29d0415ae8c6918984"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#ac122dfe6b9515f29d0415ae8c6918984">◆ </a></span>BSplineCurve()</h2>
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<div class="memitem">
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<div class="memproto">
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<table class="memname">
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<tr>
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<td class="memname">vde::curves::BSplineCurve::BSplineCurve </td>
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<td>(</td>
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<td class="paramtype">std::vector< Point3D > </td>
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<td class="paramname"><em>control_points</em>, </td>
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</tr>
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<tr>
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<td class="paramkey"></td>
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<td></td>
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<td class="paramtype">std::vector< double > </td>
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<td class="paramname"><em>knots</em>, </td>
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</tr>
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<tr>
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<td class="paramkey"></td>
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<td></td>
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<td class="paramtype">int </td>
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<td class="paramname"><em>degree</em> </td>
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</tr>
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<tr>
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<td></td>
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<td>)</td>
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<td></td><td></td>
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</tr>
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</table>
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</div><div class="memdoc">
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<p>构造 B 样条曲线 </p>
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<dl class="params"><dt>参数</dt><dd>
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<table class="params">
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<tr><td class="paramname">control_points</td><td>n+1 个控制点 </td></tr>
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<tr><td class="paramname">knots</td><td>节点向量，必须为非递减序列 </td></tr>
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<tr><td class="paramname">degree</td><td>阶次 p </td></tr>
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</table>
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</dd>
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</dl>
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<dl class="section note"><dt>注解</dt><dd>节点数量 = 控制点数 + 阶次 + 1；clamped 端点多重度为 p+1 <div class="fragment"><div class="line"><span class="comment">// 三次 B 样条，7 个控制点</span></div>
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<div class="line"><a class="code" href="classvde_1_1curves_1_1BSplineCurve.html#ac122dfe6b9515f29d0415ae8c6918984">BSplineCurve</a> curve(cps, {0,0,0,0, 0.25,0.5,0.75, 1,1,1,1}, 3);</div>
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<div class="ttc" id="aclassvde_1_1curves_1_1BSplineCurve_html_ac122dfe6b9515f29d0415ae8c6918984"><div class="ttname"><a href="classvde_1_1curves_1_1BSplineCurve.html#ac122dfe6b9515f29d0415ae8c6918984">vde::curves::BSplineCurve::BSplineCurve</a></div><div class="ttdeci">BSplineCurve(std::vector< Point3D > control_points, std::vector< double > knots, int degree)</div><div class="ttdoc">构造 B 样条曲线</div></div>
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</div><!-- fragment --> </dd></dl>
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</div>
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</div>
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<h2 class="groupheader">成员函数说明</h2>
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<a id="aacd9293d70a48686ffe89542d2343051"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#aacd9293d70a48686ffe89542d2343051">◆ </a></span>basis_derivatives()</h2>
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<div class="memitem">
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<div class="memproto">
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<table class="memname">
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<tr>
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<td class="memname">std::vector<double> vde::curves::BSplineCurve::basis_derivatives </td>
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<td>(</td>
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<td class="paramtype">double </td>
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<td class="paramname"><em>t</em>, </td>
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</tr>
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<tr>
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<td class="paramkey"></td>
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<td></td>
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<td class="paramtype">int </td>
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<td class="paramname"><em>span</em> = <code>-1</code> </td>
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</tr>
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<tr>
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<td></td>
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<td>)</td>
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<td></td><td> const</td>
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</tr>
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</table>
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</div><div class="memdoc">
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<p>在 t 处求非零基函数的一阶导数 </p>
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<dl class="params"><dt>参数</dt><dd>
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<table class="params">
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<tr><td class="paramname">t</td><td>参数值 </td></tr>
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<tr><td class="paramname">span</td><td>节点区间索引，-1 表示自动查找 </td></tr>
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</table>
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</dd>
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</dl>
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<dl class="section return"><dt>返回</dt><dd>dN_{span-p}/du, ..., dN_{span}/du 共 p+1 个值 </dd></dl>
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<dl class="section note"><dt>注解</dt><dd>B 样条基函数导数递推： N'_{i,p} = p/(t_{i+p}-t_i)·N_{i,p-1} - p/(t_{i+p+1}-t_{i+1})·N_{i+1,p-1} </dd></dl>
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<dl class="section see"><dt>参见</dt><dd><a class="el" href="classvde_1_1curves_1_1BSplineCurve.html#a233f6553350307715708fff3961e535d" title="在 t 处求非零基函数值">basis_functions</a> </dd></dl>
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</div>
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</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a233f6553350307715708fff3961e535d">◆ </a></span>basis_functions()</h2>
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<td class="memname">std::vector<double> vde::curves::BSplineCurve::basis_functions </td>
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<td>(</td>
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<td class="paramtype">double </td>
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<td class="paramname"><em>t</em>, </td>
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<td class="paramkey"></td>
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<td></td>
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<td class="paramtype">int </td>
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<td class="paramname"><em>span</em> = <code>-1</code> </td>
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<td></td>
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<td>)</td>
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<td></td><td> const</td>
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<p>在 t 处求非零基函数值 </p>
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<dl class="params"><dt>参数</dt><dd>
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<table class="params">
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<tr><td class="paramname">t</td><td>参数值 </td></tr>
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<tr><td class="paramname">span</td><td>节点区间索引，-1 表示自动查找 </td></tr>
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</dd>
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</dl>
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<dl class="section return"><dt>返回</dt><dd>N_{span-p}(t), ..., N_{span}(t) 共 p+1 个值 </dd></dl>
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<dl class="section note"><dt>注解</dt><dd>仅返回 p+1 个非零基函数；用于端点求值及插值 </dd></dl>
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<dl class="section see"><dt>参见</dt><dd><a class="el" href="classvde_1_1curves_1_1BSplineCurve.html#af4e982a7104d5a7e1df0ec94539518f1" title="查找 t 所在的节点区间">find_span</a> </dd></dl>
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<a id="a04741a3297bbd26c5ab8ac712fc86906"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#a04741a3297bbd26c5ab8ac712fc86906">◆ </a></span>control_points()</h2>
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<td class="memname">const std::vector<Point3D>& vde::curves::BSplineCurve::control_points </td>
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<td>(</td>
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<td class="paramname"></td><td>)</td>
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<td> const</td>
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</td>
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<td class="mlabels-right">
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<span class="mlabels"><span class="mlabel">inline</span></span> </td>
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<p>控制点访问 </p>
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<dl class="section return"><dt>返回</dt><dd>控制点数组只读引用 </dd></dl>
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<p class="definition">在文件 <a class="el" href="bspline__curve_8h_source.html">bspline_curve.h</a> 第 <a class="el" href="bspline__curve_8h_source.html#l00075">75</a> 行定义.</p>
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<a id="a4139f8b3b8bd315c5506136681e60273"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#a4139f8b3b8bd315c5506136681e60273">◆ </a></span>degree()</h2>
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<td class="mlabels-left">
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<td class="memname">int vde::curves::BSplineCurve::degree </td>
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<td>(</td>
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<td class="paramname"></td><td>)</td>
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<td> const</td>
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</tr>
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</table>
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</td>
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<td class="mlabels-right">
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<span class="mlabels"><span class="mlabel">inline</span></span> </td>
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</div><div class="memdoc">
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<p>曲线阶次 </p>
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<dl class="section return"><dt>返回</dt><dd>阶次 p </dd></dl>
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<p class="definition">在文件 <a class="el" href="bspline__curve_8h_source.html">bspline_curve.h</a> 第 <a class="el" href="bspline__curve_8h_source.html#l00063">63</a> 行定义.</p>
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</div>
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<a id="a2fd32824beffc6ed33e731d1aba6288f"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#a2fd32824beffc6ed33e731d1aba6288f">◆ </a></span>derivative()</h2>
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<td class="memname">Vector3D vde::curves::BSplineCurve::derivative </td>
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<td>(</td>
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<td class="paramtype">double </td>
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<td class="paramname"><em>t</em>, </td>
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<td class="paramkey"></td>
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<td></td>
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<td class="paramtype">int </td>
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<td class="paramname"><em>order</em> = <code>1</code> </td>
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<tr>
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<td></td>
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<td>)</td>
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<td></td><td> const</td>
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</div><div class="memdoc">
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<p>在参数 t 处求导数 </p>
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<dl class="params"><dt>参数</dt><dd>
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<table class="params">
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<tr><td class="paramname">t</td><td>参数值 </td></tr>
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<tr><td class="paramname">order</td><td>导数阶数，1 = 一切向量 </td></tr>
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</table>
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</dd>
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</dl>
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<dl class="section return"><dt>返回</dt><dd>导数向量 </dd></dl>
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<dl class="section note"><dt>注解</dt><dd>B 样条导数公式： P^{(k)}(t) = Σ P_i^{(k)} N_{i,p-k}(t)，其中 P_i^{(0)} = P_i， P_i^{(k)} = p·(P_i^{(k-1)} - P_{i-1}^{(k-1)}) / (t_{i+p-k+1} - t_i) </dd></dl>
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<a id="a020663a039886ef2b17009405e5db855"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#a020663a039886ef2b17009405e5db855">◆ </a></span>domain()</h2>
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<td class="memname">std::pair<double, double> vde::curves::BSplineCurve::domain </td>
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<td>(</td>
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<td class="paramname"></td><td>)</td>
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<td> const</td>
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<p>参数定义域 </p>
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<dl class="section return"><dt>返回</dt><dd>有效参数范围 [knots[p], knots[n+1]] </dd></dl>
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</div>
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</div>
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<a id="aa16013651a9b35a2f755c56b94436278"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#aa16013651a9b35a2f755c56b94436278">◆ </a></span>evaluate()</h2>
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<td class="memname">Point3D vde::curves::BSplineCurve::evaluate </td>
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<td>(</td>
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<td class="paramtype">double </td>
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<td class="paramname"><em>t</em></td><td>)</td>
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<td> const</td>
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</div><div class="memdoc">
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<p>在参数 t 处求值曲线点 </p>
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<dl class="params"><dt>参数</dt><dd>
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<table class="params">
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<tr><td class="paramname">t</td><td>参数值，必须在 <a class="el" href="classvde_1_1curves_1_1BSplineCurve.html#a020663a039886ef2b17009405e5db855" title="参数定义域">domain()</a> 范围内 </td></tr>
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</table>
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</dd>
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</dl>
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<dl class="section return"><dt>返回</dt><dd>曲线上对应 t 的点坐标 </dd></dl>
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<dl class="section note"><dt>注解</dt><dd>使用 de Boor（Cox-de Boor）递推算法 </dd></dl>
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</div>
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</div>
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<a id="af4e982a7104d5a7e1df0ec94539518f1"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#af4e982a7104d5a7e1df0ec94539518f1">◆ </a></span>find_span()</h2>
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<td class="memname">int vde::curves::BSplineCurve::find_span </td>
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<td>(</td>
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<td class="paramtype">double </td>
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<td class="paramname"><em>t</em></td><td>)</td>
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<td> const</td>
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</div><div class="memdoc">
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<p>查找 t 所在的节点区间 </p>
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<dl class="params"><dt>参数</dt><dd>
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<table class="params">
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<tr><td class="paramname">t</td><td>参数值 </td></tr>
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</table>
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</dd>
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</dl>
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<dl class="section return"><dt>返回</dt><dd>节点区间索引 i 满足 knots[i] ≤ t < knots[i+1] </dd></dl>
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<dl class="section note"><dt>注解</dt><dd>使用二分查找，O(log n)；求值和基函数的前置步骤 <div class="fragment"><div class="line"><span class="keywordtype">int</span> span = curve.find_span(0.5);</div>
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<div class="line"><span class="keyword">auto</span> N = curve.basis_functions(0.5, span);</div>
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</div><!-- fragment --> </dd></dl>
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<a id="ae2f04689aac05a5cc631daa2d4d2661b"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#ae2f04689aac05a5cc631daa2d4d2661b">◆ </a></span>knots()</h2>
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<td class="memname">const std::vector<double>& vde::curves::BSplineCurve::knots </td>
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<td>(</td>
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<td class="paramname"></td><td>)</td>
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<td> const</td>
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<td class="mlabels-right">
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<p>节点向量访问 </p>
|
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<dl class="section return"><dt>返回</dt><dd>节点向量只读引用 </dd></dl>
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<p class="definition">在文件 <a class="el" href="bspline__curve_8h_source.html">bspline_curve.h</a> 第 <a class="el" href="bspline__curve_8h_source.html#l00081">81</a> 行定义.</p>
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</div>
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</div>
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<hr/>该类的文档由以下文件生成:<ul>
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<li>include/vde/curves/<a class="el" href="bspline__curve_8h_source.html">bspline_curve.h</a></li>
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</ul>
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