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<title>自由曲线和曲面的表示方法</title>
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<font color="#FFFF00" size="5"><span style="mso-fareast-font-family: 宋体; mso-hansi-font-family: Times New Roman; layout-flow: vertical"><b>八
<a name="#自由曲线和曲面的表示方法">自由曲线和曲面的表示方法</a></b></span></font>
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<p align="left"><span style="mso-fareast-font-family: 宋体; mso-hansi-font-family: Times New Roman; layout-flow: vertical"><font size="4" color="#FFFF00"><b>Bezier</b></font></span></li>
<li>
<p align="left"><span style="mso-fareast-font-family: 宋体; mso-hansi-font-family: Times New Roman; layout-flow: vertical"><font size="4"><b>B_Spline</b></font></span></li>
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<p align="left"><font size="4"><b>Hermite,Coons</b></font></li>
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<p align="left"><b><font color="#FFFF00"><a name="#概念">1 概念</a></font></b></p>
<p align="left"><b>
能够用方程式描述的曲线，如二次曲线、曲面等。计算机可以用方程求出曲线、曲面上所有的点。而不能用方程式描述的曲线、曲面，计算机如何生成呢？</b></p>
<p align="left"><b>
曲线、曲面拟合方法：用一组离散的控制点近似地拟合曲线、曲面。</b></p>
<ul>
<li>
<p align="left"><b>插值方法：曲线、曲面通过控制点。</b></li>
<li>
<p align="left"><b>逼近方法：在某种准则下，曲线、曲面“接近”控制点。</b></li>
</ul>
<p align="left"><b>
<a name="#曲线曲面方程">平面上的曲线（显式）方程：y=f(x)</a></b></p>
<p align="left"><b>
空间上的曲线（显式）方程：z=f(x,y)</b></p>
<p align="left"><b>
空间上的曲线段（参数）方程：</b></p>
<p align="left"><b> x=X(t) y=Y(t) z=Z(t)
<font size="3">0<span style="mso-bidi-font-size: 12.0pt; font-family: 宋体; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">≦t≦1</span></font></b></p>
<p align="left"><b><font size="3"><span style="mso-bidi-font-size: 12.0pt; font-family: 宋体; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">
</span></font>空间上的曲面片（参数）方程：</b></p>
<p align="left"><b> x=X(u,w) y=Y(u,w) z=Z(u,w)
<font size="3">0<span style="mso-bidi-font-size: 12.0pt; font-family: 宋体; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">≦u≦1
</span>0<span style="mso-bidi-font-size: 12.0pt; font-family: 宋体; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">≦w≦1</span></font></b></p>
<p align="left"><b><font color="#FFFF00" size="3"><span style="mso-bidi-font-size: 12.0pt; font-family: 宋体; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA"> </span></font><font size="3"><span style="mso-bidi-font-size: 12.0pt; font-family: 宋体; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA"><font color="#FFFF00">
样条函数</font>：按照一定光滑性要求连接起来的分段多项式。</span></font></b></p>
<p align="left"><font size="3"><span style="mso-bidi-font-size: 12.0pt; font-family: 宋体; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA"><b>
<font color="#FFFF00">三次样条函数</font>：就是分段多项式为三次多项式。</b></span></font></p>
<p align="left"><font size="3"><span style="mso-bidi-font-size: 12.0pt; font-family: 宋体; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA"><b>
研究样条就是： </b></span></font></p>
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<li>
<p align="left"><font size="3"><span style="mso-bidi-font-size: 12.0pt; font-family: 宋体; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA"><b>确定一种划分（分段）</b></span></font></li>
<li>
<p align="left"><font size="3"><span style="mso-bidi-font-size: 12.0pt; font-family: 宋体; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA"><b>确定每个分段上的三次多项式</b></span></font></li>
<li>
<p align="left"><font size="3"><span style="mso-bidi-font-size: 12.0pt; font-family: 宋体; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA"><b>确定段与段之间的光滑连接关系</b></span></font></li>
</ul>
<p align="left"><font size="3" color="#FFFF00"><span style="mso-bidi-font-size: 12.0pt; font-family: 宋体; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA"><b>2
<a name="#三次Bezier多项式">三次Bezier多项式</a> </b></span></font></p>
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<p align="left"><span style="mso-text-raise: -14.0pt; mso-bidi-font-size: 12.0pt; font-family: Times New Roman; mso-fareast-font-family: 宋体; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA" lang="EN-US"><font size="3"><b>其中Pi(i=0,1,2,3)为控制点，</b></font></span></p>
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<p align="left"><b>三次Bezier曲线是用三次Bezier多项式表示段内的曲线P(t)。</b></p>
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