pnormallinemethod.htm
来自「电脑图学（Computer Graphics）是资料结构、演算法与数学的应用」· HTM 代码 · 共 202 行
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<h1><a href="ComputerGraphics.htm">Computer Graphics: 平面法线判定法</a></h1>
法线向量判定法适用于无凹陷的凸多面体，例如正多面体或水晶球，其原理为求出每个面的法线向量，如果法线向量的Z 分量大于0（即面朝向我们），则该面为可视，如果法线向量的Z分量小于0，则看不到那个面，则这是个隐藏面，所以无需绘制。 <br>
<img style="width: 317px; height: 136px;" alt="" src="images/pNormalLineMethod-1.jpg"><br>
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法线向量的选择是由面上的三个顶点来判定，三个顶点的选择是依面的逆时针方向的来选出，然后求出这三个顶点所成的两向量之外积即为法线向量，通常我们取单位法线向量（normalize后的外积），下面这个程式片段可以作为外积计算的参考：<br>
<div style="margin-left: 40px;"><span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> // 取得外积 </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> public static Point3D op(Point3D p0, </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;">
Point3D p1, </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;">
Point3D p2) { </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> p1.x = p1.x-p0.x; </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> p1.y = p1.y-p0.y; </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> p1.z = p1.z-p0.z; </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> p2.x = p2.x-p0.x; </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> p2.y = p2.y-p0.y; </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> p2.z = p2.z-p0.z; </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> double x = p1.y*p2.z - p1.z*p2.y; </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> double y = p1.z*p2.x - p1.x*p2.z; </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> double z = p1.x*p2.y - p1.y*p2.x; </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> return new Point3D(x, y, z); </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> } </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> // 计算单位向量 </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> public static Point3D ev(Point3D p) { </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> double d = Math.sqrt(p.x*p.x + p.y*p.y + p.z*p.z); </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> if(d > 0.0) </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> return new Point3D(p.x/d, p.y/d, p.z/d); </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> else </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> return p; </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> } </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> // 取得nomalize后的外积 </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> public static Point3D nop(Point3D p0, </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;">
Point3D p1, </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;">
Point3D p2) { </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> return ev(op(p0, p1, p2)); </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> }</span><br>
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配合外积，可以利用下面这个程式片段进行隐藏面的判断与非隐藏面的绘制，其中vet[]是顶点阵列，NFc?表示凸面体的平面数量，Nvt表示一个平面上的顶点数 ，ord[][]表示顶点索引阵列：<br>
<div style="margin-left: 40px;"><span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> int N = 0; </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> Point[][] pvt = new Point[NFc][Nvt]; </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> int[] n = new int[Nvt]; </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> Point3D np; </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> for(int i = 0; i < NFc; i++) { </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> for(int j = 0; j < Nvt; j++) </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> // 取同一个平面的顶点</span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> n[j] = ord[i][j]; </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> // 计算nomalize后的外积 </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> np = nop(vet[n[0]], vet[n[1]], vet[n[2]]); </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> // 如果非隐藏面 </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> if(np.z > 0) { </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> // 储存顶点 </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> pvt[N][j] = new Point(vet[n[j]].x, vet[n[j]].y); </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> // 可绘制的面+1 </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> N++; </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> } </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> } </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> // 画实心多边形 </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> for(int i = 0; i < N; i++) </span><br style="font-weight: bold; font-family: Courier New,Courier,monospace;">
<span style="font-weight: bold; font-family: Courier New,Courier,monospace;"> g.fillPolygon(pvt[i], Nvt);</span><br>
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上面这个程式只是个演算的参考，实作时得依您所使用的语言与API作调整。<br>
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