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双三次贝塞尔曲面是由16个控制点定义的曲面#xff0c;通常表示为4x4矩阵。
曲面的公式如下#xff1a; p ( u , v ) ∑ i 0 3 ∑ j 0 3 P i , j B i , 3 ( u ) B j , 3 ( v ) , ( u , v ) ∈ [ 0 , 1 ] [ 0 , 1 ] p(u,v)\sum_{i0}^3\sum_{j0}…双三次贝塞尔曲线的定义
双三次贝塞尔曲面是由16个控制点定义的曲面通常表示为4x4矩阵。
曲面的公式如下 p ( u , v ) ∑ i 0 3 ∑ j 0 3 P i , j B i , 3 ( u ) B j , 3 ( v ) , ( u , v ) ∈ [ 0 , 1 ] × [ 0 , 1 ] p(u,v)\sum_{i0}^3\sum_{j0}^3P_{i,j}B_{i,3}(u)B_{j,3}(v),\\(u,v)\in[0,1]\times[0,1] p(u,v)i0∑3j0∑3Pi,jBi,3(u)Bj,3(v),(u,v)∈[0,1]×[0,1]
其中 P i , j , i , j 0 , 1 , 2 , 3 P_{i,j},i,j0,1,2,3 Pi,j,i,j0,1,2,3是曲面上的16控制点 B i , 3 ( u ) 和 B j , 3 ( v ) B_{i, 3} ( u )和 B_{j,3}( v ) Bi,3(u)和Bj,3(v)是三次Bernstein基函数定义如下 { B 0 , 3 ( u ) ( 1 − u ) 3 B 1 , 3 ( u ) 3 u ( 1 − u ) 2 B 2 , 3 ( u ) 3 u 2 ( 1 − u ) B 3 , 3 ( u ) u 3 { B 0 , 3 ( v ) ( 1 − v ) 3 B 1 , 3 ( v ) 3 v ( 1 − v ) 2 B 2 , 3 ( v ) 3 v 2 ( 1 − v ) B 3 , 3 ( v ) v 3 \begin{cases} B_{0,3}(u)(1-u)^3\\ B_{1,3}(u)3u(1-u)^2\\ B_{2,3}(u)3u^2(1-u)\\ B_{3,3}(u)u^3 \end{cases}\begin{cases} B_{0,3}(v)(1-v)^3\\ B_{1,3}(v)3v(1-v)^2\\ B_{2,3}(v)3v^2(1-v)\\ B_{3,3}(v)v^3 \end{cases} ⎩ ⎨ ⎧B0,3(u)(1−u)3B1,3(u)3u(1−u)2B2,3(u)3u2(1−u)B3,3(u)u3⎩ ⎨ ⎧B0,3(v)(1−v)3B1,3(v)3v(1−v)2B2,3(v)3v2(1−v)B3,3(v)v3 用矩阵表示公式如下 p ( u , v ) ( B 0 , 3 ( u ) B 1 , 3 ( u ) B 2 , 3 ( u ) B 3 , 3 ( u ) ) ( P 00 P 01 P 02 P 03 P 10 P 11 P 12 P 13 P 20 P 21 P 22 P 23 P 30 P 31 P 32 P 33 ) ( B 0 , 3 ( v ) B 0 , 1 ( v ) B 0 , 2 ( v ) B 0 , 3 ( v ) ) p(u,v)\begin{pmatrix}B_{0,3}(u)B_{1,3}(u)B_{2,3}(u)B_{3,3}(u) \end{pmatrix}\begin{pmatrix} P_{00}P_{01}P_{02}P_{03}\\ P_{10}P_{11}P_{12}P_{13}\\ P_{20}P_{21}P_{22}P_{23}\\ P_{30}P_{31}P_{32}P_{33}\end{pmatrix}\begin{pmatrix} B_{0,3}(v)B_{0,1}(v)B_{0,2}(v)B_{0,3}(v) \end{pmatrix} p(u,v)(B0,3(u)B1,3(u)B2,3(u)B3,3(u)) P00P10P20P30P01P11P21P31P02P12P22P32P03P13P23P33 (B0,3(v)B0,1(v)B0,2(v)B0,3(v)) 将Bernstein基数带入得到如下公式 p ( u , v ) ( u 3 u 2 u 1 ) ( − 1 3 − 3 1 3 − 6 3 0 − 3 3 0 0 1 0 0 0 ) ( P 00 P 01 P 02 P 03 P 10 P 11 P 12 P 13 P 20 P 21 P 22 P 23 P 30 P 31 P 32 P 33 ) ( − 1 3 − 3 1 3 − 6 3 0 − 3 3 0 0 1 0 0 0 ) ( v 3 v 2 v 1 ) p(u,v)\begin{pmatrix}u^3u^2u1\end{pmatrix}\begin{pmatrix} -13-31\\3-630\\-3300\\1000 \end{pmatrix}\begin{pmatrix} P_{00}P_{01}P_{02}P_{03}\\ P_{10}P_{11}P_{12}P_{13}\\ P_{20}P_{21}P_{22}P_{23}\\ P_{30}P_{31}P_{32}P_{33}\end{pmatrix}\begin{pmatrix} -13-31\\3-630\\-3300\\1000 \end{pmatrix}\begin{pmatrix}v^3\\v^2\\v\\1\end{pmatrix} p(u,v)(u3u2u1) −13−313−630−33001000 P00P10P20P30P01P11P21P31P02P12P22P32P03P13P23P33 −13−313−630−33001000 v3v2v1 令 U ( u 3 u 2 u 1 ) , V ( v 3 v 2 v 1 ) M ( − 1 3 − 3 1 3 − 6 3 0 − 3 3 0 0 1 0 0 0 ) , P ( P 00 P 01 P 02 P 03 P 10 P 11 P 12 P 13 P 20 P 21 P 22 P 23 P 30 P 31 P 32 P 33 ) 那么 p ( u , v ) U M P M T V T 令U\begin{pmatrix}u^3u^2u1\end{pmatrix},V\begin{pmatrix}v^3v^2v1\end{pmatrix}\\M\begin{pmatrix} -13-31\\3-630\\-3300\\1000 \end{pmatrix},P\begin{pmatrix} P_{00}P_{01}P_{02}P_{03}\\ P_{10}P_{11}P_{12}P_{13}\\ P_{20}P_{21}P_{22}P_{23}\\ P_{30}P_{31}P_{32}P_{33}\end{pmatrix} \\那么p(u,v)UMPM^TV^T 令U(u3u2u1),V(v3v2v1)M −13−313−630−33001000 ,P P00P10P20P30P01P11P21P31P02P12P22P32P03P13P23P33 那么p(u,v)UMPMTVT 对得到的这个式子进行编程就可以绘制出一个双三次贝塞尔曲线。
绘制双贝塞尔曲面
双三次贝塞尔曲面可以看着一个弯曲的四面体采用四叉树递归划分法进行细分递归细分次数足够时分割出来的子曲面近似为一个平面四边形这些平面四边形拼成了双三次贝塞尔曲线。 递归绘制细分曲面
递归函数
void CBezier::Recursion(CDC* pDC, int ReNumber, CMesh Mesh)
{if (0 ReNumber){Tessellation(Mesh);//细分曲面根据公式求坐标将uv点转换为xy点DrawQuadrilateral(pDC);//绘制小平面四边形return;}else{CDimension2 Mid (Mesh.m_BottomLeft Mesh.m_TopRight) / 2.0;CMesh SubMesh[4];//一分为四个//左下子长方形SubMesh[0].m_BottomLeft Mesh.m_BottomLeft;SubMesh[0].m_BottomRight CDimension2(Mid.m_u, Mesh.m_BottomLeft.m_v);SubMesh[0].m_TopRight CDimension2(Mid.m_u, Mid.m_v);SubMesh[0].m_TopLeft CDimension2(Mesh.m_BottomLeft.m_u, Mid.m_v);//右下子长方形SubMesh[1].m_BottomLeft SubMesh[0].m_BottomRight;SubMesh[1].m_BottomRight Mesh.m_BottomRight;SubMesh[1].m_TopRight CDimension2(Mesh.m_BottomRight.m_u, Mid.m_v);SubMesh[1].m_TopLeft SubMesh[0].m_TopRight;//右上子长方形SubMesh[2].m_BottomLeft SubMesh[1].m_TopLeft;SubMesh[2].m_BottomRight SubMesh[1].m_TopRight;SubMesh[2].m_TopRight Mesh.m_TopRight;SubMesh[2].m_TopLeft CDimension2(Mid.m_u, Mesh.m_TopRight.m_v);//左上子长方形SubMesh[3].m_BottomLeft SubMesh[0].m_TopLeft;SubMesh[3].m_BottomRight SubMesh[2].m_BottomLeft;SubMesh[3].m_TopRight SubMesh[2].m_TopLeft;SubMesh[3].m_TopLeft Mesh.m_TopLeft;Recursion(pDC, ReNumber - 1, SubMesh[0]);//递归绘制4个子曲面Recursion(pDC, ReNumber - 1, SubMesh[1]);Recursion(pDC, ReNumber - 1, SubMesh[2]);Recursion(pDC, ReNumber - 1, SubMesh[3]);}
}求细分曲面四个顶点的坐标
void CBezier::Tessellation(CMesh Mesh)
{double M[4][4];//系数矩阵MM[0][0] -1, M[0][1] 3, M[0][2] -3, M[0][3] 1;M[1][0] 3, M[1][1] -6, M[1][2] 3, M[1][3] 0;M[2][0] -3, M[2][1] 3, M[2][2] 0, M[2][3] 0;M[3][0] 1, M[3][1] 0, M[3][2] 0, M[3][3] 0;CPoint3 P3[4][4];//曲线计算用控制点数组for (int i 0; i 4; i)for (int j 0; j 4; j)P3[i][j] m_CtrPt[i][j];LeftMultiplyMatrix(M, P3);//系数矩阵左乘三维点矩阵TransposeMatrix(M);//计算转置矩阵RightMultiplyMatrix(P3, M);//系数矩阵右乘三维点矩阵double u0, u1, u2, u3, v0, v1, v2, v3;//u、v参数的幂double u[4] { Mesh.m_BottomLeft.m_u,Mesh.m_BottomRight.m_u ,Mesh.m_TopRight.m_u ,Mesh.m_TopLeft.m_u };double v[4] { Mesh.m_BottomLeft.m_v,Mesh.m_BottomRight.m_v ,Mesh.m_TopRight.m_v ,Mesh.m_TopLeft.m_v };for (int i 0; i 4; i){u3 pow(u[i], 3.0), u2 pow(u[i], 2.0), u1 u[i], u0 1;v3 pow(v[i], 3.0), v2 pow(v[i], 2.0), v1 v[i], v0 1;CPoint3 Pt (u3 * P3[0][0] u2 * P3[1][0] u1 * P3[2][0] u0 * P3[3][0]) * v3 (u3 * P3[0][1] u2 * P3[1][1] u1 * P3[2][1] u0 * P3[3][1]) * v2 (u3 * P3[0][2] u2 * P3[1][2] u1 * P3[2][2] u0 * P3[3][2]) * v1 (u3 * P3[0][3] u2 * P3[1][3] u1 * P3[2][3] u0 * P3[3][3]) * v0;m_QuadrPoint[i] Pt;}
}绘制细分曲面,就是知道了四个点绘制四边形
void CBezier::DrawQuadrilateral(CDC* pDC)
{CPoint2 ScreenPoint[4];//二维投影点for (int nPoint 0; nPoint 4; nPoint)ScreenPoint[nPoint] m_QuadrPoint[nPoint];//正交投影CPen NewPen, * pOldPen;NewPen.CreatePen(PS_SOLID, 2, RGB(255, 0, 0));pOldPen pDC-SelectObject(NewPen);pDC-MoveTo(ROUND(ScreenPoint[0].m_x), ROUND(ScreenPoint[0].m_y));pDC-LineTo(ROUND(ScreenPoint[1].m_x), ROUND(ScreenPoint[1].m_y));pDC-LineTo(ROUND(ScreenPoint[2].m_x), ROUND(ScreenPoint[2].m_y));pDC-LineTo(ROUND(ScreenPoint[3].m_x), ROUND(ScreenPoint[3].m_y));pDC-LineTo(ROUND(ScreenPoint[0].m_x), ROUND(ScreenPoint[0].m_y));pDC-SelectObject(pOldPen);NewPen.DeleteObject();
}第二种方法 在四根贝塞尔曲线p(t),q(t),r(t),s(t)上去相同的t根据定义可以得到四个点P,Q,R,S这四个点又可以画一个三次贝塞尔曲线那么在t从0递增至1的过程中就会产生一系列的三次贝塞尔曲线这一组曲线就构成了一个曲面。 需要项目代码的可以评论区留言或者私信
