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C++實(shí)現(xiàn)移動(dòng)立方體示例講解

 更新時(shí)間：2022年12月21日 14:11:17   作者：Leonban  
這篇文章主要介紹了C++實(shí)現(xiàn)移動(dòng)立方體，文中通過(guò)示例代碼介紹的非常詳細(xì)，對(duì)大家的學(xué)習(xí)或者工作具有一定的參考學(xué)習(xí)價(jià)值，需要的朋友們下面隨著小編來(lái)一起學(xué)習(xí)吧

本文描述了一個(gè)創(chuàng)建三維標(biāo)量場(chǎng)等值面多邊形曲面表示的算法。這類(lèi)問(wèn)題的一個(gè)常見(jiàn)名稱(chēng)是所謂的“移動(dòng)立方體”算法。它結(jié)合了簡(jiǎn)單和高速，因?yàn)樗鼛缀跬耆糜诓檎冶怼?/p>

這種技術(shù)有很多應(yīng)用，兩個(gè)非常常見(jiàn)的是:

從醫(yī)學(xué)體積數(shù)據(jù)集重建表面。例如，MRI掃描在常規(guī)3d網(wǎng)格的頂點(diǎn)處產(chǎn)生一個(gè)3d體積的樣本。

創(chuàng)建數(shù)學(xué)標(biāo)量場(chǎng)的三維輪廓。在這種情況下，函數(shù)是已知的，但在一個(gè)常規(guī)的3D網(wǎng)格的頂點(diǎn)采樣。

1.解決方案

其基本問(wèn)題是通過(guò)在矩形三維網(wǎng)格上采樣的標(biāo)量場(chǎng)形成一個(gè)面逼近等值面。給定一個(gè)由頂點(diǎn)和每個(gè)頂點(diǎn)的標(biāo)量值定義的網(wǎng)格單元，有必要?jiǎng)?chuàng)建通過(guò)該網(wǎng)格單元最能代表等值面的平面facet。等值面可能不會(huì)穿過(guò)網(wǎng)格單元格，它可能切斷任何一個(gè)頂點(diǎn)，或者它可能通過(guò)許多更復(fù)雜的方式中的任何一種。每一種可能性都將以等值面以上或以下值的頂點(diǎn)數(shù)來(lái)表征。如果一個(gè)頂點(diǎn)在等值面上，而另一個(gè)相鄰頂點(diǎn)在等值面上，那么我們就知道等值面上切割這兩個(gè)頂點(diǎn)之間的邊。它切割邊緣的位置將被線性插值，兩個(gè)頂點(diǎn)之間的長(zhǎng)度之比將與等值面值與網(wǎng)格單元格頂點(diǎn)處值之比相同。

算法中使用的頂點(diǎn)和邊的索引約定如下所示

例如，如果頂點(diǎn)3的值低于等值面值，而所有其他頂點(diǎn)的值都高于等值面值，那么我們將創(chuàng)建一個(gè)三角形面，它穿過(guò)邊2、3和11。三角形面的頂點(diǎn)的確切位置分別取決于等值面值與頂點(diǎn)3-2、3-0、3-7上的值的關(guān)系。

使算法“困難”的是大量的可能組合(256個(gè))，以及需要為每個(gè)解決方案導(dǎo)出一致的facet組合，以便相鄰網(wǎng)格單元的facet正確地連接在一起。

算法的第一部分使用一個(gè)表(edgeTable)，它將等值面下的頂點(diǎn)映射到相交的邊緣。一個(gè)8位的索引是由每個(gè)位對(duì)應(yīng)一個(gè)頂點(diǎn)形成的。

cubeindex = 0;

if (grid.val[0] < isolevel) cubeindex |= 1;

if (grid.val[1] < isolevel) cubeindex |= 2;

if (grid.val[2] < isolevel) cubeindex |= 4;

if (grid.val[3] < isolevel) cubeindex |= 8;

if (grid.val[4] < isolevel) cubeindex |= 16;

if (grid.val[5] < isolevel) cubeindex |= 32;

if (grid.val[6] < isolevel) cubeindex |= 64;

if (grid.val[7] < isolevel) cubeindex |= 128;

查找邊緣表返回一個(gè)12位的數(shù)字，每一位對(duì)應(yīng)一條邊，0表示該邊沒(méi)有被等值面切割，1表示該邊被等值面切割。如果沒(méi)有任何邊被切割，則表返回0，當(dāng)cubeindex為0(等值面以下的所有頂點(diǎn))或0xff(等值面以上的所有頂點(diǎn))時(shí)就會(huì)發(fā)生這種情況。

使用前面的例子，只有頂點(diǎn)3低于等值面，cubeindex將等于0000 1000或8。edgeTable[8] = 1000 0000 1100。這意味著邊2,3和11與等值面相交。

交點(diǎn)現(xiàn)在由線性插值計(jì)算。如果P1和P2是切邊的兩個(gè)頂點(diǎn)，V1和V2是每個(gè)頂點(diǎn)的標(biāo)量值，那么交點(diǎn)P是

P = P1 +(等值- V1) (P2 - P1) / (V2 - V1)

算法的最后一部分涉及到從等值面與網(wǎng)格單元的邊緣相交的位置形成正確的facet。同樣使用了一個(gè)表(triTable)，這次使用相同的立方體索引，但允許查找頂點(diǎn)序列，因?yàn)樵诰W(wǎng)格單元中表示等值面需要多少三角形面就需要多少三角形面。最多需要5個(gè)三角形面。

回到我們的例子，在前面的步驟中，我們計(jì)算了沿邊2、3和11的交點(diǎn)。triTable中的第8個(gè)元素是

{3、11、2、1,1,1,1,1,1,1,1,1,1,1,1,1},

這是一個(gè)特別簡(jiǎn)單的示例，請(qǐng)確保facet組合對(duì)于表中的許多情況不是那么明顯。

2.舉例子

假設(shè)頂點(diǎn)0和3在等值面以下。Cubeindex將是0000 1001 == 9。進(jìn)入egdeTable的第9項(xiàng)是905hex == 1001 0000 0101，這意味著邊11、8、2和0被切割，所以我們計(jì)算出等面與這些邊相交的頂點(diǎn)。

接下來(lái)，triitable中的9是0,11,2,8,11,0。這對(duì)應(yīng)于兩個(gè)三角形面片，一個(gè)在邊0,11和2的交點(diǎn)之間。另一個(gè)在沿邊8 11和0的交點(diǎn)之間。

3.網(wǎng)格分辨率

當(dāng)對(duì)一個(gè)值已知或可以在空間中任意位置插值的字段進(jìn)行多邊形化時(shí)，一個(gè)非常理想的控制是采樣網(wǎng)格的分辨率。這允許根據(jù)所需的平滑度和/或顯示表面的處理能力生成等面的過(guò)程或精細(xì)近似。

4.源代碼

基于OpenGL繪制（VS可直接運(yùn)行）

#include "stdio.h"
#include "math.h"
//This program requires the OpenGL and GLUT libraries
// You can obtain them for free from http://www.opengl.org
#include "GL/glut.h"
struct GLvector
{
        GLfloat fX;
        GLfloat fY;
        GLfloat fZ;     
};
//These tables are used so that everything can be done in little loops that you can look at all at once
// rather than in pages and pages of unrolled code.
//a2fVertexOffset lists the positions, relative to vertex0, of each of the 8 vertices of a cube
static const GLfloat a2fVertexOffset[8][3] =
{
        {0.0, 0.0, 0.0},{1.0, 0.0, 0.0},{1.0, 1.0, 0.0},{0.0, 1.0, 0.0},
        {0.0, 0.0, 1.0},{1.0, 0.0, 1.0},{1.0, 1.0, 1.0},{0.0, 1.0, 1.0}
};
//a2iEdgeConnection lists the index of the endpoint vertices for each of the 12 edges of the cube
static const GLint a2iEdgeConnection[12][2] = 
{
        {0,1}, {1,2}, {2,3}, {3,0},
        {4,5}, {5,6}, {6,7}, {7,4},
        {0,4}, {1,5}, {2,6}, {3,7}
};
//a2fEdgeDirection lists the direction vector (vertex1-vertex0) for each edge in the cube
static const GLfloat a2fEdgeDirection[12][3] =
{
        {1.0, 0.0, 0.0},{0.0, 1.0, 0.0},{-1.0, 0.0, 0.0},{0.0, -1.0, 0.0},
        {1.0, 0.0, 0.0},{0.0, 1.0, 0.0},{-1.0, 0.0, 0.0},{0.0, -1.0, 0.0},
        {0.0, 0.0, 1.0},{0.0, 0.0, 1.0},{ 0.0, 0.0, 1.0},{0.0,  0.0, 1.0}
};
//a2iTetrahedronEdgeConnection lists the index of the endpoint vertices for each of the 6 edges of the tetrahedron
static const GLint a2iTetrahedronEdgeConnection[6][2] =
{
        {0,1},  {1,2},  {2,0},  {0,3},  {1,3},  {2,3}
};
//a2iTetrahedronEdgeConnection lists the index of verticies from a cube 
// that made up each of the six tetrahedrons within the cube
static const GLint a2iTetrahedronsInACube[6][4] =
{
        {0,5,1,6},
        {0,1,2,6},
        {0,2,3,6},
        {0,3,7,6},
        {0,7,4,6},
        {0,4,5,6},
};
static const GLfloat afAmbientWhite [] = {0.25, 0.25, 0.25, 1.00}; 
static const GLfloat afAmbientRed   [] = {0.25, 0.00, 0.00, 1.00}; 
static const GLfloat afAmbientGreen [] = {0.00, 0.25, 0.00, 1.00}; 
static const GLfloat afAmbientBlue  [] = {0.00, 0.00, 0.25, 1.00}; 
static const GLfloat afDiffuseWhite [] = {0.75, 0.75, 0.75, 1.00}; 
static const GLfloat afDiffuseRed   [] = {0.75, 0.00, 0.00, 1.00}; 
static const GLfloat afDiffuseGreen [] = {0.00, 0.75, 0.00, 1.00}; 
static const GLfloat afDiffuseBlue  [] = {0.00, 0.00, 0.75, 1.00}; 
static const GLfloat afSpecularWhite[] = {1.00, 1.00, 1.00, 1.00}; 
static const GLfloat afSpecularRed  [] = {1.00, 0.25, 0.25, 1.00}; 
static const GLfloat afSpecularGreen[] = {0.25, 1.00, 0.25, 1.00}; 
static const GLfloat afSpecularBlue [] = {0.25, 0.25, 1.00, 1.00}; 
GLenum    ePolygonMode = GL_LINE;
GLint     iDataSetSize = 26;
GLfloat   fStepSize = 1.0/iDataSetSize;
GLfloat   fTargetValue = 48.0;
GLfloat   fTime = 0.0;
GLvector  sSourcePoint[3];
GLboolean bSpin = true;
GLboolean bMove = true;
GLboolean bLight = true;
void vIdle();
void vDrawScene(); 
void vResize(GLsizei, GLsizei);
void vKeyboard(unsigned char cKey, int iX, int iY);
void vSpecial(int iKey, int iX, int iY);
GLvoid vPrintHelp();
GLvoid vSetTime(GLfloat fTime);
GLfloat fSample1(GLfloat fX, GLfloat fY, GLfloat fZ);
GLfloat fSample2(GLfloat fX, GLfloat fY, GLfloat fZ);
GLfloat fSample3(GLfloat fX, GLfloat fY, GLfloat fZ);
GLfloat (*fSample)(GLfloat fX, GLfloat fY, GLfloat fZ) = fSample1;
GLvoid vMarchingCubes();
GLvoid vMarchCube1(GLfloat fX, GLfloat fY, GLfloat fZ, GLfloat fScale);
GLvoid vMarchCube2(GLfloat fX, GLfloat fY, GLfloat fZ, GLfloat fScale);
GLvoid (*vMarchCube)(GLfloat fX, GLfloat fY, GLfloat fZ, GLfloat fScale) = vMarchCube1;
void main(int argc, char **argv) 
{ 
        GLfloat afPropertiesAmbient [] = {0.50, 0.50, 0.50, 1.00}; 
        GLfloat afPropertiesDiffuse [] = {0.75, 0.75, 0.75, 1.00}; 
        GLfloat afPropertiesSpecular[] = {1.00, 1.00, 1.00, 1.00}; 
        GLsizei iWidth = 640.0; 
        GLsizei iHeight = 480.0; 
        glutInit(&argc, argv);
        glutInitWindowPosition( 0, 0);
        glutInitWindowSize(iWidth, iHeight);
        glutInitDisplayMode( GLUT_RGB | GLUT_DEPTH | GLUT_DOUBLE );
        glutCreateWindow( "Marching Cubes" );
        glutDisplayFunc( vDrawScene );
        glutIdleFunc( vIdle );
        glutReshapeFunc( vResize );
        glutKeyboardFunc( vKeyboard );
        glutSpecialFunc( vSpecial );
        glClearColor( 0.0, 0.0, 0.0, 1.0 ); 
        glClearDepth( 1.0 ); 
        glEnable(GL_DEPTH_TEST); 
        glEnable(GL_LIGHTING);
        glPolygonMode(GL_FRONT_AND_BACK, ePolygonMode);
        glLightfv( GL_LIGHT0, GL_AMBIENT,  afPropertiesAmbient); 
        glLightfv( GL_LIGHT0, GL_DIFFUSE,  afPropertiesDiffuse); 
        glLightfv( GL_LIGHT0, GL_SPECULAR, afPropertiesSpecular); 
        glLightModelf(GL_LIGHT_MODEL_TWO_SIDE, 1.0); 
        glEnable( GL_LIGHT0 ); 
        glMaterialfv(GL_BACK,  GL_AMBIENT,   afAmbientGreen); 
        glMaterialfv(GL_BACK,  GL_DIFFUSE,   afDiffuseGreen); 
        glMaterialfv(GL_FRONT, GL_AMBIENT,   afAmbientBlue); 
        glMaterialfv(GL_FRONT, GL_DIFFUSE,   afDiffuseBlue); 
        glMaterialfv(GL_FRONT, GL_SPECULAR,  afSpecularWhite); 
        glMaterialf( GL_FRONT, GL_SHININESS, 25.0); 
        vResize(iWidth, iHeight); 
        vPrintHelp();
        glutMainLoop(); 
}
GLvoid vPrintHelp()
{
        printf("Marching Cubes Example by Cory Bloyd (dejaspaminacan@my-deja.com)\n\n");
        printf("+/-  increase/decrease sample density\n");
        printf("PageUp/PageDown  increase/decrease surface value\n");
        printf("s  change sample function\n");
        printf("c  toggle marching cubes / marching tetrahedrons\n");
        printf("w  wireframe on/off\n");
        printf("l  toggle lighting / color-by-normal\n");
        printf("Home  spin scene on/off\n");
        printf("End  source point animation on/off\n");
}
void vResize( GLsizei iWidth, GLsizei iHeight ) 
{ 
        GLfloat fAspect, fHalfWorldSize = (1.4142135623730950488016887242097/2); 
        glViewport( 0, 0, iWidth, iHeight ); 
        glMatrixMode (GL_PROJECTION);
        glLoadIdentity ();
        if(iWidth <= iHeight)
        {
                fAspect = (GLfloat)iHeight / (GLfloat)iWidth;
                glOrtho(-fHalfWorldSize, fHalfWorldSize, -fHalfWorldSize*fAspect,
                        fHalfWorldSize*fAspect, -10*fHalfWorldSize, 10*fHalfWorldSize);
        }
        else
        {
                fAspect = (GLfloat)iWidth / (GLfloat)iHeight; 
                glOrtho(-fHalfWorldSize*fAspect, fHalfWorldSize*fAspect, -fHalfWorldSize,
                        fHalfWorldSize, -10*fHalfWorldSize, 10*fHalfWorldSize);
        }
        glMatrixMode( GL_MODELVIEW );
}
void vKeyboard(unsigned char cKey, int iX, int iY)
{
        switch(cKey)
        {
                case 'w' :
                {
                        if(ePolygonMode == GL_LINE)
                        {
                                ePolygonMode = GL_FILL;
                        }
                        else
                        {
                                ePolygonMode = GL_LINE;
                        }
                        glPolygonMode(GL_FRONT_AND_BACK, ePolygonMode);
                } break;
                case '+' :
                case '=' :
                {
                        ++iDataSetSize;
                        fStepSize = 1.0/iDataSetSize;
                } break;
                case '-' :
                {
                        if(iDataSetSize > 1)
                        {
                                --iDataSetSize;
                                fStepSize = 1.0/iDataSetSize;
                        }
                } break;
                case 'c' :
                {
                        if(vMarchCube == vMarchCube1)
                        {
                                vMarchCube = vMarchCube1;//Use Marching Tetrahedrons
                        }
                        else
                        {
                                vMarchCube = vMarchCube1;//Use Marching Cubes
                        }
                } break;
                case 's' :
                {
                        if(fSample == fSample1)
                        {
                                fSample = fSample1;
                        }
                        else if(fSample == fSample2)
                        {
                                fSample = fSample1;
                        }
                        else
                        {
                                fSample = fSample1;
                        }
                } break;
                case 'l' :
                {
                        if(bLight)
                        {
                                glDisable(GL_LIGHTING);//use vertex colors
                        }
                        else
                        {
                                glEnable(GL_LIGHTING);//use lit material color
                        }
                        bLight = !bLight;
                };
        }
}
void vSpecial(int iKey, int iX, int iY)
{
        switch(iKey)
        {
                case GLUT_KEY_PAGE_UP :
                {
                        if(fTargetValue < 1000.0)
                        {
                                fTargetValue *= 1.1;
                        }
                } break;
                case GLUT_KEY_PAGE_DOWN :
                {
                        if(fTargetValue > 1.0)
                        {
                                fTargetValue /= 1.1;
                        }
                } break;
                case GLUT_KEY_HOME :
                {
                        bSpin = !bSpin;
                } break;
                case GLUT_KEY_END :
                {
                        bMove = !bMove;
                } break;
        }
}
void vIdle()
{
        glutPostRedisplay();
}
void vDrawScene() 
{ 
        static GLfloat fPitch = 0.0;
        static GLfloat fYaw   = 0.0;
        static GLfloat fTime = 0.0;
        glClear( GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT ); 
        glPushMatrix(); 
        if(bSpin)
        {
                fPitch += 4.0;
                fYaw   += 2.5;
        }
        if(bMove)
        {
                fTime  += 0.025;
        }
        vSetTime(fTime);
        glTranslatef(0.0, 0.0, -1.0);  
        glRotatef( -fPitch, 1.0, 0.0, 0.0);
        glRotatef(     0.0, 0.0, 1.0, 0.0);
        glRotatef(    fYaw, 0.0, 0.0, 1.0);
        glPushAttrib(GL_LIGHTING_BIT);
                glDisable(GL_LIGHTING);
                glColor3f(1.0, 1.0, 1.0);
                glutWireCube(1.0); 
        glPopAttrib(); 
        glPushMatrix(); 
        glTranslatef(-0.5, -0.5, -0.5);
        glBegin(GL_TRIANGLES);
                vMarchingCubes();
        glEnd();
        glPopMatrix(); 
        glPopMatrix(); 
        glutSwapBuffers(); 
}
//fGetOffset finds the approximate point of intersection of the surface
// between two points with the values fValue1 and fValue2
GLfloat fGetOffset(GLfloat fValue1, GLfloat fValue2, GLfloat fValueDesired)
{
        GLdouble fDelta = fValue2 - fValue1;
        if(fDelta == 0.0)
        {
                return 0.5;
        }
        return (fValueDesired - fValue1)/fDelta;
}
//vGetColor generates a color from a given position and normal of a point
GLvoid vGetColor(GLvector &rfColor, GLvector &rfPosition, GLvector &rfNormal)
{
        GLfloat fX = rfNormal.fX;
        GLfloat fY = rfNormal.fY;
        GLfloat fZ = rfNormal.fZ;
        rfColor.fX = (fX > 0.0 ? fX : 0.0) + (fY < 0.0 ? -0.5*fY : 0.0) + (fZ < 0.0 ? -0.5*fZ : 0.0);
        rfColor.fY = (fY > 0.0 ? fY : 0.0) + (fZ < 0.0 ? -0.5*fZ : 0.0) + (fX < 0.0 ? -0.5*fX : 0.0);
        rfColor.fZ = (fZ > 0.0 ? fZ : 0.0) + (fX < 0.0 ? -0.5*fX : 0.0) + (fY < 0.0 ? -0.5*fY : 0.0);
}
GLvoid vNormalizeVector(GLvector &rfVectorResult, GLvector &rfVectorSource)
{
        GLfloat fOldLength;
        GLfloat fScale;
        fOldLength = sqrtf( (rfVectorSource.fX * rfVectorSource.fX) +
                            (rfVectorSource.fY * rfVectorSource.fY) +
                            (rfVectorSource.fZ * rfVectorSource.fZ) );
        if(fOldLength == 0.0)
        {
                rfVectorResult.fX = rfVectorSource.fX;
                rfVectorResult.fY = rfVectorSource.fY;
                rfVectorResult.fZ = rfVectorSource.fZ;
        }
        else
        {
                fScale = 1.0/fOldLength;
                rfVectorResult.fX = rfVectorSource.fX*fScale;
                rfVectorResult.fY = rfVectorSource.fY*fScale;
                rfVectorResult.fZ = rfVectorSource.fZ*fScale;
        }
}
//Generate a sample data set.  fSample1(), fSample2() and fSample3() define three scalar fields whose
// values vary by the X,Y and Z coordinates and by the fTime value set by vSetTime()
GLvoid vSetTime(GLfloat fNewTime)
{
        GLfloat fOffset;
        GLint iSourceNum;
        for(iSourceNum = 0; iSourceNum < 3; iSourceNum++)
        {
                sSourcePoint[iSourceNum].fX = 0.5;
                sSourcePoint[iSourceNum].fY = 0.5;
                sSourcePoint[iSourceNum].fZ = 0.5;
        }
        fTime = fNewTime;
        fOffset = 1.0 + sinf(fTime);
        sSourcePoint[0].fX *= fOffset;
        sSourcePoint[1].fY *= fOffset;
        sSourcePoint[2].fZ *= fOffset;
}
//fSample1 finds the distance of (fX, fY, fZ) from three moving points
GLfloat fSample3(GLfloat fX, GLfloat fY, GLfloat fZ)
{
        GLdouble fResult = 0.0;
        GLdouble fDx, fDy, fDz;
        fDx = fX - sSourcePoint[0].fX;
        fDy = fY - sSourcePoint[0].fY;
        fDz = fZ - sSourcePoint[0].fZ;
        fResult += 0.5/(fDx*fDx + fDy*fDy + fDz*fDz);
        fDx = fX - sSourcePoint[1].fX;
        fDy = fY - sSourcePoint[1].fY;
        fDz = fZ - sSourcePoint[1].fZ;
        fResult += 1.0/(fDx*fDx + fDy*fDy + fDz*fDz);
        fDx = fX - sSourcePoint[2].fX;
        fDy = fY - sSourcePoint[2].fY;
        fDz = fZ - sSourcePoint[2].fZ;
        fResult += 1.5/(fDx*fDx + fDy*fDy + fDz*fDz);
        return fResult;
}
//fSample2 finds the distance of (fX, fY, fZ) from three moving lines
GLfloat fSample2(GLfloat fX, GLfloat fY, GLfloat fZ)
{
        GLdouble fResult = 0.0;
        GLdouble fDx, fDy, fDz;
        fDx = fX - sSourcePoint[0].fX;
        fDy = fY - sSourcePoint[0].fY;
        fResult += 0.5/(fDx*fDx + fDy*fDy);
        fDx = fX - sSourcePoint[1].fX;
        fDz = fZ - sSourcePoint[1].fZ;
        fResult += 0.75/(fDx*fDx + fDz*fDz);
        fDy = fY - sSourcePoint[2].fY;
        fDz = fZ - sSourcePoint[2].fZ;
        fResult += 1.0/(fDy*fDy + fDz*fDz);
        return fResult;
}
//fSample2 defines a height field by plugging the distance from the center into the sin and cos functions
GLfloat fSample1(GLfloat fX, GLfloat fY, GLfloat fZ)
{
        GLfloat fHeight = ((sqrt((0.5-fX)*(0.5-fX) + (0.5-fY)*(0.5-fY))) +48);
        GLdouble fResult = (fHeight-fZ);
        return fResult;
}
//vGetNormal() finds the gradient of the scalar field at a point
//This gradient can be used as a very accurate vertx normal for lighting calculations
GLvoid vGetNormal(GLvector &rfNormal, GLfloat fX, GLfloat fY, GLfloat fZ)
{
        rfNormal.fX = fSample(fX-0.01, fY, fZ) - fSample(fX+0.01, fY, fZ);
        rfNormal.fY = fSample(fX, fY-0.01, fZ) - fSample(fX, fY+0.01, fZ);
        rfNormal.fZ = fSample(fX, fY, fZ-0.01) - fSample(fX, fY, fZ+0.01);
        vNormalizeVector(rfNormal, rfNormal);
}
//vMarchCube1 performs the Marching Cubes algorithm on a single cube
GLvoid vMarchCube1(GLfloat fX, GLfloat fY, GLfloat fZ, GLfloat fScale)
{
        extern GLint aiCubeEdgeFlags[256];
        extern GLint a2iTriangleConnectionTable[256][16];
        GLint iCorner, iVertex, iVertexTest, iEdge, iTriangle, iFlagIndex, iEdgeFlags;
        GLfloat fOffset;
        GLvector sColor;
        GLfloat afCubeValue[8];
        GLvector asEdgeVertex[12];
        GLvector asEdgeNorm[12];
        //Make a local copy of the values at the cube's corners
        for(iVertex = 0; iVertex < 8; iVertex++)
        {
                afCubeValue[iVertex] = fSample(fX + a2fVertexOffset[iVertex][0]*fScale,
                                                   fY + a2fVertexOffset[iVertex][1]*fScale,
                                                   fZ + a2fVertexOffset[iVertex][2]*fScale);
        }
        //Find which vertices are inside of the surface and which are outside
        iFlagIndex = 0;
        for(iVertexTest = 0; iVertexTest < 8; iVertexTest++)
        {
                if(afCubeValue[iVertexTest] <= fTargetValue) 
                        iFlagIndex |= 1<<iVertexTest;
        }
        //Find which edges are intersected by the surface
        iEdgeFlags = aiCubeEdgeFlags[iFlagIndex];
        //If the cube is entirely inside or outside of the surface, then there will be no intersections
        if(iEdgeFlags == 0) 
        {
                return;
        }
        //Find the point of intersection of the surface with each edge
        //Then find the normal to the surface at those points
        for(iEdge = 0; iEdge < 12; iEdge++)
        {
                //if there is an intersection on this edge
                if(iEdgeFlags & (1<<iEdge))
                {
                        fOffset = fGetOffset(afCubeValue[ a2iEdgeConnection[iEdge][0] ], 
                                                     afCubeValue[ a2iEdgeConnection[iEdge][1] ], fTargetValue);
                        asEdgeVertex[iEdge].fX = fX + (a2fVertexOffset[ a2iEdgeConnection[iEdge][0] ][0]  +  fOffset * a2fEdgeDirection[iEdge][0]) * fScale;
                        asEdgeVertex[iEdge].fY = fY + (a2fVertexOffset[ a2iEdgeConnection[iEdge][0] ][1]  +  fOffset * a2fEdgeDirection[iEdge][1]) * fScale;
                        asEdgeVertex[iEdge].fZ = fZ + (a2fVertexOffset[ a2iEdgeConnection[iEdge][0] ][2]  +  fOffset * a2fEdgeDirection[iEdge][2]) * fScale;
                        vGetNormal(asEdgeNorm[iEdge], asEdgeVertex[iEdge].fX, asEdgeVertex[iEdge].fY, asEdgeVertex[iEdge].fZ);
                }
        }
        //Draw the triangles that were found.  There can be up to five per cube
        for(iTriangle = 0; iTriangle < 5; iTriangle++)
        {
                if(a2iTriangleConnectionTable[iFlagIndex][3*iTriangle] < 0)
                        break;
                for(iCorner = 0; iCorner < 3; iCorner++)
                {
                        iVertex = a2iTriangleConnectionTable[iFlagIndex][3*iTriangle+iCorner];
                        vGetColor(sColor, asEdgeVertex[iVertex], asEdgeNorm[iVertex]);
                        glColor3f(sColor.fX, sColor.fY, sColor.fZ);
                        glNormal3f(asEdgeNorm[iVertex].fX,   asEdgeNorm[iVertex].fY,   asEdgeNorm[iVertex].fZ);
                        glVertex3f(asEdgeVertex[iVertex].fX, asEdgeVertex[iVertex].fY, asEdgeVertex[iVertex].fZ);
                }
        }
}
//vMarchTetrahedron performs the Marching Tetrahedrons algorithm on a single tetrahedron
GLvoid vMarchTetrahedron(GLvector *pasTetrahedronPosition, GLfloat *pafTetrahedronValue)
{
        extern GLint aiTetrahedronEdgeFlags[16];
        extern GLint a2iTetrahedronTriangles[16][7];
        GLint iEdge, iVert0, iVert1, iEdgeFlags, iTriangle, iCorner, iVertex, iFlagIndex = 0;
        GLfloat fOffset, fInvOffset, fValue = 0.0;
        GLvector asEdgeVertex[6];
        GLvector asEdgeNorm[6];
        GLvector sColor;
        //Find which vertices are inside of the surface and which are outside
        for(iVertex = 0; iVertex < 4; iVertex++)
        {
                if(pafTetrahedronValue[iVertex] <= fTargetValue) 
                        iFlagIndex |= 1<<iVertex;
        }
        //Find which edges are intersected by the surface
        iEdgeFlags = aiTetrahedronEdgeFlags[iFlagIndex];
        //If the tetrahedron is entirely inside or outside of the surface, then there will be no intersections
        if(iEdgeFlags == 0)
        {
                return;
        }
        //Find the point of intersection of the surface with each edge
        // Then find the normal to the surface at those points
        for(iEdge = 0; iEdge < 6; iEdge++)
        {
                //if there is an intersection on this edge
                if(iEdgeFlags & (1<<iEdge))
                {
                        iVert0 = a2iTetrahedronEdgeConnection[iEdge][0];
                        iVert1 = a2iTetrahedronEdgeConnection[iEdge][1];
                        fOffset = fGetOffset(pafTetrahedronValue[iVert0], pafTetrahedronValue[iVert1], fTargetValue);
                        fInvOffset = 1.0 - fOffset;
                        asEdgeVertex[iEdge].fX = fInvOffset*pasTetrahedronPosition[iVert0].fX  +  fOffset*pasTetrahedronPosition[iVert1].fX;
                        asEdgeVertex[iEdge].fY = fInvOffset*pasTetrahedronPosition[iVert0].fY  +  fOffset*pasTetrahedronPosition[iVert1].fY;
                        asEdgeVertex[iEdge].fZ = fInvOffset*pasTetrahedronPosition[iVert0].fZ  +  fOffset*pasTetrahedronPosition[iVert1].fZ;
                        vGetNormal(asEdgeNorm[iEdge], asEdgeVertex[iEdge].fX, asEdgeVertex[iEdge].fY, asEdgeVertex[iEdge].fZ);
                }
        }
        //Draw the triangles that were found.  There can be up to 2 per tetrahedron
        for(iTriangle = 0; iTriangle < 2; iTriangle++)
        {
                if(a2iTetrahedronTriangles[iFlagIndex][3*iTriangle] < 0)
                        break;
                for(iCorner = 0; iCorner < 3; iCorner++)
                {
                        iVertex = a2iTetrahedronTriangles[iFlagIndex][3*iTriangle+iCorner];
                        vGetColor(sColor, asEdgeVertex[iVertex], asEdgeNorm[iVertex]);
                        glColor3f(sColor.fX, sColor.fY, sColor.fZ);
                        glNormal3f(asEdgeNorm[iVertex].fX,   asEdgeNorm[iVertex].fY,   asEdgeNorm[iVertex].fZ);
                        glVertex3f(asEdgeVertex[iVertex].fX, asEdgeVertex[iVertex].fY, asEdgeVertex[iVertex].fZ);
                }
        }
}
//vMarchCube2 performs the Marching Tetrahedrons algorithm on a single cube by making six calls to vMarchTetrahedron
GLvoid vMarchCube2(GLfloat fX, GLfloat fY, GLfloat fZ, GLfloat fScale)
{
        GLint iVertex, iTetrahedron, iVertexInACube;
        GLvector asCubePosition[8];
        GLfloat  afCubeValue[8];
        GLvector asTetrahedronPosition[4];
        GLfloat  afTetrahedronValue[4];
        //Make a local copy of the cube's corner positions
        for(iVertex = 0; iVertex < 8; iVertex++)
        {
                asCubePosition[iVertex].fX = fX + a2fVertexOffset[iVertex][0]*fScale;
                asCubePosition[iVertex].fY = fY + a2fVertexOffset[iVertex][1]*fScale;
                asCubePosition[iVertex].fZ = fZ + a2fVertexOffset[iVertex][2]*fScale;
        }
        //Make a local copy of the cube's corner values
        for(iVertex = 0; iVertex < 8; iVertex++)
        {
                afCubeValue[iVertex] = fSample(asCubePosition[iVertex].fX,
                                                   asCubePosition[iVertex].fY,
                                               asCubePosition[iVertex].fZ);
        }
        for(iTetrahedron = 0; iTetrahedron < 6; iTetrahedron++)
        {
                for(iVertex = 0; iVertex < 4; iVertex++)
                {
                        iVertexInACube = a2iTetrahedronsInACube[iTetrahedron][iVertex];
                        asTetrahedronPosition[iVertex].fX = asCubePosition[iVertexInACube].fX;
                        asTetrahedronPosition[iVertex].fY = asCubePosition[iVertexInACube].fY;
                        asTetrahedronPosition[iVertex].fZ = asCubePosition[iVertexInACube].fZ;
                        afTetrahedronValue[iVertex] = afCubeValue[iVertexInACube];
                }
                vMarchTetrahedron(asTetrahedronPosition, afTetrahedronValue);
        }
}
//vMarchingCubes iterates over the entire dataset, calling vMarchCube on each cube
GLvoid vMarchingCubes()
{
        GLint iX, iY, iZ;
        for(iX = 0; iX < iDataSetSize; iX++)
        for(iY = 0; iY < iDataSetSize; iY++)
        for(iZ = 0; iZ < iDataSetSize; iZ++)
        {
                vMarchCube(iX*fStepSize, iY*fStepSize, iZ*fStepSize, fStepSize);
        }
}
// For any edge, if one vertex is inside of the surface and the other is outside of the surface
//  then the edge intersects the surface
// For each of the 4 vertices of the tetrahedron can be two possible states : either inside or outside of the surface
// For any tetrahedron the are 2^4=16 possible sets of vertex states
// This table lists the edges intersected by the surface for all 16 possible vertex states
// There are 6 edges.  For each entry in the table, if edge #n is intersected, then bit #n is set to 1
GLint aiTetrahedronEdgeFlags[16]=
{
        0x00, 0x0d, 0x13, 0x1e, 0x26, 0x2b, 0x35, 0x38, 0x38, 0x35, 0x2b, 0x26, 0x1e, 0x13, 0x0d, 0x00, 
};
// For each of the possible vertex states listed in aiTetrahedronEdgeFlags there is a specific triangulation
// of the edge intersection points.  a2iTetrahedronTriangles lists all of them in the form of
// 0-2 edge triples with the list terminated by the invalid value -1.
//
// I generated this table by hand
GLint a2iTetrahedronTriangles[16][7] =
{
        {-1, -1, -1, -1, -1, -1, -1},
        { 0,  3,  2, -1, -1, -1, -1},
        { 0,  1,  4, -1, -1, -1, -1},
        { 1,  4,  2,  2,  4,  3, -1},
        { 1,  2,  5, -1, -1, -1, -1},
        { 0,  3,  5,  0,  5,  1, -1},
        { 0,  2,  5,  0,  5,  4, -1},
        { 5,  4,  3, -1, -1, -1, -1},
        { 3,  4,  5, -1, -1, -1, -1},
        { 4,  5,  0,  5,  2,  0, -1},
        { 1,  5,  0,  5,  3,  0, -1},
        { 5,  2,  1, -1, -1, -1, -1},
        { 3,  4,  2,  2,  4,  1, -1},
        { 4,  1,  0, -1, -1, -1, -1},
        { 2,  3,  0, -1, -1, -1, -1},
        {-1, -1, -1, -1, -1, -1, -1},
};
// For any edge, if one vertex is inside of the surface and the other is outside of the surface
//  then the edge intersects the surface
// For each of the 8 vertices of the cube can be two possible states : either inside or outside of the surface
// For any cube the are 2^8=256 possible sets of vertex states
// This table lists the edges intersected by the surface for all 256 possible vertex states
// There are 12 edges.  For each entry in the table, if edge #n is intersected, then bit #n is set to 1
GLint aiCubeEdgeFlags[256]=
{
        0x000, 0x109, 0x203, 0x30a, 0x406, 0x50f, 0x605, 0x70c, 0x80c, 0x905, 0xa0f, 0xb06, 0xc0a, 0xd03, 0xe09, 0xf00, 
        0x190, 0x099, 0x393, 0x29a, 0x596, 0x49f, 0x795, 0x69c, 0x99c, 0x895, 0xb9f, 0xa96, 0xd9a, 0xc93, 0xf99, 0xe90, 
        0x230, 0x339, 0x033, 0x13a, 0x636, 0x73f, 0x435, 0x53c, 0xa3c, 0xb35, 0x83f, 0x936, 0xe3a, 0xf33, 0xc39, 0xd30, 
        0x3a0, 0x2a9, 0x1a3, 0x0aa, 0x7a6, 0x6af, 0x5a5, 0x4ac, 0xbac, 0xaa5, 0x9af, 0x8a6, 0xfaa, 0xea3, 0xda9, 0xca0, 
        0x460, 0x569, 0x663, 0x76a, 0x066, 0x16f, 0x265, 0x36c, 0xc6c, 0xd65, 0xe6f, 0xf66, 0x86a, 0x963, 0xa69, 0xb60, 
        0x5f0, 0x4f9, 0x7f3, 0x6fa, 0x1f6, 0x0ff, 0x3f5, 0x2fc, 0xdfc, 0xcf5, 0xfff, 0xef6, 0x9fa, 0x8f3, 0xbf9, 0xaf0, 
        0x650, 0x759, 0x453, 0x55a, 0x256, 0x35f, 0x055, 0x15c, 0xe5c, 0xf55, 0xc5f, 0xd56, 0xa5a, 0xb53, 0x859, 0x950, 
        0x7c0, 0x6c9, 0x5c3, 0x4ca, 0x3c6, 0x2cf, 0x1c5, 0x0cc, 0xfcc, 0xec5, 0xdcf, 0xcc6, 0xbca, 0xac3, 0x9c9, 0x8c0, 
        0x8c0, 0x9c9, 0xac3, 0xbca, 0xcc6, 0xdcf, 0xec5, 0xfcc, 0x0cc, 0x1c5, 0x2cf, 0x3c6, 0x4ca, 0x5c3, 0x6c9, 0x7c0, 
        0x950, 0x859, 0xb53, 0xa5a, 0xd56, 0xc5f, 0xf55, 0xe5c, 0x15c, 0x055, 0x35f, 0x256, 0x55a, 0x453, 0x759, 0x650, 
        0xaf0, 0xbf9, 0x8f3, 0x9fa, 0xef6, 0xfff, 0xcf5, 0xdfc, 0x2fc, 0x3f5, 0x0ff, 0x1f6, 0x6fa, 0x7f3, 0x4f9, 0x5f0, 
        0xb60, 0xa69, 0x963, 0x86a, 0xf66, 0xe6f, 0xd65, 0xc6c, 0x36c, 0x265, 0x16f, 0x066, 0x76a, 0x663, 0x569, 0x460, 
        0xca0, 0xda9, 0xea3, 0xfaa, 0x8a6, 0x9af, 0xaa5, 0xbac, 0x4ac, 0x5a5, 0x6af, 0x7a6, 0x0aa, 0x1a3, 0x2a9, 0x3a0, 
        0xd30, 0xc39, 0xf33, 0xe3a, 0x936, 0x83f, 0xb35, 0xa3c, 0x53c, 0x435, 0x73f, 0x636, 0x13a, 0x033, 0x339, 0x230, 
        0xe90, 0xf99, 0xc93, 0xd9a, 0xa96, 0xb9f, 0x895, 0x99c, 0x69c, 0x795, 0x49f, 0x596, 0x29a, 0x393, 0x099, 0x190, 
        0xf00, 0xe09, 0xd03, 0xc0a, 0xb06, 0xa0f, 0x905, 0x80c, 0x70c, 0x605, 0x50f, 0x406, 0x30a, 0x203, 0x109, 0x000
};
//  For each of the possible vertex states listed in aiCubeEdgeFlags there is a specific triangulation
//  of the edge intersection points.  a2iTriangleConnectionTable lists all of them in the form of
//  0-5 edge triples with the list terminated by the invalid value -1.
//  For example: a2iTriangleConnectionTable[3] list the 2 triangles formed when corner[0] 
//  and corner[1] are inside of the surface, but the rest of the cube is not.
//
//  I found this table in an example program someone wrote long ago.  It was probably generated by hand
GLint a2iTriangleConnectionTable[256][16] =  
{
        {-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {0, 8, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {0, 1, 9, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {1, 8, 3, 9, 8, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {1, 2, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {0, 8, 3, 1, 2, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {9, 2, 10, 0, 2, 9, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {2, 8, 3, 2, 10, 8, 10, 9, 8, -1, -1, -1, -1, -1, -1, -1},
        {3, 11, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {0, 11, 2, 8, 11, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {1, 9, 0, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {1, 11, 2, 1, 9, 11, 9, 8, 11, -1, -1, -1, -1, -1, -1, -1},
        {3, 10, 1, 11, 10, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {0, 10, 1, 0, 8, 10, 8, 11, 10, -1, -1, -1, -1, -1, -1, -1},
        {3, 9, 0, 3, 11, 9, 11, 10, 9, -1, -1, -1, -1, -1, -1, -1},
        {9, 8, 10, 10, 8, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {4, 7, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {4, 3, 0, 7, 3, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {0, 1, 9, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {4, 1, 9, 4, 7, 1, 7, 3, 1, -1, -1, -1, -1, -1, -1, -1},
        {1, 2, 10, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {3, 4, 7, 3, 0, 4, 1, 2, 10, -1, -1, -1, -1, -1, -1, -1},
        {9, 2, 10, 9, 0, 2, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1},
        {2, 10, 9, 2, 9, 7, 2, 7, 3, 7, 9, 4, -1, -1, -1, -1},
        {8, 4, 7, 3, 11, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {11, 4, 7, 11, 2, 4, 2, 0, 4, -1, -1, -1, -1, -1, -1, -1},
        {9, 0, 1, 8, 4, 7, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1},
        {4, 7, 11, 9, 4, 11, 9, 11, 2, 9, 2, 1, -1, -1, -1, -1},
        {3, 10, 1, 3, 11, 10, 7, 8, 4, -1, -1, -1, -1, -1, -1, -1},
        {1, 11, 10, 1, 4, 11, 1, 0, 4, 7, 11, 4, -1, -1, -1, -1},
        {4, 7, 8, 9, 0, 11, 9, 11, 10, 11, 0, 3, -1, -1, -1, -1},
        {4, 7, 11, 4, 11, 9, 9, 11, 10, -1, -1, -1, -1, -1, -1, -1},
        {9, 5, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {9, 5, 4, 0, 8, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {0, 5, 4, 1, 5, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {8, 5, 4, 8, 3, 5, 3, 1, 5, -1, -1, -1, -1, -1, -1, -1},
        {1, 2, 10, 9, 5, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {3, 0, 8, 1, 2, 10, 4, 9, 5, -1, -1, -1, -1, -1, -1, -1},
        {5, 2, 10, 5, 4, 2, 4, 0, 2, -1, -1, -1, -1, -1, -1, -1},
        {2, 10, 5, 3, 2, 5, 3, 5, 4, 3, 4, 8, -1, -1, -1, -1},
        {9, 5, 4, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {0, 11, 2, 0, 8, 11, 4, 9, 5, -1, -1, -1, -1, -1, -1, -1},
        {0, 5, 4, 0, 1, 5, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1},
        {2, 1, 5, 2, 5, 8, 2, 8, 11, 4, 8, 5, -1, -1, -1, -1},
        {10, 3, 11, 10, 1, 3, 9, 5, 4, -1, -1, -1, -1, -1, -1, -1},
        {4, 9, 5, 0, 8, 1, 8, 10, 1, 8, 11, 10, -1, -1, -1, -1},
        {5, 4, 0, 5, 0, 11, 5, 11, 10, 11, 0, 3, -1, -1, -1, -1},
        {5, 4, 8, 5, 8, 10, 10, 8, 11, -1, -1, -1, -1, -1, -1, -1},
        {9, 7, 8, 5, 7, 9, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {9, 3, 0, 9, 5, 3, 5, 7, 3, -1, -1, -1, -1, -1, -1, -1},
        {0, 7, 8, 0, 1, 7, 1, 5, 7, -1, -1, -1, -1, -1, -1, -1},
        {1, 5, 3, 3, 5, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {9, 7, 8, 9, 5, 7, 10, 1, 2, -1, -1, -1, -1, -1, -1, -1},
        {10, 1, 2, 9, 5, 0, 5, 3, 0, 5, 7, 3, -1, -1, -1, -1},
        {8, 0, 2, 8, 2, 5, 8, 5, 7, 10, 5, 2, -1, -1, -1, -1},
        {2, 10, 5, 2, 5, 3, 3, 5, 7, -1, -1, -1, -1, -1, -1, -1},
        {7, 9, 5, 7, 8, 9, 3, 11, 2, -1, -1, -1, -1, -1, -1, -1},
        {9, 5, 7, 9, 7, 2, 9, 2, 0, 2, 7, 11, -1, -1, -1, -1},
        {2, 3, 11, 0, 1, 8, 1, 7, 8, 1, 5, 7, -1, -1, -1, -1},
        {11, 2, 1, 11, 1, 7, 7, 1, 5, -1, -1, -1, -1, -1, -1, -1},
        {9, 5, 8, 8, 5, 7, 10, 1, 3, 10, 3, 11, -1, -1, -1, -1},
        {5, 7, 0, 5, 0, 9, 7, 11, 0, 1, 0, 10, 11, 10, 0, -1},
        {11, 10, 0, 11, 0, 3, 10, 5, 0, 8, 0, 7, 5, 7, 0, -1},
        {11, 10, 5, 7, 11, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {10, 6, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {0, 8, 3, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {9, 0, 1, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {1, 8, 3, 1, 9, 8, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1},
        {1, 6, 5, 2, 6, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {1, 6, 5, 1, 2, 6, 3, 0, 8, -1, -1, -1, -1, -1, -1, -1},
        {9, 6, 5, 9, 0, 6, 0, 2, 6, -1, -1, -1, -1, -1, -1, -1},
        {5, 9, 8, 5, 8, 2, 5, 2, 6, 3, 2, 8, -1, -1, -1, -1},
        {2, 3, 11, 10, 6, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {11, 0, 8, 11, 2, 0, 10, 6, 5, -1, -1, -1, -1, -1, -1, -1},
        {0, 1, 9, 2, 3, 11, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1},
        {5, 10, 6, 1, 9, 2, 9, 11, 2, 9, 8, 11, -1, -1, -1, -1},
        {6, 3, 11, 6, 5, 3, 5, 1, 3, -1, -1, -1, -1, -1, -1, -1},
        {0, 8, 11, 0, 11, 5, 0, 5, 1, 5, 11, 6, -1, -1, -1, -1},
        {3, 11, 6, 0, 3, 6, 0, 6, 5, 0, 5, 9, -1, -1, -1, -1},
        {6, 5, 9, 6, 9, 11, 11, 9, 8, -1, -1, -1, -1, -1, -1, -1},
        {5, 10, 6, 4, 7, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {4, 3, 0, 4, 7, 3, 6, 5, 10, -1, -1, -1, -1, -1, -1, -1},
        {1, 9, 0, 5, 10, 6, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1},
        {10, 6, 5, 1, 9, 7, 1, 7, 3, 7, 9, 4, -1, -1, -1, -1},
        {6, 1, 2, 6, 5, 1, 4, 7, 8, -1, -1, -1, -1, -1, -1, -1},
        {1, 2, 5, 5, 2, 6, 3, 0, 4, 3, 4, 7, -1, -1, -1, -1},
        {8, 4, 7, 9, 0, 5, 0, 6, 5, 0, 2, 6, -1, -1, -1, -1},
        {7, 3, 9, 7, 9, 4, 3, 2, 9, 5, 9, 6, 2, 6, 9, -1},
        {3, 11, 2, 7, 8, 4, 10, 6, 5, -1, -1, -1, -1, -1, -1, -1},
        {5, 10, 6, 4, 7, 2, 4, 2, 0, 2, 7, 11, -1, -1, -1, -1},
        {0, 1, 9, 4, 7, 8, 2, 3, 11, 5, 10, 6, -1, -1, -1, -1},
        {9, 2, 1, 9, 11, 2, 9, 4, 11, 7, 11, 4, 5, 10, 6, -1},
        {8, 4, 7, 3, 11, 5, 3, 5, 1, 5, 11, 6, -1, -1, -1, -1},
        {5, 1, 11, 5, 11, 6, 1, 0, 11, 7, 11, 4, 0, 4, 11, -1},
        {0, 5, 9, 0, 6, 5, 0, 3, 6, 11, 6, 3, 8, 4, 7, -1},
        {6, 5, 9, 6, 9, 11, 4, 7, 9, 7, 11, 9, -1, -1, -1, -1},
        {10, 4, 9, 6, 4, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {4, 10, 6, 4, 9, 10, 0, 8, 3, -1, -1, -1, -1, -1, -1, -1},
        {10, 0, 1, 10, 6, 0, 6, 4, 0, -1, -1, -1, -1, -1, -1, -1},
        {8, 3, 1, 8, 1, 6, 8, 6, 4, 6, 1, 10, -1, -1, -1, -1},
        {1, 4, 9, 1, 2, 4, 2, 6, 4, -1, -1, -1, -1, -1, -1, -1},
        {3, 0, 8, 1, 2, 9, 2, 4, 9, 2, 6, 4, -1, -1, -1, -1},
        {0, 2, 4, 4, 2, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {8, 3, 2, 8, 2, 4, 4, 2, 6, -1, -1, -1, -1, -1, -1, -1},
        {10, 4, 9, 10, 6, 4, 11, 2, 3, -1, -1, -1, -1, -1, -1, -1},
        {0, 8, 2, 2, 8, 11, 4, 9, 10, 4, 10, 6, -1, -1, -1, -1},
        {3, 11, 2, 0, 1, 6, 0, 6, 4, 6, 1, 10, -1, -1, -1, -1},
        {6, 4, 1, 6, 1, 10, 4, 8, 1, 2, 1, 11, 8, 11, 1, -1},
        {9, 6, 4, 9, 3, 6, 9, 1, 3, 11, 6, 3, -1, -1, -1, -1},
        {8, 11, 1, 8, 1, 0, 11, 6, 1, 9, 1, 4, 6, 4, 1, -1},
        {3, 11, 6, 3, 6, 0, 0, 6, 4, -1, -1, -1, -1, -1, -1, -1},
        {6, 4, 8, 11, 6, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {7, 10, 6, 7, 8, 10, 8, 9, 10, -1, -1, -1, -1, -1, -1, -1},
        {0, 7, 3, 0, 10, 7, 0, 9, 10, 6, 7, 10, -1, -1, -1, -1},
        {10, 6, 7, 1, 10, 7, 1, 7, 8, 1, 8, 0, -1, -1, -1, -1},
        {10, 6, 7, 10, 7, 1, 1, 7, 3, -1, -1, -1, -1, -1, -1, -1},
        {1, 2, 6, 1, 6, 8, 1, 8, 9, 8, 6, 7, -1, -1, -1, -1},
        {2, 6, 9, 2, 9, 1, 6, 7, 9, 0, 9, 3, 7, 3, 9, -1},
        {7, 8, 0, 7, 0, 6, 6, 0, 2, -1, -1, -1, -1, -1, -1, -1},
        {7, 3, 2, 6, 7, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {2, 3, 11, 10, 6, 8, 10, 8, 9, 8, 6, 7, -1, -1, -1, -1},
        {2, 0, 7, 2, 7, 11, 0, 9, 7, 6, 7, 10, 9, 10, 7, -1},
        {1, 8, 0, 1, 7, 8, 1, 10, 7, 6, 7, 10, 2, 3, 11, -1},
        {11, 2, 1, 11, 1, 7, 10, 6, 1, 6, 7, 1, -1, -1, -1, -1},
        {8, 9, 6, 8, 6, 7, 9, 1, 6, 11, 6, 3, 1, 3, 6, -1},
        {0, 9, 1, 11, 6, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {7, 8, 0, 7, 0, 6, 3, 11, 0, 11, 6, 0, -1, -1, -1, -1},
        {7, 11, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {7, 6, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {3, 0, 8, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {0, 1, 9, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {8, 1, 9, 8, 3, 1, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1},
        {10, 1, 2, 6, 11, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {1, 2, 10, 3, 0, 8, 6, 11, 7, -1, -1, -1, -1, -1, -1, -1},
        {2, 9, 0, 2, 10, 9, 6, 11, 7, -1, -1, -1, -1, -1, -1, -1},
        {6, 11, 7, 2, 10, 3, 10, 8, 3, 10, 9, 8, -1, -1, -1, -1},
        {7, 2, 3, 6, 2, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {7, 0, 8, 7, 6, 0, 6, 2, 0, -1, -1, -1, -1, -1, -1, -1},
        {2, 7, 6, 2, 3, 7, 0, 1, 9, -1, -1, -1, -1, -1, -1, -1},
        {1, 6, 2, 1, 8, 6, 1, 9, 8, 8, 7, 6, -1, -1, -1, -1},
        {10, 7, 6, 10, 1, 7, 1, 3, 7, -1, -1, -1, -1, -1, -1, -1},
        {10, 7, 6, 1, 7, 10, 1, 8, 7, 1, 0, 8, -1, -1, -1, -1},
        {0, 3, 7, 0, 7, 10, 0, 10, 9, 6, 10, 7, -1, -1, -1, -1},
        {7, 6, 10, 7, 10, 8, 8, 10, 9, -1, -1, -1, -1, -1, -1, -1},
        {6, 8, 4, 11, 8, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {3, 6, 11, 3, 0, 6, 0, 4, 6, -1, -1, -1, -1, -1, -1, -1},
        {8, 6, 11, 8, 4, 6, 9, 0, 1, -1, -1, -1, -1, -1, -1, -1},
        {9, 4, 6, 9, 6, 3, 9, 3, 1, 11, 3, 6, -1, -1, -1, -1},
        {6, 8, 4, 6, 11, 8, 2, 10, 1, -1, -1, -1, -1, -1, -1, -1},
        {1, 2, 10, 3, 0, 11, 0, 6, 11, 0, 4, 6, -1, -1, -1, -1},
        {4, 11, 8, 4, 6, 11, 0, 2, 9, 2, 10, 9, -1, -1, -1, -1},
        {10, 9, 3, 10, 3, 2, 9, 4, 3, 11, 3, 6, 4, 6, 3, -1},
        {8, 2, 3, 8, 4, 2, 4, 6, 2, -1, -1, -1, -1, -1, -1, -1},
        {0, 4, 2, 4, 6, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {1, 9, 0, 2, 3, 4, 2, 4, 6, 4, 3, 8, -1, -1, -1, -1},
        {1, 9, 4, 1, 4, 2, 2, 4, 6, -1, -1, -1, -1, -1, -1, -1},
        {8, 1, 3, 8, 6, 1, 8, 4, 6, 6, 10, 1, -1, -1, -1, -1},
        {10, 1, 0, 10, 0, 6, 6, 0, 4, -1, -1, -1, -1, -1, -1, -1},
        {4, 6, 3, 4, 3, 8, 6, 10, 3, 0, 3, 9, 10, 9, 3, -1},
        {10, 9, 4, 6, 10, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {4, 9, 5, 7, 6, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {0, 8, 3, 4, 9, 5, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1},
        {5, 0, 1, 5, 4, 0, 7, 6, 11, -1, -1, -1, -1, -1, -1, -1},
        {11, 7, 6, 8, 3, 4, 3, 5, 4, 3, 1, 5, -1, -1, -1, -1},
        {9, 5, 4, 10, 1, 2, 7, 6, 11, -1, -1, -1, -1, -1, -1, -1},
        {6, 11, 7, 1, 2, 10, 0, 8, 3, 4, 9, 5, -1, -1, -1, -1},
        {7, 6, 11, 5, 4, 10, 4, 2, 10, 4, 0, 2, -1, -1, -1, -1},
        {3, 4, 8, 3, 5, 4, 3, 2, 5, 10, 5, 2, 11, 7, 6, -1},
        {7, 2, 3, 7, 6, 2, 5, 4, 9, -1, -1, -1, -1, -1, -1, -1},
        {9, 5, 4, 0, 8, 6, 0, 6, 2, 6, 8, 7, -1, -1, -1, -1},
        {3, 6, 2, 3, 7, 6, 1, 5, 0, 5, 4, 0, -1, -1, -1, -1},
        {6, 2, 8, 6, 8, 7, 2, 1, 8, 4, 8, 5, 1, 5, 8, -1},
        {9, 5, 4, 10, 1, 6, 1, 7, 6, 1, 3, 7, -1, -1, -1, -1},
        {1, 6, 10, 1, 7, 6, 1, 0, 7, 8, 7, 0, 9, 5, 4, -1},
        {4, 0, 10, 4, 10, 5, 0, 3, 10, 6, 10, 7, 3, 7, 10, -1},
        {7, 6, 10, 7, 10, 8, 5, 4, 10, 4, 8, 10, -1, -1, -1, -1},
        {6, 9, 5, 6, 11, 9, 11, 8, 9, -1, -1, -1, -1, -1, -1, -1},
        {3, 6, 11, 0, 6, 3, 0, 5, 6, 0, 9, 5, -1, -1, -1, -1},
        {0, 11, 8, 0, 5, 11, 0, 1, 5, 5, 6, 11, -1, -1, -1, -1},
        {6, 11, 3, 6, 3, 5, 5, 3, 1, -1, -1, -1, -1, -1, -1, -1},
        {1, 2, 10, 9, 5, 11, 9, 11, 8, 11, 5, 6, -1, -1, -1, -1},
        {0, 11, 3, 0, 6, 11, 0, 9, 6, 5, 6, 9, 1, 2, 10, -1},
        {11, 8, 5, 11, 5, 6, 8, 0, 5, 10, 5, 2, 0, 2, 5, -1},
        {6, 11, 3, 6, 3, 5, 2, 10, 3, 10, 5, 3, -1, -1, -1, -1},
        {5, 8, 9, 5, 2, 8, 5, 6, 2, 3, 8, 2, -1, -1, -1, -1},
        {9, 5, 6, 9, 6, 0, 0, 6, 2, -1, -1, -1, -1, -1, -1, -1},
        {1, 5, 8, 1, 8, 0, 5, 6, 8, 3, 8, 2, 6, 2, 8, -1},
        {1, 5, 6, 2, 1, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {1, 3, 6, 1, 6, 10, 3, 8, 6, 5, 6, 9, 8, 9, 6, -1},
        {10, 1, 0, 10, 0, 6, 9, 5, 0, 5, 6, 0, -1, -1, -1, -1},
        {0, 3, 8, 5, 6, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {10, 5, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {11, 5, 10, 7, 5, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {11, 5, 10, 11, 7, 5, 8, 3, 0, -1, -1, -1, -1, -1, -1, -1},
        {5, 11, 7, 5, 10, 11, 1, 9, 0, -1, -1, -1, -1, -1, -1, -1},
        {10, 7, 5, 10, 11, 7, 9, 8, 1, 8, 3, 1, -1, -1, -1, -1},
        {11, 1, 2, 11, 7, 1, 7, 5, 1, -1, -1, -1, -1, -1, -1, -1},
        {0, 8, 3, 1, 2, 7, 1, 7, 5, 7, 2, 11, -1, -1, -1, -1},
        {9, 7, 5, 9, 2, 7, 9, 0, 2, 2, 11, 7, -1, -1, -1, -1},
        {7, 5, 2, 7, 2, 11, 5, 9, 2, 3, 2, 8, 9, 8, 2, -1},
        {2, 5, 10, 2, 3, 5, 3, 7, 5, -1, -1, -1, -1, -1, -1, -1},
        {8, 2, 0, 8, 5, 2, 8, 7, 5, 10, 2, 5, -1, -1, -1, -1},
        {9, 0, 1, 5, 10, 3, 5, 3, 7, 3, 10, 2, -1, -1, -1, -1},
        {9, 8, 2, 9, 2, 1, 8, 7, 2, 10, 2, 5, 7, 5, 2, -1},
        {1, 3, 5, 3, 7, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {0, 8, 7, 0, 7, 1, 1, 7, 5, -1, -1, -1, -1, -1, -1, -1},
        {9, 0, 3, 9, 3, 5, 5, 3, 7, -1, -1, -1, -1, -1, -1, -1},
        {9, 8, 7, 5, 9, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {5, 8, 4, 5, 10, 8, 10, 11, 8, -1, -1, -1, -1, -1, -1, -1},
        {5, 0, 4, 5, 11, 0, 5, 10, 11, 11, 3, 0, -1, -1, -1, -1},
        {0, 1, 9, 8, 4, 10, 8, 10, 11, 10, 4, 5, -1, -1, -1, -1},
        {10, 11, 4, 10, 4, 5, 11, 3, 4, 9, 4, 1, 3, 1, 4, -1},
        {2, 5, 1, 2, 8, 5, 2, 11, 8, 4, 5, 8, -1, -1, -1, -1},
        {0, 4, 11, 0, 11, 3, 4, 5, 11, 2, 11, 1, 5, 1, 11, -1},
        {0, 2, 5, 0, 5, 9, 2, 11, 5, 4, 5, 8, 11, 8, 5, -1},
        {9, 4, 5, 2, 11, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {2, 5, 10, 3, 5, 2, 3, 4, 5, 3, 8, 4, -1, -1, -1, -1},
        {5, 10, 2, 5, 2, 4, 4, 2, 0, -1, -1, -1, -1, -1, -1, -1},
        {3, 10, 2, 3, 5, 10, 3, 8, 5, 4, 5, 8, 0, 1, 9, -1},
        {5, 10, 2, 5, 2, 4, 1, 9, 2, 9, 4, 2, -1, -1, -1, -1},
        {8, 4, 5, 8, 5, 3, 3, 5, 1, -1, -1, -1, -1, -1, -1, -1},
        {0, 4, 5, 1, 0, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {8, 4, 5, 8, 5, 3, 9, 0, 5, 0, 3, 5, -1, -1, -1, -1},
        {9, 4, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {4, 11, 7, 4, 9, 11, 9, 10, 11, -1, -1, -1, -1, -1, -1, -1},
        {0, 8, 3, 4, 9, 7, 9, 11, 7, 9, 10, 11, -1, -1, -1, -1},
        {1, 10, 11, 1, 11, 4, 1, 4, 0, 7, 4, 11, -1, -1, -1, -1},
        {3, 1, 4, 3, 4, 8, 1, 10, 4, 7, 4, 11, 10, 11, 4, -1},
        {4, 11, 7, 9, 11, 4, 9, 2, 11, 9, 1, 2, -1, -1, -1, -1},
        {9, 7, 4, 9, 11, 7, 9, 1, 11, 2, 11, 1, 0, 8, 3, -1},
        {11, 7, 4, 11, 4, 2, 2, 4, 0, -1, -1, -1, -1, -1, -1, -1},
        {11, 7, 4, 11, 4, 2, 8, 3, 4, 3, 2, 4, -1, -1, -1, -1},
        {2, 9, 10, 2, 7, 9, 2, 3, 7, 7, 4, 9, -1, -1, -1, -1},
        {9, 10, 7, 9, 7, 4, 10, 2, 7, 8, 7, 0, 2, 0, 7, -1},
        {3, 7, 10, 3, 10, 2, 7, 4, 10, 1, 10, 0, 4, 0, 10, -1},
        {1, 10, 2, 8, 7, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {4, 9, 1, 4, 1, 7, 7, 1, 3, -1, -1, -1, -1, -1, -1, -1},
        {4, 9, 1, 4, 1, 7, 0, 8, 1, 8, 7, 1, -1, -1, -1, -1},
        {4, 0, 3, 7, 4, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {4, 8, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {9, 10, 8, 10, 11, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {3, 0, 9, 3, 9, 11, 11, 9, 10, -1, -1, -1, -1, -1, -1, -1},
        {0, 1, 10, 0, 10, 8, 8, 10, 11, -1, -1, -1, -1, -1, -1, -1},
        {3, 1, 10, 11, 3, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {1, 2, 11, 1, 11, 9, 9, 11, 8, -1, -1, -1, -1, -1, -1, -1},
        {3, 0, 9, 3, 9, 11, 1, 2, 9, 2, 11, 9, -1, -1, -1, -1},
        {0, 2, 11, 8, 0, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {3, 2, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {2, 3, 8, 2, 8, 10, 10, 8, 9, -1, -1, -1, -1, -1, -1, -1},
        {9, 10, 2, 0, 9, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {2, 3, 8, 2, 8, 10, 0, 1, 8, 1, 10, 8, -1, -1, -1, -1},
        {1, 10, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {1, 3, 8, 9, 1, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {0, 9, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {0, 3, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
        {-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1}
};

親測(cè)運(yùn)行效果如下：

5.算法細(xì)節(jié)

這個(gè)算法有兩個(gè)主要組成部分。第一個(gè)是決定如何定義切割單個(gè)立方體的截面或表面截面。如果我們將每個(gè)角分類(lèi)為低于或高于等值，就有256種可能的角分類(lèi)構(gòu)型。其中兩個(gè)是無(wú)關(guān)緊要的;所有點(diǎn)都在立方體內(nèi)部或外部的位置對(duì)等值面沒(méi)有影響。對(duì)于所有其他構(gòu)型，我們需要確定在每個(gè)立方體邊緣上等值面相交的位置，并使用這些邊緣交點(diǎn)為等值面創(chuàng)建一個(gè)或多個(gè)三角形塊。

如果考慮對(duì)稱(chēng)性，在剩下的254種可能中，實(shí)際上只有14種構(gòu)型是唯一的。當(dāng)只有一個(gè)角小于等值時(shí)，這就形成了一個(gè)三角形，它與在這個(gè)角上相交的邊緣相交，而補(bǔ)丁法線朝向這個(gè)角。顯然，這種類(lèi)型有8個(gè)相關(guān)配置(例如，配置2 -你可能需要調(diào)整colormap來(lái)查看球體/像素之間的平面)。通過(guò)反轉(zhuǎn)法線，我們得到8種構(gòu)型它們有7個(gè)角比等值小。然而，我們并不認(rèn)為這些是獨(dú)一無(wú)二的。對(duì)于兩個(gè)角小于等值的構(gòu)型，有3種獨(dú)特的構(gòu)型(例如，構(gòu)型12)，這取決于這些角是否屬于同一條邊，屬于立方體的同一面，或相對(duì)于彼此的對(duì)角位置。對(duì)于3個(gè)角小于等值的構(gòu)型，也有3個(gè)獨(dú)特的構(gòu)型(例如構(gòu)型14)，這取決于是否有0、1或2個(gè)共享邊(2個(gè)共享邊會(huì)給你一個(gè)L形)。有7獨(dú)特配置有4個(gè)角小于等值時(shí),根據(jù)是否有0,2、3(3變體在這一點(diǎn)),或4共享邊緣(例如配置30 -你可能需要調(diào)整顏色看到孤立(遠(yuǎn))范圍內(nèi)的三角形/像素)。

每一種非平凡構(gòu)型都會(huì)導(dǎo)致1到4個(gè)三角形被添加到等值面上。實(shí)際的頂點(diǎn)本身可以通過(guò)沿邊的插值來(lái)計(jì)算，或者默認(rèn)它們的位置在邊的中間。插值位置顯然會(huì)給你更好的陰影計(jì)算和更平滑的表面。

現(xiàn)在我們可以為單個(gè)體素創(chuàng)建表面補(bǔ)丁，我們可以將這個(gè)過(guò)程應(yīng)用到整個(gè)體素。我們可以在板中處理體積，其中每個(gè)板由2個(gè)像素片組成。我們可以獨(dú)立對(duì)待每個(gè)立方體，或者我們可以傳播共享邊的立方體之間的邊相交。這種共享也可以在相鄰的板之間完成，這增加了一些存儲(chǔ)和復(fù)雜性，但節(jié)省了計(jì)算時(shí)間。邊緣/頂點(diǎn)信息的共享也導(dǎo)致了一個(gè)更緊湊的模型，一個(gè)更易于插值著色。

6.總結(jié)

Marching Cubes是一種用于繪制立體數(shù)據(jù)中等值面的算法?；靖拍钍?，我們可以通過(guò)立方體8個(gè)角的像素值來(lái)定義體素(立方體)。如果一個(gè)立方體中的一個(gè)或多個(gè)像素的值小于用戶指定的等值，并且一個(gè)或多個(gè)像素的值大于這個(gè)值，我們知道體素必須貢獻(xiàn)等值面的某個(gè)分量。通過(guò)確定立方體的哪些邊與等值面相交，我們可以創(chuàng)建三角形塊，將立方體劃分為等值面內(nèi)部區(qū)域和外部區(qū)域。通過(guò)連接等值面邊界上所有立方體的小塊，我們得到了一個(gè)表面表示法。

到此這篇關(guān)于C++實(shí)現(xiàn)移動(dòng)立方體示例講解的文章就介紹到這了,更多相關(guān)C++移動(dòng)立方體內(nèi)容請(qǐng)搜索腳本之家以前的文章或繼續(xù)瀏覽下面的相關(guān)文章希望大家以后多多支持腳本之家！

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