0，引入

其实，这一章节的内容是之前落下的。
当时，觉得“磨边物体”只是磨边前的物体与部分圆环、部分球面组成的复合物体（Compound Objects）。感觉太简单了，所以没有仔细看。
我们在前面的某些章节中已经有用到过“磨边”的概念，比如：
“Q91：真实地模拟透明材质（Realistic Transparency）（4）——Fish Bowl”中的鱼缸的边沿；
（http://blog.csdn.net/libing_zeng/article/details/65442187）
“ Q101：真实地模拟一个玻璃酒杯（Wine Glass）（回旋曲面）”中酒杯的边沿；
（http://blog.csdn.net/libing_zeng/article/details/69791261）

1，简单的磨边的物体

另外，我们再看一些简单几何图形的磨边后的图形（如下三张图片来自《Ray Tracing from the Ground Up》官网）：
磨边前：

磨边后：

看看其中磨边封闭圆柱面的各个部分分离的图形：

注意到：
“磨边”的意思就是：
面与面相交为曲线时，添加一个圆环面来过渡；
面与面相交为直线时，添加一个开放圆柱面来过渡；
线与线相交时，添加一个球面来过渡；

所以，
磨边的封闭圆柱面是由：1个开放圆柱面、2个底面、2个圆环组成；
磨边的长方体是由：磨边前长方体对应的8个矩形、长方体的边对应的12个圆环、长方体的顶点对应的8个球面组成。

2，磨边的楔形物体（Beveled Wedges）

2.1 理论分析

我们在前面看到的都是对简单物体的磨边。但是，楔形物体的磨边貌似没有如此简单。

（相关参数：内经r0=1.5，外径r1=3，磨边半径rb=0.25）

这样一个磨边的楔形是由哪些基本图形组成的呢？
第一部分：磨边前的面：前后2个矩形、左右2个部分圆柱面、上下2个annulus；
第二部分：面与面的过渡：圆柱面和上下两个annulus相交过渡的4个圆环、圆柱面和矩形相交过渡的4个圆柱面、矩形和annulus相交过渡的4个圆柱面；
第三部分：线与线的过渡：8个小球面
总共是2+2+2+4+4+4+8=26个基本图形。

但是，书上说是一共是30个基本图形，另外4个是什么呢？在哪呢？
书上说，另外4个是上下面上的patch annuli（annuli补丁）。
“annuli补丁”，也就是说，磨边后的楔形的上下表面若分别只用一个annulus的话，则是缺一块的。所以，才需要“补丁”。

若将代码中的补丁图形注释掉一个，则得到的图形是：

看到木有？上表面是不是缺一块哈？
为了，看得更清晰些，咱改一下楔形的参数：内经r0=0.5，外径r1=3，磨边半径rb=0.5。得到的图形是：

磨边楔形俯视图如下：

参数的具体计算，咱在此不做推导（初中几何的常见内容）。
咱特别指出的是patch annulus，因为这个特别容易被忽略。

2.2 C++代码实现

// ------------------------------------------------------------------------------ constructorBeveledWedge::BeveledWedge( const double _y0,       // minimum y valueconst double _y1,       // maximum y valueconst double _r0,       // inner radiusconst double _r1,       // outer radiusconst double _rb,       // bevel radiusconst double _phi0,     // minimum azimuth angle in degreesconst double _phi1)     // maximum azimuth angle in degrees:   y0(_y0),y1(_y1),r0(_r0),r1(_r1),rb(_rb),phi0(_phi0),phi1(_phi1)
{double sin_phi0 = sin(phi0 * PI_ON_180);  // in radiansdouble cos_phi0 = cos(phi0 * PI_ON_180);  // in radiansdouble sin_phi1 = sin(phi1 * PI_ON_180);  // in radiansdouble cos_phi1 = cos(phi1 * PI_ON_180);  // in radiansdouble sin_alpha = rb / (r0 + rb);double cos_alpha = sqrt(r0 * r0 + 2.0 * r0 * rb) / (r0 + rb);double sin_beta = rb / (r1 - rb);double cos_beta = sqrt(r1 * r1 - 2.0 * r1 * rb) / (r1 - rb);double xc1 = (r0 + rb) * (sin_phi0 * cos_alpha + cos_phi0 * sin_alpha);double zc1 = (r0 + rb) * (cos_phi0 * cos_alpha - sin_phi0 * sin_alpha);double xc2 = (r1 - rb) * (sin_phi0 * cos_beta + cos_phi0 * sin_beta);double zc2 = (r1 - rb) * (cos_phi0 * cos_beta - sin_phi0 * sin_beta);double xc3 = (r0 + rb) * (sin_phi1 * cos_alpha - cos_phi1 * sin_alpha);double zc3 = (r0 + rb) * (cos_phi1 * cos_alpha + sin_phi1 * sin_alpha);double xc4 = (r1 - rb) * (sin_phi1 * cos_beta - cos_phi1 * sin_beta);double zc4 = (r1 - rb) * (cos_phi1 * cos_beta + sin_phi1 * sin_beta);// corner spheres -------------------------------------------------------------------------------// bottom spheresSphere* bottom_c1 = new Sphere(Point3D(xc1, y0 + rb, zc1), rb);objects.push_back(bottom_c1);Sphere* bottom_c2 = new Sphere(Point3D(xc2, y0 + rb, zc2), rb);objects.push_back(bottom_c2);Sphere* bottom_c3 = new Sphere(Point3D(xc3, y0 + rb, zc3), rb);objects.push_back(bottom_c3);Sphere* bottom_c4 = new Sphere(Point3D(xc4, y0 + rb, zc4), rb);objects.push_back(bottom_c4);// top spheresSphere* top_c1 = new Sphere(Point3D(xc1, y1 - rb, zc1), rb);objects.push_back(top_c1);Sphere* top_c2 = new Sphere(Point3D(xc2, y1 - rb, zc2), rb);objects.push_back(top_c2);Sphere* top_c3 = new Sphere(Point3D(xc3, y1 - rb, zc3), rb);objects.push_back(top_c3);Sphere* top_c4 = new Sphere(Point3D(xc4, y1 - rb, zc4), rb);objects.push_back(top_c4);// vertical cylinders ------------------------------------------------------------------------------Instance* bottom_c1_cylinder = new Instance(new OpenCylinder(y0 + rb, y1 - rb, rb));bottom_c1_cylinder->translate(xc1, 0.0, zc1);bottom_c1_cylinder->transform_texture(false);objects.push_back(bottom_c1_cylinder);Instance* bottom_c2_cylinder = new Instance(new OpenCylinder(y0 + rb, y1 - rb, rb));bottom_c2_cylinder->translate(xc2, 0.0, zc2);bottom_c2_cylinder->transform_texture(false);objects.push_back(bottom_c2_cylinder);Instance* bottom_c3_cylinder = new Instance(new OpenCylinder(y0 + rb, y1 - rb, rb));bottom_c3_cylinder->translate(xc3, 0.0, zc3);bottom_c3_cylinder->transform_texture(false);objects.push_back(bottom_c3_cylinder);Instance* bottom_c4_cylinder = new Instance(new OpenCylinder(y0 + rb, y1 - rb, rb));bottom_c4_cylinder->translate(xc4, 0.0, zc4);bottom_c4_cylinder->transform_texture(false);objects.push_back(bottom_c4_cylinder);// inner curved surface ---------------------------------------------------------------------------------// the azimuth angle range has to be specified in degreesdouble alpha = acos(cos_alpha);  // radiansdouble phi_min = phi0 + alpha * 180.0 / PI;double phi_max = phi1 - alpha * 180.0 / PI;OpenCylinderPartConcave* inner_cylinder_ptr = new OpenCylinderPartConcave(y0 + rb, y1 - rb, r0, phi_min, phi_max);objects.push_back(inner_cylinder_ptr);// outer curved surface -----------------------------------------------------------------------------------// the azimuth angle range has to be specified in degreesdouble beta = acos(cos_beta);  // radiansphi_min = phi0 + beta * 180.0 / PI;phi_max = phi1 - beta * 180.0 / PI;OpenCylinderPartConvex* outer_cylinder_ptr = new OpenCylinderPartConvex(y0 + rb, y1 - rb, r1, phi_min, phi_max);objects.push_back(outer_cylinder_ptr);// phi0 vertical rectangledouble s1 = sqrt(r0 * r0 + 2.0 * r0 * rb);double s2 = sqrt(r1 * r1 - 2.0 * r1 * rb);Point3D p1(s1 * sin_phi0, y0 + rb, s1 * cos_phi0);Point3D p2(s2 * sin_phi0, y0 + rb, s2 * cos_phi0);Vector3D a = p2 - p1;Vector3D b(0, y1 - y0 - 2.0 * rb, 0);Rectangle* phi0_rectangle_ptr = new Rectangle(p1, a, b);objects.push_back(phi0_rectangle_ptr);// phi1 vertical rectanglePoint3D p3(s1 * sin_phi1, y0 + rb, s1 * cos_phi1);Point3D p4(s2 * sin_phi1, y0 + rb, s2 * cos_phi1);a = p3 - p4;Rectangle* phi1_rectangle_ptr = new Rectangle(p4, a, b);objects.push_back(phi1_rectangle_ptr);// the tori --------------------------------------------------------------------------------------------// inner bottomphi_min = phi0 - 90 + alpha * 180.0 / PI; // "-90" transform angle of against z-axis to x-axisphi_max = phi1 - 90 - alpha * 180.0 / PI;Instance* inner_bottom_torus = new Instance(new TorusPartConvex(r0 + rb, rb, phi_min, phi_max, 0, 360));inner_bottom_torus->translate(0.0, y0 + rb, 0.0);inner_bottom_torus->transform_texture(false);objects.push_back(inner_bottom_torus);// inner topInstance* inner_top_torus = new Instance(new TorusPartConvex(r0 + rb, rb, phi_min, phi_max, 0, 360));inner_top_torus->translate(0.0, y1 - rb, 0.0);inner_top_torus->transform_texture(false);objects.push_back(inner_top_torus);// outer bottomphi_min = phi0 - 90 + beta * 180.0 / PI;phi_max = phi1 - 90 - beta * 180.0 / PI;Instance* outer_bottom_torus = new Instance(new TorusPartConvex(r1 - rb, rb, phi_min, phi_max, 0, 360));outer_bottom_torus->translate(0.0, y0 + rb, 0.0);outer_bottom_torus->transform_texture(false);objects.push_back(outer_bottom_torus);// outer topInstance* outer_top_torus = new Instance(new TorusPartConvex(r1 - rb, rb, phi_min, phi_max, 0, 360));outer_top_torus->translate(0.0, y1 - rb, 0.0);outer_top_torus->transform_texture(false);objects.push_back(outer_top_torus);// horizontal cylinders ----------------------------------------------------------------------------------// phi0 bottom cylinderInstance* phi0_bottom_cylinder_ptr = new Instance(new OpenCylinder(0, s2 - s1, rb));phi0_bottom_cylinder_ptr->rotate_x(90);phi0_bottom_cylinder_ptr->rotate_y(phi0);phi0_bottom_cylinder_ptr->translate(xc1, y0 + rb, zc1);phi0_bottom_cylinder_ptr->transform_texture(false);objects.push_back(phi0_bottom_cylinder_ptr);// phi0 top cylinderInstance* phi0_top_cylinder_ptr = new Instance(new OpenCylinder(0, s2 - s1, rb));phi0_top_cylinder_ptr->rotate_x(90);phi0_top_cylinder_ptr->rotate_y(phi0);phi0_top_cylinder_ptr->translate(xc1, y1 - rb, zc1);phi0_top_cylinder_ptr->transform_texture(false);objects.push_back(phi0_top_cylinder_ptr);// phi1 bottom cylinderInstance* phi1_bottom_cylinder_ptr = new Instance(new OpenCylinder(0, s2 - s1, rb));phi1_bottom_cylinder_ptr->rotate_x(90);phi1_bottom_cylinder_ptr->rotate_y(phi1);phi1_bottom_cylinder_ptr->translate(xc3, y0 + rb, zc3);phi1_bottom_cylinder_ptr->transform_texture(false);objects.push_back(phi1_bottom_cylinder_ptr);// phi1 top cylinderInstance* phi1_top_cylinder_ptr = new Instance(new OpenCylinder(0, s2 - s1, rb));phi1_top_cylinder_ptr->rotate_x(90);phi1_top_cylinder_ptr->rotate_y(phi1);phi1_top_cylinder_ptr->translate(xc3, y1 - rb, zc3);phi1_top_cylinder_ptr->transform_texture(false);objects.push_back(phi1_top_cylinder_ptr);// top flat surface -----------------------------------------------------------------------------------// main partPoint3D center(0, y1, 0);Normal normal(0, 1, 0);double r_min = r0 + rb;double r_max = r1 - rb;phi_min = phi0 + alpha * 180.0 / PI;phi_max = phi1 - alpha * 180.0 / PI;AnnulusPart* top_main_part_ptr = new AnnulusPart(center, normal, r_min, r_max, phi_min, phi_max);objects.push_back(top_main_part_ptr);// small phi0 side patchr_min = 0.0;r_max = s2 - s1;phi_min = 0.0;phi_max = alpha * 180.0 / PI;Instance* top_phi0_patch_ptr = new Instance(new AnnulusPart(center, normal, r_min, r_max, phi_min, phi_max));top_phi0_patch_ptr->rotate_y(phi0);top_phi0_patch_ptr->translate(xc1, 0.0, zc1);top_phi0_patch_ptr->transform_texture(false);objects.push_back(top_phi0_patch_ptr);// small phi1 side patchphi_min = 360.0 - alpha * 180.0 / PI;phi_max = 360.0;Instance* top_phi1_patch_ptr = new Instance(new AnnulusPart(center, normal, r_min, r_max, phi_min, phi_max));top_phi1_patch_ptr->rotate_y(phi1);top_phi1_patch_ptr->translate(xc3, 0.0, zc3);top_phi1_patch_ptr->transform_texture(false);objects.push_back(top_phi1_patch_ptr);// bottom flat surface ---------------------------------------------------------------------------------// main partcenter = Point3D(0, y0, 0);normal = Normal(0, -1, 0);r_min = r0 + rb;r_max = r1 - rb;phi_min = phi0 + alpha * 180.0 / PI;phi_max = phi1 - alpha * 180.0 / PI;AnnulusPart* bottom_main_part_ptr = new AnnulusPart(center, normal, r_min, r_max, phi_min, phi_max);objects.push_back(bottom_main_part_ptr);// small phi0 side patchr_min = 0.0;r_max = s2 - s1;phi_min = 0.0;phi_max = alpha * 180.0 / PI;Instance* bottom_phi0_patch_ptr = new Instance(new AnnulusPart(center, normal, r_min, r_max, phi_min, phi_max));bottom_phi0_patch_ptr->rotate_y(phi0);bottom_phi0_patch_ptr->translate(xc1, 0.0, zc1);bottom_phi0_patch_ptr->transform_texture(false);objects.push_back(bottom_phi0_patch_ptr);// small phi1 side patchphi_min = 360.0 - alpha * 180.0 / PI;phi_max = 360.0;Instance* bottom_phi1_patch_ptr = new Instance(new AnnulusPart(center, normal, r_min, r_max, phi_min, phi_max));bottom_phi1_patch_ptr->rotate_y(phi1);bottom_phi1_patch_ptr->translate(xc3, 0.0, zc3);bottom_phi1_patch_ptr->transform_texture(false);objects.push_back(bottom_phi1_patch_ptr);// compute the bounding boxdouble x[4] = {xc1, xc2, xc3, xc4};double z[4] = {zc1, zc2, zc3, zc4};// first, assume that the wedge is completely inside a quadrant, which will be true for most wedges// work out the maximum and minimum valuesdouble x0 = kHugeValue;double z0 = kHugeValue;for (int j = 0; j <= 3; j++)  {if (x[j] < x0)x0 = x[j];}for (int j = 0; j <= 3; j++) {if (z[j] < z0)z0 = z[j];}double x1 = -kHugeValue;double z1 = -kHugeValue;for (int j = 0; j <= 3; j++) {if (x[j] > x1)x1 = x[j];}for (int j = 0; j <= 3; j++) {if (z[j] > z1)z1 = z[j];}// assign values to the bounding boxbbox.x0 = x0 - rb;bbox.y0 = y0;bbox.z0 = z0 - rb;bbox.x1 = x1 + rb;bbox.y1 = y1;bbox.z1 = z1 + rb;bool spans90 = phi0 < 90 and phi1 > 90;bool spans180 = phi0 < 180 and phi1 > 180;bool spans270 = phi0 < 270 and phi1 > 270;if (spans90 && spans180 && spans270) {bbox.x0 = -r1;bbox.z0 = -r1;bbox.x1 = r1;bbox.z1 = max(zc2, zc4);}else if (spans90 && spans180) {bbox.x0 = xc4 - rb;bbox.z0 = -r1;bbox.x1 = r1;bbox.z1 = zc2 + rb;}else if (spans180 && spans270) {bbox.x0 = -r1;bbox.z0 = -r1;bbox.x1 = xc2 + rb;bbox.z1 = zc4 + rb;}else if (spans90) {bbox.x0 = min(xc1, xc3);bbox.z0 = zc4 - rb;bbox.x1 = r1;bbox.z1 = zc2 + rb;}else if (spans180) {bbox.x0 = xc4 - rb;bbox.z0 = -r1;bbox.x1 = xc2 + rb;bbox.z1 = max(zc1, zc3);}else if (spans270) {bbox.x0 = -r1;bbox.z0 = zc2 - rb;bbox.x1 = max(xc1, xc3);bbox.z1 = zc4 + rb;}
}

3，其他说明

完整代码下载链接：
http://download.csdn.net/detail/libing_zeng/9815179

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Q103：磨边的物体（Beveled Objects）