334 lines
10 KiB
Markdown
334 lines
10 KiB
Markdown
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# SDF 隐式建模教程
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有符号距离函数（Signed Distance Function）是 ViewDesignEngine 的核心建模方式之一。
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通过数学函数直接定义几何体，无需依赖网格或曲面表示。
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## 核心概念
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SDF 函数 `f(p)` 返回点 `p` 到最近表面的有符号距离：
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- `f(p) < 0`：点在几何体**内部**
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- `f(p) = 0`：点在几何体**表面**上
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- `f(p) > 0`：点在几何体**外部**
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## 基本图元（球、盒、环）
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ViewDesignEngine 提供的内建 SDF 图元，定义在 `vde/sdf/sdf_primitives.h`：
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```cpp
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#include <vde/sdf/sdf_primitives.h>
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using namespace vde::sdf;
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using namespace vde::core;
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// 球体：位于原点，半径 1.0
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double d = sphere(Point3D(1.0, 0.0, 0.0), 1.0);
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// d ≈ 0.0（点在球面上）
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// 轴对齐盒子：以原点为中心，半长为 halfExtent
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auto box_fn = [](double x, double y, double z) {
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return box(Point3D(x, y, z), Point3D(1.5, 1.5, 0.5));
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};
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// 环面（torus）：major_radius = 环半径，minor_radius = 管半径
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auto torus_fn = [](double x, double y, double z) {
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return torus(Point3D(x, y, z), 2.0, 0.4);
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};
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// 椭球：指定各轴半径
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auto ellipsoid_fn = [](double x, double y, double z) {
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return ellipsoid(Point3D(x, y, z), Point3D(2.0, 1.0, 1.5));
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};
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// 圆柱（沿 Y 轴）
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auto cylinder_fn = [](double x, double y, double z) {
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return cylinder(Point3D(x, y, z), 0.5, 3.0);
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};
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// 胶囊体：两点 + 半径
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auto capsule_fn = [](double x, double y, double z) {
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return capsule(Point3D(x, y, z),
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Point3D(0, -2, 0), Point3D(0, 2, 0), 0.4);
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};
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// 无限长圆柱：沿任意轴
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auto inf_cyl = [](double x, double y, double z) {
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return infinite_cylinder(Point3D(x, y, z),
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Vector3D(0, 0, 1), 0.5);
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};
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```
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## CSG 操作（并、交、差、光滑并集）
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布尔组合运算定义在 `vde/sdf/sdf_operations.h`：
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```cpp
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#include <vde/sdf/sdf_primitives.h>
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#include <vde/sdf/sdf_operations.h>
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using namespace vde::sdf;
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// === 标准布尔运算 ===
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// 并集 A ∪ B：取最小值（两个几何体的合并）
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auto union_shape = [](double x, double y, double z) {
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Point3D p(x, y, z);
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return op_union(sphere(p, 1.0),
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box(p, Point3D(0.8, 0.8, 0.8)));
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};
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// 交集 A ∩ B：取最大值（两个几何体的共同区域）
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auto intersect_shape = [](double x, double y, double z) {
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Point3D p(x, y, z);
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return op_intersection(sphere(p, 1.0),
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box(p, Point3D(0.6, 0.6, 0.6)));
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};
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// 差集 A \ B：从 A 中减去 B
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auto diff_shape = [](double x, double y, double z) {
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Point3D p(x, y, z);
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return op_difference(sphere(p, 1.0),
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box(p, Point3D(0.4, 0.4, 0.4)));
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};
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// === 光滑布尔运算（带有机融合过渡） ===
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// smooth union，k 控制过渡弧度
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auto smooth_union_shape = [](double x, double y, double z) {
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Point3D p(x, y, z);
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double d1 = sphere(p, 1.0);
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double d2 = sphere(p - Point3D(1.5, 0, 0), 1.0);
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return op_smooth_union(d1, d2, 0.3); // k=0.3 适中的融合
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};
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// smooth intersection
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auto smooth_isect_shape = [](double x, double y, double z) {
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Point3D p(x, y, z);
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double d1 = sphere(p, 1.0);
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double d2 = box(p, Point3D(0.8, 0.8, 0.8));
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return op_smooth_intersection(d1, d2, 0.2);
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};
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// smooth difference
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auto smooth_diff_shape = [](double x, double y, double z) {
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Point3D p(x, y, z);
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double d1 = box(p, Point3D(1.5, 1.5, 1.5));
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double d2 = sphere(p, 0.8);
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return op_smooth_difference(d1, d2, 0.2);
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};
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```
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## 域变形（重复、镜像、扭转）
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域变形算子对采样点做空间变换，再传入 SDF 函数求值：
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```cpp
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#include <vde/sdf/sdf_operations.h>
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using namespace vde::sdf;
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// 无限重复：将空间单元化，在每个单元内放置几何体
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auto repeated_shape = [](double x, double y, double z) {
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Point3D p(x, y, z);
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// 以 (2, 2, 2) 为周期重复
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Point3D q = op_repeat(p, Point3D(2.0, 2.0, 2.0));
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return sphere(q, 0.5);
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};
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// 镜像：沿 YZ 平面对称（x 方向镜像，offset 为对称轴位置）
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auto mirrored_shape = [](double x, double y, double z) {
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Point3D p(x, y, z);
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Point3D q = op_mirror_x(p, 0.0); // 以 x=0 为镜像轴
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return sphere(q - Point3D(1.5, 0, 0), 0.6);
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};
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// 多维镜像组合
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auto quad_mirror = [](double x, double y, double z) {
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Point3D p(x, y, z);
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Point3D q = op_mirror_x(op_mirror_y(p));
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return sphere(q - Point3D(2.0, 2.0, 0), 0.5);
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};
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// 平移 / 旋转 / 缩放
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auto transformed_shape = [](double x, double y, double z) {
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Point3D p(x, y, z);
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// sdf_operations.h 提供 op_translate, op_rotate, op_scale 等
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// 注意：是逆变换——要移动几何体，需反向移动采样点
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Point3D q = op_translate(p, Point3D(2.0, 0, 0)); // 几何体向右移 2
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return sphere(q, 1.0);
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};
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// 扭转（沿 Y 轴）
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#include <cmath>
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auto twisted_shape = [](double x, double y, double z) -> double {
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double angle = y * 0.5; // 扭转角度与 y 坐标成正比
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double sx = std::sin(angle);
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double cx = std::cos(angle);
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double rx = cx * x - sx * z;
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double rz = sx * x + cx * z;
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Point3D q(rx, y, rz);
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return box(q, Point3D(0.8, 3.0, 0.3));
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};
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// 修饰算子：圆角、壳
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auto decorated_shape = [](double x, double y, double z) {
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Point3D p(x, y, z);
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double d = box(p, Point3D(1.0, 1.0, 1.0));
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// op_round: 对所有边做圆角（近似 round_box）
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d = op_round(d, 0.1);
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// op_onion: 创建壳层（空心厚度）
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// d = op_onion(d, 0.05);
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return d;
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};
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```
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## 从 SDF 到网格（Marching Cubes）
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```cpp
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#include <vde/sdf/sdf_primitives.h>
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#include <vde/sdf/sdf_operations.h>
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#include <vde/mesh/marching_cubes.h>
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#include <vde/mesh/halfedge_mesh.h>
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#include <vde/foundation/io_gltf.h>
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using namespace vde::core;
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using namespace vde::sdf;
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using namespace vde::mesh;
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using namespace vde::foundation;
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int main() {
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// 定义 SDF 函数
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auto shape = [](double x, double y, double z) -> double {
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Point3D p(x, y, z);
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double d = box(p, Point3D(2.0, 2.0, 2.0));
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d = op_round(d, 0.15); // 圆角
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return d;
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};
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// 采样区域：[-3,3]³
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Point3D bmin(-3, -3, -3), bmax(3, 3, 3);
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// Marching Cubes：64³ 分辨率
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auto mc = marching_cubes(shape, 0.0, bmin, bmax, 64);
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// 构建半边网格并导出
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HalfedgeMesh mesh;
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mesh.build_from_triangles(mc.vertices, mc.triangles);
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GltfOptions opts;
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opts.binary = true;
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opts.include_normals = true;
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write_gltf("rounded_box.glb", mesh, opts);
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std::cout << "Vertices: " << mesh.num_vertices()
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<< ", Faces: " << mesh.num_faces() << "\n";
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return 0;
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}
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```
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**分辨率选择指南：**
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| 分辨率 | 适用场景 | 输出网格规模 |
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|--------|----------|-------------|
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| 32³ | 快速预览 | ~2K 顶点 |
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| 64³ | 一般用途 | ~8K 顶点 |
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| 128³ | 精细表面 | ~30K 顶点 |
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| 256³ | 高精度需求 | ~100K+ 顶点 |
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## 完整示例：齿轮形状
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```cpp
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#include <vde/sdf/sdf_primitives.h>
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#include <vde/sdf/sdf_operations.h>
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#include <vde/mesh/marching_cubes.h>
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#include <vde/mesh/halfedge_mesh.h>
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#include <vde/foundation/io_gltf.h>
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#include <iostream>
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#include <cmath>
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using namespace vde::core;
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using namespace vde::sdf;
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using namespace vde::mesh;
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using namespace vde::foundation;
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int main() {
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const double M_PI = 3.14159265358979323846;
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const double gear_radius = 2.0; // 齿轮半径
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const double tooth_count = 12.0; // 齿数
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const double tooth_depth = 0.3; // 齿深
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const double gear_thickness = 0.5; // 齿轮厚度
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auto gear_sdf = [&](double x, double y, double z) -> double {
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Point3D p(x, y, z);
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// 圆柱主体（沿 Y 轴的薄圆盘）
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double d_cylinder = cylinder(Point3D(x, y, z), gear_radius, gear_thickness);
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// 中心孔
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double d_hole = cylinder(Point3D(x, y, z), 0.4, gear_thickness * 2);
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// 差集：打孔
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double d = op_difference(d_cylinder, d_hole);
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// 齿轮齿：沿圆周等距分布
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double angle = std::atan2(z, x);
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double radius = std::sqrt(x * x + z * z);
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// 每个齿是一个小 Box 沿径向放置
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double teeth_union = 1e10;
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for (int i = 0; i < tooth_count; i++) {
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double theta = i * 2.0 * M_PI / tooth_count;
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double ca = std::cos(theta);
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double sa = std::sin(theta);
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// 旋转采样点到该齿的局部空间
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double lx = ca * x + sa * z;
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double lz = -sa * x + ca * z;
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Point3D lp(lx, y, lz);
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double tooth = box(lp,
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Point3D(0.3, 0.6, gear_radius + tooth_depth));
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// 向圆心方向偏移
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double d_tooth = op_translate_along_x(lp,
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gear_radius + tooth_depth * 0.5).norm() > 0
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? box(Point3D(lx - (gear_radius + tooth_depth * 0.5), y, lz),
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Point3D(0.15, gear_thickness * 1.1, tooth_depth * 0.5))
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: 1e10;
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// 用旋转矩阵的逆变换做齿的径向放置
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double r = std::sqrt(x * x + z * z);
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double dx = r - (gear_radius + tooth_depth * 0.5);
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double dy_abs = std::abs(y);
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double dz = std::abs(std::atan2(z, x) - theta);
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if (dz > M_PI) dz = 2 * M_PI - dz;
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dz = dz * gear_radius;
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double tooth_dist = std::sqrt(dx * dx + dy_abs * dy_abs + dz * dz)
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- tooth_depth * 0.5;
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if (dy_abs <= gear_thickness * 0.55 && r >= gear_radius - 0.05) {
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teeth_union = std::min(teeth_union, tooth_dist);
|
|||
|
|
}
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
d = op_union(d, teeth_union);
|
|||
|
|
return d;
|
|||
|
|
};
|
|||
|
|
|
|||
|
|
// Marching Cubes
|
|||
|
|
Point3D bmin(-3, -3, -3), bmax(3, 3, 3);
|
|||
|
|
auto mc = marching_cubes(gear_sdf, 0.0, bmin, bmax, 128);
|
|||
|
|
|
|||
|
|
HalfedgeMesh mesh;
|
|||
|
|
mesh.build_from_triangles(mc.vertices, mc.triangles);
|
|||
|
|
|
|||
|
|
GltfOptions opts;
|
|||
|
|
opts.binary = true;
|
|||
|
|
opts.include_normals = true;
|
|||
|
|
write_gltf("gear.glb", mesh, opts);
|
|||
|
|
|
|||
|
|
std::cout << "Gear exported: " << mesh.num_vertices()
|
|||
|
|
<< " vertices, " << mesh.num_faces() << " faces\n";
|
|||
|
|
return 0;
|
|||
|
|
}
|
|||
|
|
```
|
|||
|
|
|
|||
|
|
编译运行，用 `viewer/index.html` 查看效果。
|