feat(sdf): S12 — 可微分几何完整模块

S12-A: 自动微分引擎
- sdf_gradient.h: 前向/反向梯度、解析梯度（sphere/box/torus/cylinder/plane/capsule）
- 链式法则: union/intersection/difference/smooth + translate/rotate/scale/twist/repeat
- 参数梯度: sphere/box/cylinder 的 param_grad
- test_sdf_gradient.cpp: 604 行测试

S12-B: 梯度驱动优化 + 应用
- sdf_optimize.h: 形状拟合/碰撞避免/可达性/对称检测/体积计算
- sdf_optimize.cpp: 682 行实现（数值梯度 + GradientDescent）
- test_sdf_optimize.cpp: 342 行，20 项测试

S12-C: PyTorch 集成 + Python
- sdf_torch.h/cpp: 批量 SDF 求值 + 梯度
- python/vde/sdf.py: 高级 Python API
- python/vde/torch_sdf.py: torch.autograd.Function
- test_sdf_torch_bridge.cpp: C++ 桥梁测试

include/vde/sdf/sdf_optimize.h

@@ -0,0 +1,178 @@
+#pragma once
+#include "vde/sdf/sdf_tree.h"
+#include "vde/core/point.h"
+#include <vector>
+#include <functional>
+
+namespace vde::sdf {
+
+using core::Point3D;
+using core::Vector3D;
+
+// ────────────────────────────────────────────────
+//  Shape Optimization
+// ────────────────────────────────────────────────
+
+/// Optimization result
+struct OptimizeResult {
+    SdfNodePtr optimized_shape;
+    double final_loss;
+    int iterations;
+    bool converged;
+    std::vector<double> loss_history;
+};
+
+/// Loss function type: takes SDF evaluator + target, returns scalar loss
+using LossFn = std::function<double(const std::function<double(const Point3D&)>&)>;
+
+// ── Shape Fitting ───────────────────────────────
+
+/// Fit a parameterized shape to a target point cloud by minimizing
+/// distance of surface to target points.
+/// @param initial Initial shape (sphere, box, cylinder, etc.)
+/// @param target_points Target point cloud (surface samples)
+/// @param learning_rate Gradient descent step size
+/// @param max_iterations Maximum iterations
+/// @return Optimized shape + convergence info
+[[nodiscard]] OptimizeResult fit_to_point_cloud(
+    const SdfNodePtr& initial,
+    const std::vector<Point3D>& target_points,
+    double learning_rate = 0.01,
+    int max_iterations = 100);
+
+/// Fit a shape to minimize SDF values at target points (drive surface to points)
+/// Loss = mean(|sdf(p_i)|) for p_i in targets
+[[nodiscard]] OptimizeResult fit_surface_to_points(
+    const SdfNodePtr& initial,
+    const std::vector<Point3D>& surface_points,
+    double learning_rate = 0.01,
+    int max_iterations = 100);
+
+/// Fit a shape to match another SDF (shape matching)
+/// Loss = mean((sdf_a(p_i) - sdf_b(p_i))^2) over sample grid
+[[nodiscard]] OptimizeResult fit_to_sdf(
+    const SdfNodePtr& source,
+    const std::function<double(const Point3D&)>& target_sdf,
+    const Point3D& bmin, const Point3D& bmax,
+    int grid_resolution = 16,
+    double learning_rate = 0.01,
+    int max_iterations = 100);
+
+// ── Collision Avoidance ─────────────────────────
+
+/// Push shapes apart to resolve interpenetration.
+/// Modifies shapes in-place by translating them.
+/// @param shape_a First shape
+/// @param shape_b Second shape
+/// @param step_size Translation step per iteration
+/// @param max_iterations Max iterations
+/// @return True if collision resolved
+[[nodiscard]] bool resolve_collision(
+    SdfNodePtr& shape_a, SdfNodePtr& shape_b,
+    double step_size = 0.1,
+    int max_iterations = 50);
+
+/// Find minimum translation distance to avoid collision
+[[nodiscard]] double penetration_depth(
+    const SdfNodePtr& shape_a,
+    const SdfNodePtr& shape_b,
+    const Point3D& bmin, const Point3D& bmax,
+    int samples = 1000);
+
+// ── Reachability / Accessibility ─────────────────
+
+/// Compute accessibility score at point p on surface of shape.
+/// Returns 0 (inaccessible) to 1 (fully accessible).
+/// Uses hemisphere sampling + occlusion testing.
+[[nodiscard]] double accessibility(
+    const SdfNodePtr& shape,
+    const Point3D& p,           // surface point
+    const Vector3D& direction,  // approach direction
+    int hemisphere_samples = 64);
+
+/// Find point with maximum accessibility
+[[nodiscard]] Point3D find_accessible_point(
+    const SdfNodePtr& shape,
+    const Vector3D& approach_dir,
+    const Point3D& bmin, const Point3D& bmax,
+    int grid_res = 32);
+
+// ── Symmetry Detection ──────────────────────────
+
+/// Detect if shape has reflective symmetry in given direction.
+/// Returns symmetry score: 0 = asymmetric, 1 = perfectly symmetric.
+[[nodiscard]] double symmetry_score(
+    const SdfNodePtr& shape,
+    const Point3D& bmin, const Point3D& bmax,
+    const Vector3D& plane_normal,
+    double plane_offset = 0.0,
+    int samples = 1000);
+
+/// Find the best symmetry plane
+struct SymmetryResult {
+    double score;
+    Vector3D normal;
+    double offset;
+};
+[[nodiscard]] SymmetryResult find_symmetry_plane(
+    const SdfNodePtr& shape,
+    const Point3D& bmin, const Point3D& bmax,
+    int samples = 1000);
+
+// ── SDF Volume Computation ──────────────────────
+
+/// Approximate volume of implicit shape via Monte Carlo sampling
+[[nodiscard]] double estimate_volume(
+    const SdfNodePtr& shape,
+    const Point3D& bmin, const Point3D& bmax,
+    int samples = 10000);
+
+/// Compute center of mass via Monte Carlo
+[[nodiscard]] Point3D center_of_mass(
+    const SdfNodePtr& shape,
+    const Point3D& bmin, const Point3D& bmax,
+    int samples = 10000);
+
+// ── Parameter Collection ────────────────────────
+
+/// A reference to a mutable double parameter inside an SDF tree
+struct ParamRef {
+    double* value;
+    std::string name;
+};
+
+/// Collect all mutable numeric parameters from primitive leaf nodes
+[[nodiscard]] std::vector<ParamRef> collect_params(SdfNodePtr& root);
+
+/// Compute numerical gradient of a scalar loss w.r.t. collected parameters
+[[nodiscard]] std::vector<double> numerical_gradient(
+    const std::vector<ParamRef>& params,
+    const std::function<double()>& loss_fn,
+    double eps = 1e-6);
+
+// ── Gradient Descent Utility ────────────────────
+
+/// Simple gradient descent with fixed learning rate
+class GradientDescent {
+public:
+    explicit GradientDescent(double lr) : lr_(lr) {}
+
+    /// Step parameters toward minimizing loss.
+    /// @param params Parameter vector (mutable)
+    /// @param grad_fn Function that computes gradient for given params;
+    ///                returns loss as scalar and fills gradient vector.
+    /// @return Current loss
+    double step(std::vector<double>& params,
+                const std::function<double(const std::vector<double>&,
+                                           std::vector<double>&)>& grad_fn);
+
+    void set_learning_rate(double lr) { lr_ = lr; }
+    [[nodiscard]] double learning_rate() const { return lr_; }
+    [[nodiscard]] int iteration() const { return iteration_; }
+
+private:
+    double lr_;
+    int iteration_ = 0;
+};
+
+} // namespace vde::sdf