本文實現一個經典的物理算法：剛體碰撞檢測與響應。這個算法用于檢測兩個剛體（如矩形或圓形）是否發生碰撞，并在碰撞時更新它們的速度和位置。我們將使用C++來實現這個算法，并結合**邊界框（Bounding Box）**數據結構來管理剛體的碰撞檢測。

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1. 問題描述

剛體碰撞檢測與響應是物理引擎中的核心問題。我們將實現一個簡單的二維剛體碰撞系統，支持矩形和圓形剛體的碰撞檢測，并在碰撞時應用基本的物理響應（如動量守恒）。

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2. 數據結構設計

• 剛體（RigidBody）：包含位置、速度、質量、形狀（矩形或圓形）等屬性。
• 邊界框（BoundingBox）：用于快速檢測剛體是否可能發生碰撞。
• 系統（CollisionSystem）：管理所有剛體，并檢測和處理碰撞。

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3. 算法實現

#include <iostream>
#include <vector>
#include <cmath>// 定義二維向量
struct Vector2 {double x, y;Vector2(double x = 0, double y = 0) : x(x), y(y) {}Vector2 operator+(const Vector2& other) const {return Vector2(x + other.x, y + other.y);}Vector2 operator-(const Vector2& other) const {return Vector2(x - other.x, y - other.y);}Vector2 operator*(double scalar) const {return Vector2(x * scalar, y * scalar);}double length() const {return std::sqrt(x * x + y * y);}
};// 剛體形狀基類
struct Shape {virtual bool collidesWith(const Shape* other) const = 0;virtual ~Shape() = default;
};// 圓形剛體
struct Circle : Shape {double x, y; // 圓形的位置double radius;Circle(double x, double y, double r) : x(x), y(y), radius(r) {}bool collidesWith(const Shape* other) const override {if (const Circle* circle = dynamic_cast<const Circle*>(other)) {double distance = (Vector2(x, y) - Vector2(circle->x, circle->y)).length();return distance <= (radius + circle->radius);}return false;}
};// 矩形剛體
struct Rectangle : Shape {double x, y; // 矩形的位置double width, height;Rectangle(double x, double y, double w, double h) : x(x), y(y), width(w), height(h) {}bool collidesWith(const Shape* other) const override {if (const Rectangle* rect = dynamic_cast<const Rectangle*>(other)) {return !(x + width < rect->x || rect->x + rect->width < x ||y + height < rect->y || rect->y + rect->height < y);}return false;}
};// 剛體類
struct RigidBody {Vector2 position;Vector2 velocity;double mass;Shape* shape;RigidBody(Vector2 pos, Vector2 vel, double m, Shape* s): position(pos), velocity(vel), mass(m), shape(s) {// 更新形狀的位置if (Circle* circle = dynamic_cast<Circle*>(shape)) {circle->x = pos.x;circle->y = pos.y;} else if (Rectangle* rect = dynamic_cast<Rectangle*>(shape)) {rect->x = pos.x;rect->y = pos.y;}}void update(double dt) {position = position + velocity * dt;// 更新形狀的位置if (Circle* circle = dynamic_cast<Circle*>(shape)) {circle->x = position.x;circle->y = position.y;} else if (Rectangle* rect = dynamic_cast<Rectangle*>(shape)) {rect->x = position.x;rect->y = position.y;}}
};// 碰撞系統類
struct CollisionSystem {std::vector<RigidBody*> bodies;void addBody(RigidBody* body) {bodies.push_back(body);}void checkCollisions() {for (size_t i = 0; i < bodies.size(); ++i) {for (size_t j = i + 1; j < bodies.size(); ++j) {if (bodies[i]->shape->collidesWith(bodies[j]->shape)) {resolveCollision(bodies[i], bodies[j]);}}}}void resolveCollision(RigidBody* body1, RigidBody* body2) {// 簡單動量守恒碰撞響應Vector2 relativeVelocity = body2->velocity - body1->velocity;Vector2 normal = (body2->position - body1->position).length() > 0? (body2->position - body1->position) * (1.0 / (body2->position - body1->position).length()): Vector2(1, 0);double velocityAlongNormal = relativeVelocity.x * normal.x + relativeVelocity.y * normal.y;if (velocityAlongNormal > 0) return; // 已經分離，不需要處理double e = 1.0; // 恢復系數（完全彈性碰撞）double j = -(1 + e) * velocityAlongNormal;j /= (1 / body1->mass + 1 / body2->mass);Vector2 impulse = normal * j;body1->velocity = body1->velocity - impulse * (1 / body1->mass);body2->velocity = body2->velocity + impulse * (1 / body2->mass);}void update(double dt) {for (auto body : bodies) {body->update(dt);}checkCollisions();}
};int main() {// 創建碰撞系統CollisionSystem system;// 創建剛體RigidBody* body1 = new RigidBody(Vector2(0, 0), Vector2(1, 0), 1.0, new Circle(0, 0, 1.0));RigidBody* body2 = new RigidBody(Vector2(3, 0), Vector2(-1, 0), 1.0, new Circle(3, 0, 1.0));system.addBody(body1);system.addBody(body2);// 模擬double dt = 0.01;for (int i = 0; i < 100; i++) {system.update(dt);std::cout << "Body1 Position: (" << body1->position.x << ", " << body1->position.y << ")\n";std::cout << "Body2 Position: (" << body2->position.x << ", " << body2->position.y << ")\n";}// 釋放內存delete body1->shape;delete body2->shape;delete body1;delete body2;return 0;
}

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4. 代碼解釋

1. Vector2：表示二維向量，支持加減乘等操作。
2. Shape：剛體形狀基類，支持圓形和矩形的碰撞檢測。
3. Circle 和 Rectangle：分別表示圓形和矩形剛體，實現碰撞檢測邏輯。
4. RigidBody：表示剛體，包含位置、速度、質量和形狀。
5. CollisionSystem：管理所有剛體，檢測碰撞并應用碰撞響應。
6. main函數：創建系統、剛體，并進行模擬。

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5. 運行結果

程序會輸出兩個剛體的位置隨時間的變化，模擬它們的碰撞和運動。

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