在一维弹性物体基础上设计实现二维弹性物体模拟,并加入和环境的交互;OpenGL

要实现二维弹性物体模拟，需要进行以下步骤：

1. 定义物体的形状和大小，可以通过多边形网格来表示物体的形状。

2. 定义物体的质量和弹性系数，这些参数可以影响物体的运动状态和形变程度。

3. 根据物体的形状和质量，计算物体的受力情况，包括重力、弹性力和摩擦力等。

4. 根据物体受力情况，更新物体的位置和速度状态，这些状态可以通过欧拉法或者其他数值方法进行模拟。

5. 加入与环境的交互，例如碰撞检测和反弹，可以使用OpenGL的碰撞检测函数和反弹函数来实现。

下面是一个简单的二维物体模拟的示例代码：

#include <iostream>
#include <cmath>
#include <GL/glut.h>

const int WIDTH = 800, HEIGHT = 600;
const int GRID_SIZE = 20;
const float MASS = 1.0f;
const float K = 100.0f;
const float DAMPING = 0.1f;
const float GRAVITY = 9.8f;

struct Vector2 {
    float x, y;

    Vector2() : x(0.0f), y(0.0f) {}

    Vector2(float x, float y) : x(x), y(y) {}

    Vector2 operator+(const Vector2& v) const {
        return Vector2(x + v.x, y + v.y);
    }

    Vector2 operator-(const Vector2& v) const {
        return Vector2(x - v.x, y - v.y);
    }

    Vector2 operator*(float s) const {
        return Vector2(x * s, y * s);
    }

    Vector2 operator/(float s) const {
        return Vector2(x / s, y / s);
    }

    float length() const {
        return std::sqrt(x * x + y * y);
    }

    Vector2 normalize() const {
        float len = length();
        if (len > 0.0f) {
            return (*this) / len;
        }
        return Vector2();
    }
};

struct Particle {
    Vector2 position;
    Vector2 velocity;
    Vector2 force;
    float mass;

    Particle() : position(), velocity(), force(), mass(MASS) {}

    Particle(Vector2 position, Vector2 velocity, float mass) :
        position(position), velocity(velocity), force(), mass(mass) {}

    void applyForce(const Vector2& f) {
        force = force + f;
    }

    void update(float dt) {
        Vector2 acceleration = force / mass;
        velocity = velocity + acceleration * dt;
        position = position + velocity * dt;
        force = Vector2();
    }

    void draw() {
        glColor3f(1.0f, 1.0f, 1.0f);
        glPointSize(6.0f);
        glBegin(GL_POINTS);
        glVertex2f(position.x, position.y);
        glEnd();
    }
};

struct Spring {
    Particle* p1;
    Particle* p2;
    float restLength;
    float k;
    float damping;

    Spring() : p1(nullptr), p2(nullptr), restLength(0.0f), k(K), damping(DAMPING) {}

    Spring(Particle* p1, Particle* p2, float restLength, float k) :
        p1(p1), p2(p2), restLength(restLength), k(k), damping(DAMPING) {}

    void update() {
        Vector2 delta = p2->position - p1->position;
        float length = delta.length();
        Vector2 direction = delta.normalize();
        Vector2 relativeVelocity = p2->velocity - p1->velocity;
        float velocityAlongAxis = relativeVelocity.x * direction.x + relativeVelocity.y * direction.y;
        float forceMagnitude = -k * (length - restLength);
        forceMagnitude -= damping * velocityAlongAxis;
        Vector2 force = direction * forceMagnitude;
        p1->applyForce(force);
        p2->applyForce(-force);
    }

    void draw() {
        glColor3f(0.0f, 1.0f, 1.0f);
        glLineWidth(2.0f);
        glBegin(GL_LINES);
        glVertex2f(p1->position.x, p1->position.y);
        glVertex2f(p2->position.x, p2->position.y);
        glEnd();
    }
};

Particle* particles[GRID_SIZE][GRID_SIZE];
Spring* springs[GRID_SIZE][GRID_SIZE];

void initParticlesAndSprings() {
    // create particles
    for (int i = 0; i < GRID_SIZE; i++) {
        for (int j = 0; j < GRID_SIZE; j++) {
            particles[i][j] = new Particle(Vector2(i * 20.0f, j * 20.0f), Vector2(), MASS);
        }
    }

    // create springs
    for (int i = 0; i < GRID_SIZE; i++) {
        for (int j = 0; j < GRID_SIZE; j++) {
            if (i < GRID_SIZE - 1) {
                springs[i][j] = new Spring(particles[i][j], particles[i + 1][j], 20.0f, K);
            }
            if (j < GRID_SIZE - 1) {
                springs[i][j] = new Spring(particles[i][j], particles[i][j + 1], 20.0f, K);
            }
        }
    }
}

void update(float dt) {
    // apply gravity
    for (int i = 0; i < GRID_SIZE; i++) {
        for (int j = 0; j < GRID_SIZE; j++) {
            particles[i][j]->applyForce(Vector2(0.0f, -particles[i][j]->mass * GRAVITY));
        }
    }

    // update particles
    for (int i = 0; i < GRID_SIZE; i++) {
        for (int j = 0; j < GRID_SIZE; j++) {
            particles[i][j]->update(dt);
        }
    }

    // update springs
    for (int i = 0; i < GRID_SIZE; i++) {
        for (int j = 0; j < GRID_SIZE; j++) {
            if (i < GRID_SIZE - 1) {
                springs[i][j]->update();
            }
            if (j < GRID_SIZE - 1) {
                springs[i][j]->update();
            }
        }
    }
}

void draw() {
    glClear(GL_COLOR_BUFFER_BIT);

    // draw particles
    for (int i = 0; i < GRID_SIZE; i++) {
        for (int j = 0; j < GRID_SIZE; j++) {
            particles[i][j]->draw();
        }
    }

    // draw springs
    for (int i = 0; i < GRID_SIZE; i++) {
        for (int j = 0; j < GRID_SIZE; j++) {
            if (i < GRID_SIZE - 1) {
                springs[i][j]->draw();
            }
            if (j < GRID_SIZE - 1) {
                springs[i][j]->draw();
            }
        }
    }

    glutSwapBuffers();
}

void reshape(int w, int h) {
    glViewport(0, 0, w, h);
    glMatrixMode(GL_PROJECTION);
    glLoadIdentity();
    gluOrtho2D(0, w, h, 0);
    glMatrixMode(GL_MODELVIEW);
}

void timer(int value) {
    const float dt = 0.01f;
    update(dt);
    glutPostRedisplay();
    glutTimerFunc(16, timer, 0);
}

int main(int argc, char** argv) {
    glutInit(&argc, argv);
    glutInitDisplayMode(GLUT_DOUBLE | GLUT_RGB);
    glutInitWindowSize(WIDTH, HEIGHT);
    glutCreateWindow("Elastic Body Simulation");
    glutDisplayFunc(draw);
    glutReshapeFunc(reshape);
    glutTimerFunc(0, timer, 0);
    initParticlesAndSprings();
    glutMainLoop();
    return 0;
}

在这个示例代码中，定义了一个二维网格状的物体，并通过粒子和弹簧模型来模拟物体的弹性特性。在每个时间步中，先计算物体的受力情况，然后根据欧拉法来更新物体的位置和速度状态。最后在OpenGL中绘制物体的形状和弹簧模型，形成一个二维弹性物体模拟系统。