简单的二维碰撞问题

Question

我想找出静态球和运动球之间发生碰撞的时间，但我提出的算法有时无法检测到碰撞，而运动球会穿过静态球。运动的球受重力影响，静止的则不受。

这是我的碰撞检测代码：

GLfloat whenSpheresCollide(const sphere2d &firstSphere, const sphere2d &secondSphere)
{
    Vector2f relativePosition = subtractVectors(firstSphere.vPosition, secondSphere.vPosition);
    Vector2f relativeVelocity = subtractVectors(firstSphere.vVelocity, secondSphere.vVelocity);

    GLfloat radiusSum = firstSphere.radius + secondSphere.radius;

    //We'll find the time when objects collide if a collision takes place

    //r(t) = P[0] + t * V[0]
    //
    //d^2(t) = P[0]^2 + 2 * t * P[0] * V[0] + t^2 * V[0]^2
    //
    //d^2(t) = V[0]^2 * t^2 + 2t( P[0] . V[0] ) + P[0]^2
    //
    //d(t) = R
    //
    //d(t)^2 = R^2
    //
    //V[0]^2 * t^2 + 2t( P[0] . V[0] ) + P[0]^2 - R^2 = 0
    //
    //delta = ( P[0] . V[0] )^2 - V[0]^2 * (P[0]^2 - R^2)
    //
    //  We are interested in the lowest t:
    //
    //t = ( -( P[0] . V[0] ) - sqrt(delta) ) / V[0]^2
    //

    GLfloat equationDelta = squaref( dotProduct(relativePosition, relativeVelocity) ) - squarev( relativeVelocity ) * ( squarev( relativePosition ) - squaref(radiusSum)  );

    if (equationDelta >= 0.0f)
    {
        GLfloat collisionTime = ( - dotProduct(relativePosition, relativeVelocity) - sqrtf(equationDelta) ) / squarev(relativeVelocity);

        if (collisionTime >= 0.0f && collisionTime <= 1.0f / GAME_FPS)
        {
            return collisionTime;
        }
    }

    return -1.0f;
}

这里是调用碰撞检测的更新函数：

void GamePhysicsManager::updateBallPhysics()
{
    //
    //Update velocity
    vVelocity->y -= constG / GAME_FPS;  //v = a * t = g * 1 sec / (updates per second)

    shouldApplyForcesToBall = TRUE;

    vPosition->x += vVelocity->x / GAME_FPS;
    vPosition->y += vVelocity->y / GAME_FPS;

    if ( distanceBetweenVectors( *pBall->getPositionVector(), *pBasket->getPositionVector() ) <= pBasket->getRadius() + vectorLength(*vVelocity) / GAME_FPS )
    {
        //Ball sphere
        sphere2d ballSphere;
        ballSphere.radius = pBall->getRadius();
        ballSphere.mass = 1.0f;
        ballSphere.vPosition = *( pBall->getPositionVector() );
        ballSphere.vVelocity = *( pBall->getVelocityVector() );


        sphere2d ringSphereRight;
        ringSphereRight.radius = 0.05f;
        ringSphereRight.mass = -1.0f;
        ringSphereRight.vPosition = *( pBasket->getPositionVector() );
        //ringSphereRight.vPosition.x += pBasket->getRadius();
        ringSphereRight.vPosition.x += (pBasket->getRadius() - ringSphereRight.radius);
        ringSphereRight.vVelocity = zeroVector();


        GLfloat collisionTime = whenSpheresCollide(ballSphere, ringSphereRight);

        if ( collisionTime >= 0.0f )
        {
            DebugLog("collision");
            respondToCollision(&ballSphere, &ringSphereRight, collisionTime, pBall->getRestitution() * 0.75f );
        }

        //
        //Implement selection of the results that are first to collide collision

        vVelocity->x = ballSphere.vVelocity.x;
        vVelocity->y = ballSphere.vVelocity.y;

        vPosition->x = ballSphere.vPosition.x;
        vPosition->y = ballSphere.vPosition.y;
    }

为什么没有 100% 的情况下检测到碰撞？它仅在 70% 的病例中被发现。 谢谢。

更新：当我将 FPS 从 30 更改为 10 时，问题似乎得到了解决。FPS 如何影响我的碰撞检测？

Answer 1

delta = ( P[0] . V[0] )^2 - V[0]^2 * (P[0]^2 - R^2)

那不应该是 delta = b² - 4 ac 吗？

---

[编辑]哦，我明白了，你把 4 排除在外了。在这种情况下，您确定要同时考虑 t 的两种解决方案吗？

t = ( -( P[0] . V[0] ) - sqrt(delta) ) / V[0]^2

和

t = ( -( P[0] . V[0] ) + sqrt(delta) ) / V[0]^2

Answer 2

球体有多大，它们移动的速度有多快？一个球体可以在一帧中“跳过”第二个球体（即，它的速度矢量是否比它的宽度长？）。

按照这些思路，如果您在此处删除上限会发生什么：

if (collisionTime >= 0.0f && collisionTime <= 1.0f / GAME_FPS)
{
    return collisionTime;
}

如果球体移动太快，可能是您的算法检测到发生在一帧以上之前的碰撞..（？）