如何分辨圆和矩形在二维欧几里得空间中是否相交?(即经典二维几何)
当前回答
这个函数检测Circle和Rectangle之间的碰撞(交集)。他的回答类似于e.James的方法,但这个方法检测矩形的所有角(不仅仅是右角)的碰撞。
注意:
aRect.origin.x和aRect.origin.y是矩形左下角的坐标!
aCircle。x和圆。y为圆心坐标!
static inline BOOL RectIntersectsCircle(CGRect aRect, Circle aCircle) {
float testX = aCircle.x;
float testY = aCircle.y;
if (testX < aRect.origin.x)
testX = aRect.origin.x;
if (testX > (aRect.origin.x + aRect.size.width))
testX = (aRect.origin.x + aRect.size.width);
if (testY < aRect.origin.y)
testY = aRect.origin.y;
if (testY > (aRect.origin.y + aRect.size.height))
testY = (aRect.origin.y + aRect.size.height);
return ((aCircle.x - testX) * (aCircle.x - testX) + (aCircle.y - testY) * (aCircle.y - testY)) < aCircle.radius * aCircle.radius;
}
其他回答
这里有一个快速的单行测试:
if (length(max(abs(center - rect_mid) - rect_halves, 0)) <= radius ) {
// They intersect.
}
这是轴对齐的情况,其中rect_二分之一是一个正向量,从矩形的中间指向一个角。length()中的表达式是一个从矩形中心到最近点的增量向量。这适用于任何维度。
我想出的最简单的解决办法非常直接。
它的工作原理是在矩形中找到离圆最近的点,然后比较距离。
您可以通过一些操作来完成所有这些操作,甚至可以避免使用平方根函数。
public boolean intersects(float cx, float cy, float radius, float left, float top, float right, float bottom)
{
float closestX = (cx < left ? left : (cx > right ? right : cx));
float closestY = (cy < top ? top : (cy > bottom ? bottom : cy));
float dx = closestX - cx;
float dy = closestY - cy;
return ( dx * dx + dy * dy ) <= radius * radius;
}
就是这样!上面的解决方案假设原点在世界的左上方,x轴指向下方。
如果你想要一个解决移动的圆形和矩形之间碰撞的解决方案,这要复杂得多,并且包含在我的另一个答案中。
首先检查矩形和与圆相切的正方形是否重叠(简单)。如果它们不重叠,就不会碰撞。 检查圆的中心是否在矩形内(简单)。如果它在里面,它们就会碰撞。 计算矩形边到圆中心的最小平方距离(略硬)。如果小于半径的平方,它们就会碰撞,否则不会。
它是有效的,因为:
首先,它用一个便宜的算法检查最常见的场景,当它确定它们没有碰撞时,它就结束了。 然后它用一个廉价的算法检查下一个最常见的场景(不要计算平方根,使用平方值),当它确定它们碰撞时,它就结束了。 然后它执行更昂贵的算法来检查与矩形边框的碰撞。
假设你有矩形的四条边,检查从这些边到圆心的距离,如果小于半径,那么这些形状是相交的。
if sqrt((rectangleRight.x - circleCenter.x)^2 +
(rectangleBottom.y - circleCenter.y)^2) < radius
// then they intersect
if sqrt((rectangleRight.x - circleCenter.x)^2 +
(rectangleTop.y - circleCenter.y)^2) < radius
// then they intersect
if sqrt((rectangleLeft.x - circleCenter.x)^2 +
(rectangleTop.y - circleCenter.y)^2) < radius
// then they intersect
if sqrt((rectangleLeft.x - circleCenter.x)^2 +
(rectangleBottom.y - circleCenter.y)^2) < radius
// then they intersect
下面是我的C代码,用于解决球体和非轴对齐的盒子之间的碰撞。它依赖于我自己的几个库例程,但它可能对某些人有用。我在游戏中使用了它,效果非常好。
float physicsProcessCollisionBetweenSelfAndActorRect(SPhysics *self, SPhysics *actor)
{
float diff = 99999;
SVector relative_position_of_circle = getDifference2DBetweenVectors(&self->worldPosition, &actor->worldPosition);
rotateVector2DBy(&relative_position_of_circle, -actor->axis.angleZ); // This aligns the coord system so the rect becomes an AABB
float x_clamped_within_rectangle = relative_position_of_circle.x;
float y_clamped_within_rectangle = relative_position_of_circle.y;
LIMIT(x_clamped_within_rectangle, actor->physicsRect.l, actor->physicsRect.r);
LIMIT(y_clamped_within_rectangle, actor->physicsRect.b, actor->physicsRect.t);
// Calculate the distance between the circle's center and this closest point
float distance_to_nearest_edge_x = relative_position_of_circle.x - x_clamped_within_rectangle;
float distance_to_nearest_edge_y = relative_position_of_circle.y - y_clamped_within_rectangle;
// If the distance is less than the circle's radius, an intersection occurs
float distance_sq_x = SQUARE(distance_to_nearest_edge_x);
float distance_sq_y = SQUARE(distance_to_nearest_edge_y);
float radius_sq = SQUARE(self->physicsRadius);
if(distance_sq_x + distance_sq_y < radius_sq)
{
float half_rect_w = (actor->physicsRect.r - actor->physicsRect.l) * 0.5f;
float half_rect_h = (actor->physicsRect.t - actor->physicsRect.b) * 0.5f;
CREATE_VECTOR(push_vector);
// If we're at one of the corners of this object, treat this as a circular/circular collision
if(fabs(relative_position_of_circle.x) > half_rect_w && fabs(relative_position_of_circle.y) > half_rect_h)
{
SVector edges;
if(relative_position_of_circle.x > 0) edges.x = half_rect_w; else edges.x = -half_rect_w;
if(relative_position_of_circle.y > 0) edges.y = half_rect_h; else edges.y = -half_rect_h;
push_vector = relative_position_of_circle;
moveVectorByInverseVector2D(&push_vector, &edges);
// We now have the vector from the corner of the rect to the point.
float delta_length = getVector2DMagnitude(&push_vector);
float diff = self->physicsRadius - delta_length; // Find out how far away we are from our ideal distance
// Normalise the vector
push_vector.x /= delta_length;
push_vector.y /= delta_length;
scaleVector2DBy(&push_vector, diff); // Now multiply it by the difference
push_vector.z = 0;
}
else // Nope - just bouncing against one of the edges
{
if(relative_position_of_circle.x > 0) // Ball is to the right
push_vector.x = (half_rect_w + self->physicsRadius) - relative_position_of_circle.x;
else
push_vector.x = -((half_rect_w + self->physicsRadius) + relative_position_of_circle.x);
if(relative_position_of_circle.y > 0) // Ball is above
push_vector.y = (half_rect_h + self->physicsRadius) - relative_position_of_circle.y;
else
push_vector.y = -((half_rect_h + self->physicsRadius) + relative_position_of_circle.y);
if(fabs(push_vector.x) < fabs(push_vector.y))
push_vector.y = 0;
else
push_vector.x = 0;
}
diff = 0; // Cheat, since we don't do anything with the value anyway
rotateVector2DBy(&push_vector, actor->axis.angleZ);
SVector *from = &self->worldPosition;
moveVectorBy2D(from, push_vector.x, push_vector.y);
}
return diff;
}