圆弧球的碰撞检测
Collision detection of a ball with an arc
我正在制作一个简单的游戏，其中有一个球和一条围绕中心旋转的弧线。当用户触摸屏幕时，球会向指针方向移动并击中弧线。但是我找不到任何方法来检测该碰撞
附上一张图片以便更好地理解
游戏图片
调试快照..
我的球周围有一个圆圈...
我正在做的是
Detecting the point of intersection of ball center and circle on which arc is >revolving.
But i am not able to detect whether the arc was there when ball intersected the circle??
please help...:'(
制作圆弧的代码：
public void arc (float x, float y, float radius, float start, float degrees,int segments) {
// int segments = (int)(6 * (float)Math.cbrt(radius) * (degrees / 360.0f));
if (segments <= 0) throw new IllegalArgumentException("segments must be > 0.");
float colorBits = color.toFloatBits();
float theta = (2 * MathUtils.PI * (degrees / 360.0f)) / segments;
float cos = MathUtils.cos(theta);
float sin = MathUtils.sin(theta);
float cx = radius * MathUtils.cos(start * MathUtils.degreesToRadians);
float cy = radius * MathUtils.sin(start * MathUtils.degreesToRadians);
for (int i = 0; i < segments; i++) {
renderer.color(colorBits);
Gdx.gl20.glLineWidth(10);
Gdx.gl.glEnable(GL20.GL_BLEND);
renderer.vertex(x + cx, y + cy, 0);
float temp = cx;
cx = cos * cx - sin * cy;
cy = sin * temp + cos * cy;
renderer.color(colorBits);
renderer.vertex(x + cx, y + cy, 0);
}
}
如果您可以找到组成圆弧的线段，那么您可以使用 Intersector class 检查圆是否与每个线段相撞。我不知道你是怎么画弧线的，所以我不知道你能不能得到这些片段。
arc (float x, float y, float radius, float start, float degrees
好像x,y是圆心，start是起始角，degrees是掠角，所以结束角是end = start + degrees
如果你的交点是ix, iy，那么你可以查看这些叉积的符号：
cp_1 = (ix-x)*sin(start)-(iy-y)*Cos(start)
cp_2 = (ix-x)*sin(end)-(iy-y)*Cos(end)
如果cp1为负号，cp2为正（对于正弧方向度数），则(ix,iy)交点在(start,end)弧端点之间。 （适用于 Abs(degrees) < Pi=180）
Delphi 示例生成从-15 度到+30 度的弧，并检查从cx,cy 到某些点的光线是否与此弧相交：
function IsPointInArcLimits(cx, cy, ix, iy, StartAngle, Deg: Double): Boolean;
var
EndAngle, cp_1, cp_2: Double;
begin
EndAngle := DegToRad(StartAngle + Deg);
StartAngle := degToRad(StartAngle);
cp_1 := (ix - cx) * Sin(StartAngle) - (iy - cy) * Cos(StartAngle);
cp_2 := (ix - cx) * Sin(EndAngle) - (iy - cy) * Cos(EndAngle);
Result := (cp_1 <= 0) and (cp_2 >= 0);
end;
var
cx, cy, ix, iy, StartAngle, Degrees: Integer;
begin
cx := 0;
cy := 0;
ix := 10;
StartAngle := -15;
Degrees := 45;
for iy := -5 to 7 do
if IsPointInArcLimits(cx, cy, ix, iy, StartAngle, Degrees) then
Memo1.Lines.Add(Format('Yes y=%d an=%f arc %d+%d',
[iy, RadToDeg(ArcTan(iy / ix)), StartAngle, Degrees]))
else
Memo1.Lines.Add(Format('No y=%d an=%f arc %d+%d',
[iy, RadToDeg(ArcTan(iy / ix)), StartAngle, Degrees]));
输出：
No y=-5 an=-26.57 arc -15+45
No y=-4 an=-21.80 arc -15+45
No y=-3 an=-16.70 arc -15+45
Yes y=-2 an=-11.31 arc -15+45
Yes y=-1 an=-5.71 arc -15+45
Yes y=0 an=0.00 arc -15+45
Yes y=1 an=5.71 arc -15+45
Yes y=2 an=11.31 arc -15+45
Yes y=3 an=16.70 arc -15+45
Yes y=4 an=21.80 arc -15+45
Yes y=5 an=26.57 arc -15+45
No y=6 an=30.96 arc -15+45
No y=7 an=34.99 arc -15+45
什么是圆弧？很简单：两个圆的差，限制在两个向量（或三角形）内。
图表可能会有帮助；
大红圈的半径等于圆弧的外半径。较小的蓝色圆圈的半径等于圆弧的内半径减去球的直径。三角形显示圆弧的边缘。
从这里开始，只需测试球[距中心]的欧几里得距离与圆的半径的关系，然后找到从原点到球的两条切线，看看它们中的任何一条是否落在 angular 弧度的测量。
编辑：意识到我在自己的项目中需要这样的东西，所以我决定写下来；
double ball_radius = //Your radius of the ball
//the start and end angles of the arc
double start = //i.e -PI/4;
double end = //i.e PI/4;
double innerRadius = //inner radius of arc
double outerRadius = innerRadius + [width of lines, 10 in your code]
/* Now all the fun mathsy stuff */
boolean collides = false;
double dx = bx - cx; //[bx, by] = ball coords
double dy = by - cy; //[cx, cy] = center coords
//get distance and direction to ball from center
double dist = Math.sqrt(dx * dx + dy * dy);
double dir = Math.atan2(dy, dx);
//angles for tangents to ball from center
double tangent_angle = Math.asin(ball_radius/ dist);
double dir0 = dir + tangent_angle;
double dir1 = dir - tangent_angle;
//check if distance is good
if (dist + ball_radius> innerRadius && dist - ball_radius < outerRadius)
{
//check edges of ball against start and end of arc
boolean d = dir > start && dir < end;
boolean d0 = dir0 > start && dir0 < end;
boolean d1 = dir1 > start && dir1 < end;
//if both tangents are inside the angular measure
if (d || d0 && d1)
{
collides = true;
}
//otherwise if one tangent is inside
//We need to test the outside corners more precisely
else if (d0 != d1)
{
double x0 = cx + outerRadius * Math.cos(start) - bx;
double y0 = cy + outerRadius * Math.sin(start) - by;
double x1 = cx + outerRadius * Math.cos(end) - bx;
double y1 = cy + outerRadius * Math.sin(end) - by;
/** And so on for the other 2 corners */
/** If the arc is very thick, you will need to test against
the actual line segments at the ends of the arc */
if (x0 * x0 + y0 * y0 < ball_radius * ball_radius
|| x1 * x1 + y1 * y1 < ball_radius * ball_radius)
collides = true;
}
}
如果球只会击中弧线内侧，或者击中弧线角时有 3-4 个像素的不精确度是可以的，那么您可以替换整个 if-condition上面的代码，这样效率更高（但在角落里有点乱）；
if (dist > innerRadius - ball_radius && dist + ball_radius < outerRadius)
{
//if any tangent falls within the arc
collides = ((dir0 > start && dir0 < end) || (dir1 > start && dir1 < end));
}
最终结果：
我正在制作一个简单的游戏，其中有一个球和一条围绕中心旋转的弧线。当用户触摸屏幕时，球会向指针方向移动并击中弧线。但是我找不到任何方法来检测该碰撞 附上一张图片以便更好地理解
调试快照..
我的球周围有一个圆圈... 我正在做的是
Detecting the point of intersection of ball center and circle on which arc is >revolving. But i am not able to detect whether the arc was there when ball intersected the circle?? please help...:'(
制作圆弧的代码：
public void arc (float x, float y, float radius, float start, float degrees,int segments) {
// int segments = (int)(6 * (float)Math.cbrt(radius) * (degrees / 360.0f));
if (segments <= 0) throw new IllegalArgumentException("segments must be > 0.");
float colorBits = color.toFloatBits();
float theta = (2 * MathUtils.PI * (degrees / 360.0f)) / segments;
float cos = MathUtils.cos(theta);
float sin = MathUtils.sin(theta);
float cx = radius * MathUtils.cos(start * MathUtils.degreesToRadians);
float cy = radius * MathUtils.sin(start * MathUtils.degreesToRadians);
for (int i = 0; i < segments; i++) {
renderer.color(colorBits);
Gdx.gl20.glLineWidth(10);
Gdx.gl.glEnable(GL20.GL_BLEND);
renderer.vertex(x + cx, y + cy, 0);
float temp = cx;
cx = cos * cx - sin * cy;
cy = sin * temp + cos * cy;
renderer.color(colorBits);
renderer.vertex(x + cx, y + cy, 0);
}
}
如果您可以找到组成圆弧的线段，那么您可以使用 Intersector class 检查圆是否与每个线段相撞。我不知道你是怎么画弧线的，所以我不知道你能不能得到这些片段。
arc (float x, float y, float radius, float start, float degrees
好像x,y是圆心，start是起始角，degrees是掠角，所以结束角是end = start + degrees
如果你的交点是ix, iy，那么你可以查看这些叉积的符号：
cp_1 = (ix-x)*sin(start)-(iy-y)*Cos(start)
cp_2 = (ix-x)*sin(end)-(iy-y)*Cos(end)
如果cp1为负号，cp2为正（对于正弧方向度数），则(ix,iy)交点在(start,end)弧端点之间。 （适用于 Abs(degrees) < Pi=180）
Delphi 示例生成从-15 度到+30 度的弧，并检查从cx,cy 到某些点的光线是否与此弧相交：
function IsPointInArcLimits(cx, cy, ix, iy, StartAngle, Deg: Double): Boolean;
var
EndAngle, cp_1, cp_2: Double;
begin
EndAngle := DegToRad(StartAngle + Deg);
StartAngle := degToRad(StartAngle);
cp_1 := (ix - cx) * Sin(StartAngle) - (iy - cy) * Cos(StartAngle);
cp_2 := (ix - cx) * Sin(EndAngle) - (iy - cy) * Cos(EndAngle);
Result := (cp_1 <= 0) and (cp_2 >= 0);
end;
var
cx, cy, ix, iy, StartAngle, Degrees: Integer;
begin
cx := 0;
cy := 0;
ix := 10;
StartAngle := -15;
Degrees := 45;
for iy := -5 to 7 do
if IsPointInArcLimits(cx, cy, ix, iy, StartAngle, Degrees) then
Memo1.Lines.Add(Format('Yes y=%d an=%f arc %d+%d',
[iy, RadToDeg(ArcTan(iy / ix)), StartAngle, Degrees]))
else
Memo1.Lines.Add(Format('No y=%d an=%f arc %d+%d',
[iy, RadToDeg(ArcTan(iy / ix)), StartAngle, Degrees]));
输出：
No y=-5 an=-26.57 arc -15+45
No y=-4 an=-21.80 arc -15+45
No y=-3 an=-16.70 arc -15+45
Yes y=-2 an=-11.31 arc -15+45
Yes y=-1 an=-5.71 arc -15+45
Yes y=0 an=0.00 arc -15+45
Yes y=1 an=5.71 arc -15+45
Yes y=2 an=11.31 arc -15+45
Yes y=3 an=16.70 arc -15+45
Yes y=4 an=21.80 arc -15+45
Yes y=5 an=26.57 arc -15+45
No y=6 an=30.96 arc -15+45
No y=7 an=34.99 arc -15+45
什么是圆弧？很简单：两个圆的差，限制在两个向量（或三角形）内。
图表可能会有帮助；
大红圈的半径等于圆弧的外半径。较小的蓝色圆圈的半径等于圆弧的内半径减去球的直径。三角形显示圆弧的边缘。
从这里开始，只需测试球[距中心]的欧几里得距离与圆的半径的关系，然后找到从原点到球的两条切线，看看它们中的任何一条是否落在 angular 弧度的测量。
编辑：意识到我在自己的项目中需要这样的东西，所以我决定写下来；
double ball_radius = //Your radius of the ball
//the start and end angles of the arc
double start = //i.e -PI/4;
double end = //i.e PI/4;
double innerRadius = //inner radius of arc
double outerRadius = innerRadius + [width of lines, 10 in your code]
/* Now all the fun mathsy stuff */
boolean collides = false;
double dx = bx - cx; //[bx, by] = ball coords
double dy = by - cy; //[cx, cy] = center coords
//get distance and direction to ball from center
double dist = Math.sqrt(dx * dx + dy * dy);
double dir = Math.atan2(dy, dx);
//angles for tangents to ball from center
double tangent_angle = Math.asin(ball_radius/ dist);
double dir0 = dir + tangent_angle;
double dir1 = dir - tangent_angle;
//check if distance is good
if (dist + ball_radius> innerRadius && dist - ball_radius < outerRadius)
{
//check edges of ball against start and end of arc
boolean d = dir > start && dir < end;
boolean d0 = dir0 > start && dir0 < end;
boolean d1 = dir1 > start && dir1 < end;
//if both tangents are inside the angular measure
if (d || d0 && d1)
{
collides = true;
}
//otherwise if one tangent is inside
//We need to test the outside corners more precisely
else if (d0 != d1)
{
double x0 = cx + outerRadius * Math.cos(start) - bx;
double y0 = cy + outerRadius * Math.sin(start) - by;
double x1 = cx + outerRadius * Math.cos(end) - bx;
double y1 = cy + outerRadius * Math.sin(end) - by;
/** And so on for the other 2 corners */
/** If the arc is very thick, you will need to test against
the actual line segments at the ends of the arc */
if (x0 * x0 + y0 * y0 < ball_radius * ball_radius
|| x1 * x1 + y1 * y1 < ball_radius * ball_radius)
collides = true;
}
}
如果球只会击中弧线内侧，或者击中弧线角时有 3-4 个像素的不精确度是可以的，那么您可以替换整个 if-condition上面的代码，这样效率更高（但在角落里有点乱）；
if (dist > innerRadius - ball_radius && dist + ball_radius < outerRadius)
{
//if any tangent falls within the arc
collides = ((dir0 > start && dir0 < end) || (dir1 > start && dir1 < end));
}
最终结果：
