我试图创建一个快速的2D点内多边形算法,用于命中测试(例如多边形.contains(p:点))。对有效技术的建议将不胜感激。
当前回答
nirg的c#版本的答案在这里:我只分享代码。这可能会节省一些时间。
public static bool IsPointInPolygon(IList<Point> polygon, Point testPoint) {
bool result = false;
int j = polygon.Count() - 1;
for (int i = 0; i < polygon.Count(); i++) {
if (polygon[i].Y < testPoint.Y && polygon[j].Y >= testPoint.Y || polygon[j].Y < testPoint.Y && polygon[i].Y >= testPoint.Y) {
if (polygon[i].X + (testPoint.Y - polygon[i].Y) / (polygon[j].Y - polygon[i].Y) * (polygon[j].X - polygon[i].X) < testPoint.X) {
result = !result;
}
}
j = i;
}
return result;
}
其他回答
简单的解决方案是将多边形划分为三角形,并按这里解释的那样对三角形进行测试
如果你的多边形是凸多边形,可能有更好的方法。把这个多边形看作是无限条线的集合。每一行将空间一分为二。对于每一个点,很容易判断它是在直线的一边还是另一边。如果一个点在所有直线的同一侧,那么它在多边形内。
VBA版本:
注意:请记住,如果你的多边形是地图中的一个区域,纬度/经度是Y/X值,而不是X/Y(纬度= Y,经度= X),因为从我的理解来看,这是历史含义,因为经度不是一个测量值。
类模块:CPoint
Private pXValue As Double
Private pYValue As Double
'''''X Value Property'''''
Public Property Get X() As Double
X = pXValue
End Property
Public Property Let X(Value As Double)
pXValue = Value
End Property
'''''Y Value Property'''''
Public Property Get Y() As Double
Y = pYValue
End Property
Public Property Let Y(Value As Double)
pYValue = Value
End Property
模块:
Public Function isPointInPolygon(p As CPoint, polygon() As CPoint) As Boolean
Dim i As Integer
Dim j As Integer
Dim q As Object
Dim minX As Double
Dim maxX As Double
Dim minY As Double
Dim maxY As Double
minX = polygon(0).X
maxX = polygon(0).X
minY = polygon(0).Y
maxY = polygon(0).Y
For i = 1 To UBound(polygon)
Set q = polygon(i)
minX = vbMin(q.X, minX)
maxX = vbMax(q.X, maxX)
minY = vbMin(q.Y, minY)
maxY = vbMax(q.Y, maxY)
Next i
If p.X < minX Or p.X > maxX Or p.Y < minY Or p.Y > maxY Then
isPointInPolygon = False
Exit Function
End If
' SOURCE: http://www.ecse.rpi.edu/Homepages/wrf/Research/Short_Notes/pnpoly.html
isPointInPolygon = False
i = 0
j = UBound(polygon)
Do While i < UBound(polygon) + 1
If (polygon(i).Y > p.Y) Then
If (polygon(j).Y < p.Y) Then
If p.X < (polygon(j).X - polygon(i).X) * (p.Y - polygon(i).Y) / (polygon(j).Y - polygon(i).Y) + polygon(i).X Then
isPointInPolygon = True
Exit Function
End If
End If
ElseIf (polygon(i).Y < p.Y) Then
If (polygon(j).Y > p.Y) Then
If p.X < (polygon(j).X - polygon(i).X) * (p.Y - polygon(i).Y) / (polygon(j).Y - polygon(i).Y) + polygon(i).X Then
isPointInPolygon = True
Exit Function
End If
End If
End If
j = i
i = i + 1
Loop
End Function
Function vbMax(n1, n2) As Double
vbMax = IIf(n1 > n2, n1, n2)
End Function
Function vbMin(n1, n2) As Double
vbMin = IIf(n1 > n2, n2, n1)
End Function
Sub TestPointInPolygon()
Dim i As Integer
Dim InPolygon As Boolean
' MARKER Object
Dim p As CPoint
Set p = New CPoint
p.X = <ENTER X VALUE HERE>
p.Y = <ENTER Y VALUE HERE>
' POLYGON OBJECT
Dim polygon() As CPoint
ReDim polygon(<ENTER VALUE HERE>) 'Amount of vertices in polygon - 1
For i = 0 To <ENTER VALUE HERE> 'Same value as above
Set polygon(i) = New CPoint
polygon(i).X = <ASSIGN X VALUE HERE> 'Source a list of values that can be looped through
polgyon(i).Y = <ASSIGN Y VALUE HERE> 'Source a list of values that can be looped through
Next i
InPolygon = isPointInPolygon(p, polygon)
MsgBox InPolygon
End Sub
Like David Segonds' answer suggests I use an approach of angle summation derived from my concave polygon drawing algorithm. It relies of adding up the approximate angles of subtriangles around the point to obtain a weight. A weight around 1.0 means the point is inside the triangle, a weight around 0.0 means outside, a weight around -1.0 is what happens when inside the polygon but in reverse order (like with one of the halves of a bowtie-shaped tetragon) and a weight of NAN if exactly on an edge. The reason it's not slow is that angles don't need to be estimated accurately at all. Holes can be handled by treating them as separate polygons and subtracting the weights.
typedef struct { double x, y; } xy_t;
xy_t sub_xy(xy_t a, xy_t b)
{
a.x -= b.x;
a.y -= b.y;
return a;
}
double calc_sharp_subtriangle_pixel_weight(xy_t p0, xy_t p1)
{
xy_t rot, r0, r1;
double weight;
// Rotate points (unnormalised)
rot = sub_xy(p1, p0);
r0.x = rot.x*p0.y - rot.y*p0.x;
r0.y = rot.x*p0.x + rot.y*p0.y;
r1.y = rot.x*p1.x + rot.y*p1.y;
// Calc weight
weight = subtriangle_angle_approx(r1.y, r0.x) - subtriangle_angle_approx(r0.y, r0.x);
return weight;
}
double calc_sharp_polygon_pixel_weight(xy_t p, xy_t *corner, int corner_count)
{
int i;
xy_t p0, p1;
double weight = 0.;
p0 = sub_xy(corner[corner_count-1], p);
for (i=0; i < corner_count; i++)
{
// Transform corner coordinates
p1 = sub_xy(corner[i], p);
// Calculate weight for each subtriangle
weight += calc_sharp_subtriangle_pixel_weight(p0, p1);
p0 = p1;
}
return weight;
}
因此,对于多边形的每一段,都形成一个子三角形,并计算点,然后旋转每个子三角形以计算其近似角度并添加到权重。
调用subtriangle_angle_approx(y, x)可以替换为atan2(y, x) / (2.*pi),但是一个非常粗略的近似值就足够精确了:
double subtriangle_angle_approx(double y, double x)
{
double angle, d;
int obtuse;
if (x == 0.)
return NAN;
obtuse = fabs(y) > fabs(x);
if (obtuse)
swap_double(&y, &x);
// Core of the approximation, a very loosely approximate atan(y/x) / (2.*pi) over ]-1 , 1[
d = y / x;
angle = 0.13185 * d;
if (obtuse)
angle = sign(d)*0.25 - angle;
return angle;
}
当我还是Michael Stonebraker手下的一名研究员时,我做了一些关于这方面的工作——你知道,就是那位提出了Ingres、PostgreSQL等的教授。
我们意识到最快的方法是首先做一个边界框,因为它非常快。如果它在边界框之外,它就在外面。否则,你就得做更辛苦的工作……
如果你想要一个伟大的算法,看看开源项目PostgreSQL的源代码的地理工作…
我想指出的是,我们从来没有深入了解过左撇子和右撇子(也可以表达为“内”和“外”的问题……
更新
BKB's link provided a good number of reasonable algorithms. I was working on Earth Science problems and therefore needed a solution that works in latitude/longitude, and it has the peculiar problem of handedness - is the area inside the smaller area or the bigger area? The answer is that the "direction" of the verticies matters - it's either left-handed or right handed and in this way you can indicate either area as "inside" any given polygon. As such, my work used solution three enumerated on that page.
此外,我的工作使用单独的函数进行“在线”测试。
...因为有人问:我们发现当垂直的数量超过某个数字时,边界盒测试是最好的——如果有必要,在做更长的测试之前做一个非常快速的测试……边界框是通过简单地将最大的x,最小的x,最大的y和最小的y放在一起,组成一个框的四个点来创建的……
另一个提示是:我们在网格空间中进行了所有更复杂的“调光”计算,都是在平面上的正点上进行的,然后重新投影到“真实”的经度/纬度上,从而避免了在经度180线交叉时和处理极地时可能出现的环绕错误。工作好了!
为了完整性,这里是nirg提供的算法的lua实现,由Mecki讨论:
function pnpoly(area, test)
local inside = false
local tx, ty = table.unpack(test)
local j = #area
for i=1, #area do
local vxi, vyi = table.unpack(area[i])
local vxj, vyj = table.unpack(area[j])
if (vyi > ty) ~= (vyj > ty)
and tx < (vxj - vxi)*(ty - vyi)/(vyj - vyi) + vxi
then
inside = not inside
end
j = i
end
return inside
end
变量区域是一个点的表,这些点依次存储为2D表。例子:
> A = {{2, 1}, {1, 2}, {15, 3}, {3, 4}, {5, 3}, {4, 1.5}}
> T = {2, 1.1}
> pnpoly(A, T)
true
GitHub Gist的链接。
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