BOUNTY STATUS UPDATE:
I discovered how to map a linear lens, from destination coordinates to source coordinates.
How do you calculate the radial distance from the centre to go from fisheye to rectilinear?
1). I actually struggle to reverse it, and to map source coordinates to destination coordinates. What is the inverse, in code in the style of the converting functions I posted?
2). I also see that my undistortion is imperfect on some lenses - presumably those that are not strictly linear. What is the equivalent to-and-from source-and-destination coordinates for those lenses? Again, more code than just mathematical formulae please...
Question as originally stated:
I have some points that describe positions in a picture taken with a fisheye lens.
I want to convert these points to rectilinear coordinates. I want to undistort the image.
I've found this description of how to generate a fisheye effect, but not how to reverse it.
There's also a blog post that describes how to use tools to do it; these pictures are from that:
(1) : SOURCE Original photo link
Input : Original image with fish-eye distortion to fix.
(2) : DESTINATION Original photo link
Output : Corrected image (technically also with perspective correction, but that's a separate step).
How do you calculate the radial distance from the centre to go from fisheye to rectilinear?
My function stub looks like this:
Point correct_fisheye(const Point& p,const Size& img) {
// to polar
const Point centre = {img.width/2,img.height/2};
const Point rel = {p.x-centre.x,p.y-centre.y};
const double theta = atan2(rel.y,rel.x);
double R = sqrt((rel.x*rel.x)+(rel.y*rel.y));
// fisheye undistortion in here please
//... change R ...
// back to rectangular
const Point ret = Point(centre.x+R*cos(theta),centre.y+R*sin(theta));
fprintf(stderr,"(%d,%d) in (%d,%d) = %f,%f = (%d,%d)\n",p.x,p.y,img.width,img.height,theta,R,ret.x,ret.y);
return ret;
}
Alternatively, I could somehow convert the image from fisheye to rectilinear before finding the points, but I'm completely befuddled by the OpenCV documentation. Is there a straightforward way to do it in OpenCV, and does it perform well enough to do it to a live video feed?
The description you mention states that the projection by a pin-hole camera (one that does not introduce lens distortion) is modeled by
R_u = f*tan(theta)
and the projection by common fisheye lens cameras (that is, distorted) is modeled by
R_d = 2*f*sin(theta/2)
You already know R_d and theta and if you knew the camera's focal length (represented by f) then correcting the image would amount to computing R_u in terms of R_d and theta. In other words,
R_u = f*tan(2*asin(R_d/(2*f)))
is the formula you're looking for. Estimating the focal length f can be solved by calibrating the camera or other means such as letting the user provide feedback on how well the image is corrected or using knowledge from the original scene.
In order to solve the same problem using OpenCV, you would have to obtain the camera's intrinsic parameters and lens distortion coefficients. See, for example, Chapter 11 of Learning OpenCV (don't forget to check the correction). Then you can use a program such as this one (written with the Python bindings for OpenCV) in order to reverse lens distortion:
#!/usr/bin/python
# ./undistort 0_0000.jpg 1367.451167 1367.451167 0 0 -0.246065 0.193617 -0.002004 -0.002056
import sys
import cv
def main(argv):
if len(argv) < 10:
print 'Usage: %s input-file fx fy cx cy k1 k2 p1 p2 output-file' % argv[0]
sys.exit(-1)
src = argv[1]
fx, fy, cx, cy, k1, k2, p1, p2, output = argv[2:]
intrinsics = cv.CreateMat(3, 3, cv.CV_64FC1)
cv.Zero(intrinsics)
intrinsics[0, 0] = float(fx)
intrinsics[1, 1] = float(fy)
intrinsics[2, 2] = 1.0
intrinsics[0, 2] = float(cx)
intrinsics[1, 2] = float(cy)
dist_coeffs = cv.CreateMat(1, 4, cv.CV_64FC1)
cv.Zero(dist_coeffs)
dist_coeffs[0, 0] = float(k1)
dist_coeffs[0, 1] = float(k2)
dist_coeffs[0, 2] = float(p1)
dist_coeffs[0, 3] = float(p2)
src = cv.LoadImage(src)
dst = cv.CreateImage(cv.GetSize(src), src.depth, src.nChannels)
mapx = cv.CreateImage(cv.GetSize(src), cv.IPL_DEPTH_32F, 1)
mapy = cv.CreateImage(cv.GetSize(src), cv.IPL_DEPTH_32F, 1)
cv.InitUndistortMap(intrinsics, dist_coeffs, mapx, mapy)
cv.Remap(src, dst, mapx, mapy, cv.CV_INTER_LINEAR + cv.CV_WARP_FILL_OUTLIERS, cv.ScalarAll(0))
# cv.Undistort2(src, dst, intrinsics, dist_coeffs)
cv.SaveImage(output, dst)
if __name__ == '__main__':
main(sys.argv)
Also note that OpenCV uses a very different lens distortion model to the one in the web page you linked to.
(Original poster, providing an alternative)
The following function maps destination (rectilinear) coordinates to source (fisheye-distorted) coordinates. (I'd appreciate help in reversing it)
I got to this point through trial-and-error: I don't fundamentally grasp why this code is working, explanations and improved accuracy appreciated!
def dist(x,y):
return sqrt(x*x+y*y)
def correct_fisheye(src_size,dest_size,dx,dy,factor):
""" returns a tuple of source coordinates (sx,sy)
(note: values can be out of range)"""
# convert dx,dy to relative coordinates
rx, ry = dx-(dest_size[0]/2), dy-(dest_size[1]/2)
# calc theta
r = dist(rx,ry)/(dist(src_size[0],src_size[1])/factor)
if 0==r:
theta = 1.0
else:
theta = atan(r)/r
# back to absolute coordinates
sx, sy = (src_size[0]/2)+theta*rx, (src_size[1]/2)+theta*ry
# done
return (int(round(sx)),int(round(sy)))
When used with a factor of 3.0, it successfully undistorts the images used as examples (I made no attempt at quality interpolation):
Dead link
(And this is from the blog post, for comparison:)
If you think your formulas are exact, you can comput an exact formula with trig, like so:
Rin = 2 f sin(w/2) -> sin(w/2)= Rin/2f
Rout= f tan(w) -> tan(w)= Rout/f
(Rin/2f)^2 = [sin(w/2)]^2 = (1 - cos(w))/2 -> cos(w) = 1 - 2(Rin/2f)^2
(Rout/f)^2 = [tan(w)]^2 = 1/[cos(w)]^2 - 1
-> (Rout/f)^2 = 1/(1-2[Rin/2f]^2)^2 - 1
However, as #jmbr says, the actual camera distortion will depend on the lens and the zoom. Rather than rely on a fixed formula, you might want to try a polynomial expansion:
Rout = Rin*(1 + A*Rin^2 + B*Rin^4 + ...)
By tweaking first A, then higher-order coefficients, you can compute any reasonable local function (the form of the expansion takes advantage of the symmetry of the problem). In particular, it should be possible to compute initial coefficients to approximate the theoretical function above.
Also, for good results, you will need to use an interpolation filter to generate your corrected image. As long as the distortion is not too great, you can use the kind of filter you would use to rescale the image linearly without much problem.
Edit: as per your request, the equivalent scaling factor for the above formula:
(Rout/f)^2 = 1/(1-2[Rin/2f]^2)^2 - 1
-> Rout/f = [Rin/f] * sqrt(1-[Rin/f]^2/4)/(1-[Rin/f]^2/2)
If you plot the above formula alongside tan(Rin/f), you can see that they are very similar in shape. Basically, distortion from the tangent becomes severe before sin(w) becomes much different from w.
The inverse formula should be something like:
Rin/f = [Rout/f] / sqrt( sqrt(([Rout/f]^2+1) * (sqrt([Rout/f]^2+1) + 1) / 2 )
I blindly implemented the formulas from here, so I cannot guarantee it would do what you need.
Use auto_zoom to get the value for the zoom parameter.
def dist(x,y):
return sqrt(x*x+y*y)
def fisheye_to_rectilinear(src_size,dest_size,sx,sy,crop_factor,zoom):
""" returns a tuple of dest coordinates (dx,dy)
(note: values can be out of range)
crop_factor is ratio of sphere diameter to diagonal of the source image"""
# convert sx,sy to relative coordinates
rx, ry = sx-(src_size[0]/2), sy-(src_size[1]/2)
r = dist(rx,ry)
# focal distance = radius of the sphere
pi = 3.1415926535
f = dist(src_size[0],src_size[1])*factor/pi
# calc theta 1) linear mapping (older Nikon)
theta = r / f
# calc theta 2) nonlinear mapping
# theta = asin ( r / ( 2 * f ) ) * 2
# calc new radius
nr = tan(theta) * zoom
# back to absolute coordinates
dx, dy = (dest_size[0]/2)+rx/r*nr, (dest_size[1]/2)+ry/r*nr
# done
return (int(round(dx)),int(round(dy)))
def fisheye_auto_zoom(src_size,dest_size,crop_factor):
""" calculate zoom such that left edge of source image matches left edge of dest image """
# Try to see what happens with zoom=1
dx, dy = fisheye_to_rectilinear(src_size, dest_size, 0, src_size[1]/2, crop_factor, 1)
# Calculate zoom so the result is what we wanted
obtained_r = dest_size[0]/2 - dx
required_r = dest_size[0]/2
zoom = required_r / obtained_r
return zoom
I took what JMBR did and basically reversed it. He took the radius of the distorted image (Rd, that is, the distance in pixels from the center of the image) and found a formula for Ru, the radius of the undistorted image.
You want to go the other way. For each pixel in the undistorted (processed image), you want to know what the corresponding pixel is in the distorted image.
In other words, given (xu, yu) --> (xd, yd). You then replace each pixel in the undistorted image with its corresponding pixel from the distorted image.
Starting where JMBR did, I do the reverse, finding Rd as a function of Ru. I get:
Rd = f * sqrt(2) * sqrt( 1 - 1/sqrt(r^2 +1))
where f is the focal length in pixels (I'll explain later), and r = Ru/f.
The focal length for my camera was 2.5 mm. The size of each pixel on my CCD was 6 um square. f was therefore 2500/6 = 417 pixels. This can be found by trial and error.
Finding Rd allows you to find the corresponding pixel in the distorted image using polar coordinates.
The angle of each pixel from the center point is the same:
theta = arctan( (yu-yc)/(xu-xc) ) where xc, yc are the center points.
Then,
xd = Rd * cos(theta) + xc
yd = Rd * sin(theta) + yc
Make sure you know which quadrant you are in.
Here is the C# code I used
public class Analyzer
{
private ArrayList mFisheyeCorrect;
private int mFELimit = 1500;
private double mScaleFESize = 0.9;
public Analyzer()
{
//A lookup table so we don't have to calculate Rdistorted over and over
//The values will be multiplied by focal length in pixels to
//get the Rdistorted
mFisheyeCorrect = new ArrayList(mFELimit);
//i corresponds to Rundist/focalLengthInPixels * 1000 (to get integers)
for (int i = 0; i < mFELimit; i++)
{
double result = Math.Sqrt(1 - 1 / Math.Sqrt(1.0 + (double)i * i / 1000000.0)) * 1.4142136;
mFisheyeCorrect.Add(result);
}
}
public Bitmap RemoveFisheye(ref Bitmap aImage, double aFocalLinPixels)
{
Bitmap correctedImage = new Bitmap(aImage.Width, aImage.Height);
//The center points of the image
double xc = aImage.Width / 2.0;
double yc = aImage.Height / 2.0;
Boolean xpos, ypos;
//Move through the pixels in the corrected image;
//set to corresponding pixels in distorted image
for (int i = 0; i < correctedImage.Width; i++)
{
for (int j = 0; j < correctedImage.Height; j++)
{
//which quadrant are we in?
xpos = i > xc;
ypos = j > yc;
//Find the distance from the center
double xdif = i-xc;
double ydif = j-yc;
//The distance squared
double Rusquare = xdif * xdif + ydif * ydif;
//the angle from the center
double theta = Math.Atan2(ydif, xdif);
//find index for lookup table
int index = (int)(Math.Sqrt(Rusquare) / aFocalLinPixels * 1000);
if (index >= mFELimit) index = mFELimit - 1;
//calculated Rdistorted
double Rd = aFocalLinPixels * (double)mFisheyeCorrect[index]
/mScaleFESize;
//calculate x and y distances
double xdelta = Math.Abs(Rd*Math.Cos(theta));
double ydelta = Math.Abs(Rd * Math.Sin(theta));
//convert to pixel coordinates
int xd = (int)(xc + (xpos ? xdelta : -xdelta));
int yd = (int)(yc + (ypos ? ydelta : -ydelta));
xd = Math.Max(0, Math.Min(xd, aImage.Width-1));
yd = Math.Max(0, Math.Min(yd, aImage.Height-1));
//set the corrected pixel value from the distorted image
correctedImage.SetPixel(i, j, aImage.GetPixel(xd, yd));
}
}
return correctedImage;
}
}
I found this pdf file and I have proved that the maths are correct (except for the line vd = *xd**fv+v0 which should say vd = **yd**+fv+v0).
http://perception.inrialpes.fr/CAVA_Dataset/Site/files/Calibration_OpenCV.pdf
It does not use all of the latest co-efficients that OpenCV has available but I am sure that it could be adapted fairly easily.
double k1 = cameraIntrinsic.distortion[0];
double k2 = cameraIntrinsic.distortion[1];
double p1 = cameraIntrinsic.distortion[2];
double p2 = cameraIntrinsic.distortion[3];
double k3 = cameraIntrinsic.distortion[4];
double fu = cameraIntrinsic.focalLength[0];
double fv = cameraIntrinsic.focalLength[1];
double u0 = cameraIntrinsic.principalPoint[0];
double v0 = cameraIntrinsic.principalPoint[1];
double u, v;
u = thisPoint->x; // the undistorted point
v = thisPoint->y;
double x = ( u - u0 )/fu;
double y = ( v - v0 )/fv;
double r2 = (x*x) + (y*y);
double r4 = r2*r2;
double cDist = 1 + (k1*r2) + (k2*r4);
double xr = x*cDist;
double yr = y*cDist;
double a1 = 2*x*y;
double a2 = r2 + (2*(x*x));
double a3 = r2 + (2*(y*y));
double dx = (a1*p1) + (a2*p2);
double dy = (a3*p1) + (a1*p2);
double xd = xr + dx;
double yd = yr + dy;
double ud = (xd*fu) + u0;
double vd = (yd*fv) + v0;
thisPoint->x = ud; // the distorted point
thisPoint->y = vd;
This can be solved as an optimization problem. Simply draw on curves in images that are supposed to be straight lines. Store the contour points for each of those curves. Now we can solve the fish eye matrix as a minimization problem. Minimize the curve in points and that will give us a fisheye matrix. It works.
It can be done manually by adjusting the fish eye matrix using trackbars! Here is a fish eye GUI code using OpenCV for manual calibration.
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Lets say you have this:
P1 = (x=2, y=50)
P2 = (x=9, y=40)
P3 = (x=5, y=20)
Assume that P1 is the center point of a circle. It is always the same.
I want the angle that is made up by P2 and P3, or in other words the angle that is next to P1. The inner angle to be precise. It will always be an acute angle, so less than -90 degrees.
I thought: Man, that's simple geometry math. But I have looked for a formula for around 6 hours now, and only find people talking about complicated NASA stuff like arccos and vector scalar product stuff. My head feels like it's in a fridge.
Some math gurus here that think this is a simple problem? I don't think the programming language matters here, but for those who think it does: java and objective-c. I need it for both, but haven't tagged it for these.
If you mean the angle that P1 is the vertex of then using the Law of Cosines should work:
arccos((P122
+ P132 - P232) / (2 *
P12 * P13))
where P12 is the length of the segment from P1 to P2, calculated by
sqrt((P1x -
P2x)2 +
(P1y -
P2y)2)
It gets very simple if you think it as two vectors, one from point P1 to P2 and one from P1 to P3
so:
a = (p1.x - p2.x, p1.y - p2.y)
b = (p1.x - p3.x, p1.y - p3.y)
You can then invert the dot product formula:
to get the angle:
Remember that just means:
a1*b1 + a2*b2 (just 2 dimensions here...)
The best way to deal with angle computation is to use atan2(y, x) that given a point x, y returns the angle from that point and the X+ axis in respect to the origin.
Given that the computation is
double result = atan2(P3.y - P1.y, P3.x - P1.x) -
atan2(P2.y - P1.y, P2.x - P1.x);
i.e. you basically translate the two points by -P1 (in other words you translate everything so that P1 ends up in the origin) and then you consider the difference of the absolute angles of P3 and of P2.
The advantages of atan2 is that the full circle is represented (you can get any number between -π and π) where instead with acos you need to handle several cases depending on the signs to compute the correct result.
The only singular point for atan2 is (0, 0)... meaning that both P2 and P3 must be different from P1 as in that case doesn't make sense to talk about an angle.
Let me give an example in JavaScript, I've fought a lot with that:
/**
* Calculates the angle (in radians) between two vectors pointing outward from one center
*
* #param p0 first point
* #param p1 second point
* #param c center point
*/
function find_angle(p0,p1,c) {
var p0c = Math.sqrt(Math.pow(c.x-p0.x,2)+
Math.pow(c.y-p0.y,2)); // p0->c (b)
var p1c = Math.sqrt(Math.pow(c.x-p1.x,2)+
Math.pow(c.y-p1.y,2)); // p1->c (a)
var p0p1 = Math.sqrt(Math.pow(p1.x-p0.x,2)+
Math.pow(p1.y-p0.y,2)); // p0->p1 (c)
return Math.acos((p1c*p1c+p0c*p0c-p0p1*p0p1)/(2*p1c*p0c));
}
Bonus: Example with HTML5-canvas
Basically what you have is two vectors, one vector from P1 to P2 and another from P1 to P3. So all you need is an formula to calculate the angle between two vectors.
Have a look here for a good explanation and the formula.
If you are thinking of P1 as the center of a circle, you are thinking too complicated.
You have a simple triangle, so your problem is solveable with the law of cosines. No need for any polar coordinate tranformation or somesuch. Say the distances are P1-P2 = A, P2-P3 = B and P3-P1 = C:
Angle = arccos ( (B^2-A^2-C^2) / 2AC )
All you need to do is calculate the length of the distances A, B and C.
Those are easily available from the x- and y-coordinates of your points and
Pythagoras' theorem
Length = sqrt( (X2-X1)^2 + (Y2-Y1)^2 )
I ran into a similar problem recently, only I needed to differentiate between a positive and negative angles. In case this is of use to anyone, I recommend the code snippet I grabbed from this mailing list about detecting rotation over a touch event for Android:
#Override
public boolean onTouchEvent(MotionEvent e) {
float x = e.getX();
float y = e.getY();
switch (e.getAction()) {
case MotionEvent.ACTION_MOVE:
//find an approximate angle between them.
float dx = x-cx;
float dy = y-cy;
double a=Math.atan2(dy,dx);
float dpx= mPreviousX-cx;
float dpy= mPreviousY-cy;
double b=Math.atan2(dpy, dpx);
double diff = a-b;
this.bearing -= Math.toDegrees(diff);
this.invalidate();
}
mPreviousX = x;
mPreviousY = y;
return true;
}
Very Simple Geometric Solution with Explanation
Few days ago, a fell into the same problem & had to sit with the math book. I solved the problem by combining and simplifying some basic formulas.
Lets consider this figure-
We want to know ϴ, so we need to find out α and β first. Now, for any straight line-
y = m * x + c
Let- A = (ax, ay), B = (bx, by), and O = (ox, oy). So for the line OA-
oy = m1 * ox + c ⇒ c = oy - m1 * ox ...(eqn-1)
ay = m1 * ax + c ⇒ ay = m1 * ax + oy - m1 * ox [from eqn-1]
⇒ ay = m1 * ax + oy - m1 * ox
⇒ m1 = (ay - oy) / (ax - ox)
⇒ tan α = (ay - oy) / (ax - ox) [m = slope = tan ϴ] ...(eqn-2)
In the same way, for line OB-
tan β = (by - oy) / (bx - ox) ...(eqn-3)
Now, we need ϴ = β - α. In trigonometry we have a formula-
tan (β-α) = (tan β + tan α) / (1 - tan β * tan α) ...(eqn-4)
After replacing the value of tan α (from eqn-2) and tan b (from eqn-3) in eqn-4, and applying simplification we get-
tan (β-α) = ( (ax-ox)*(by-oy)+(ay-oy)*(bx-ox) ) / ( (ax-ox)*(bx-ox)-(ay-oy)*(by-oy) )
So,
ϴ = β-α = tan^(-1) ( ((ax-ox)*(by-oy)+(ay-oy)*(bx-ox)) / ((ax-ox)*(bx-ox)-(ay-oy)*(by-oy)) )
That is it!
Now, take following figure-
This C# or, Java method calculates the angle (ϴ)-
private double calculateAngle(double P1X, double P1Y, double P2X, double P2Y,
double P3X, double P3Y){
double numerator = P2Y*(P1X-P3X) + P1Y*(P3X-P2X) + P3Y*(P2X-P1X);
double denominator = (P2X-P1X)*(P1X-P3X) + (P2Y-P1Y)*(P1Y-P3Y);
double ratio = numerator/denominator;
double angleRad = Math.Atan(ratio);
double angleDeg = (angleRad*180)/Math.PI;
if(angleDeg<0){
angleDeg = 180+angleDeg;
}
return angleDeg;
}
In Objective-C you could do this by
float xpoint = (((atan2((newPoint.x - oldPoint.x) , (newPoint.y - oldPoint.y)))*180)/M_PI);
Or read more here
You mentioned a signed angle (-90). In many applications angles may have signs (positive and negative, see http://en.wikipedia.org/wiki/Angle). If the points are (say) P2(1,0), P1(0,0), P3(0,1) then the angle P3-P1-P2 is conventionally positive (PI/2) whereas the angle P2-P1-P3 is negative. Using the lengths of the sides will not distinguish between + and - so if this matters you will need to use vectors or a function such as Math.atan2(a, b).
Angles can also extend beyond 2*PI and while this is not relevant to the current question it was sufficiently important that I wrote my own Angle class (also to make sure that degrees and radians did not get mixed up). The questions as to whether angle1 is less than angle2 depends critically on how angles are defined. It may also be important to decide whether a line (-1,0)(0,0)(1,0) is represented as Math.PI or -Math.PI
Recently, I too have the same problem... In Delphi
It's very similar to Objective-C.
procedure TForm1.FormPaint(Sender: TObject);
var ARect: TRect;
AWidth, AHeight: Integer;
ABasePoint: TPoint;
AAngle: Extended;
begin
FCenter := Point(Width div 2, Height div 2);
AWidth := Width div 4;
AHeight := Height div 4;
ABasePoint := Point(FCenter.X+AWidth, FCenter.Y);
ARect := Rect(Point(FCenter.X - AWidth, FCenter.Y - AHeight),
Point(FCenter.X + AWidth, FCenter.Y + AHeight));
AAngle := ArcTan2(ClickPoint.Y-Center.Y, ClickPoint.X-Center.X) * 180 / pi;
AngleLabel.Caption := Format('Angle is %5.2f', [AAngle]);
Canvas.Ellipse(ARect);
Canvas.MoveTo(FCenter.X, FCenter.Y);
Canvas.LineTo(FClickPoint.X, FClickPoint.Y);
Canvas.MoveTo(FCenter.X, FCenter.Y);
Canvas.LineTo(ABasePoint.X, ABasePoint.Y);
end;
Here's a C# method to return the angle (0-360) anticlockwise from the horizontal for a point on a circle.
public static double GetAngle(Point centre, Point point1)
{
// Thanks to Dave Hill
// Turn into a vector (from the origin)
double x = point1.X - centre.X;
double y = point1.Y - centre.Y;
// Dot product u dot v = mag u * mag v * cos theta
// Therefore theta = cos -1 ((u dot v) / (mag u * mag v))
// Horizontal v = (1, 0)
// therefore theta = cos -1 (u.x / mag u)
// nb, there are 2 possible angles and if u.y is positive then angle is in first quadrant, negative then second quadrant
double magnitude = Math.Sqrt(x * x + y * y);
double angle = 0;
if(magnitude > 0)
angle = Math.Acos(x / magnitude);
angle = angle * 180 / Math.PI;
if (y < 0)
angle = 360 - angle;
return angle;
}
Cheers,
Paul
function p(x, y) {return {x,y}}
function normaliseToInteriorAngle(angle) {
if (angle < 0) {
angle += (2*Math.PI)
}
if (angle > Math.PI) {
angle = 2*Math.PI - angle
}
return angle
}
function angle(p1, center, p2) {
const transformedP1 = p(p1.x - center.x, p1.y - center.y)
const transformedP2 = p(p2.x - center.x, p2.y - center.y)
const angleToP1 = Math.atan2(transformedP1.y, transformedP1.x)
const angleToP2 = Math.atan2(transformedP2.y, transformedP2.x)
return normaliseToInteriorAngle(angleToP2 - angleToP1)
}
function toDegrees(radians) {
return 360 * radians / (2 * Math.PI)
}
console.log(toDegrees(angle(p(-10, 0), p(0, 0), p(0, -10))))
there IS a simple answer for this using high school math..
Let say that you have 3 points
To get angle from point A to B
angle = atan2(A.x - B.x, B.y - A.y)
To get angle from point B to C
angle2 = atan2(B.x - C.x, C.y - B.y)
Answer = 180 + angle2 - angle
If (answer < 0){
return answer + 360
}else{
return answer
}
I just used this code in the recent project that I made, change the B to P1.. you might as well remove the "180 +" if you want
well, the other answers seem to cover everything required, so I would like to just add this if you are using JMonkeyEngine:
Vector3f.angleBetween(otherVector)
as that is what I came here looking for :)
Atan2 output in degrees
PI/2 +90
| |
| |
PI ---.--- 0 +180 ---.--- 0
| |
| |
-PI/2 +270
public static double CalculateAngleFromHorizontal(double startX, double startY, double endX, double endY)
{
var atan = Math.Atan2(endY - startY, endX - startX); // Angle in radians
var angleDegrees = atan * (180 / Math.PI); // Angle in degrees (can be +/-)
if (angleDegrees < 0.0)
{
angleDegrees = 360.0 + angleDegrees;
}
return angleDegrees;
}
// Angle from point2 to point 3 counter clockwise
public static double CalculateAngle0To360(double centerX, double centerY, double x2, double y2, double x3, double y3)
{
var angle2 = CalculateAngleFromHorizontal(centerX, centerY, x2, y2);
var angle3 = CalculateAngleFromHorizontal(centerX, centerY, x3, y3);
return (360.0 + angle3 - angle2)%360;
}
// Smaller angle from point2 to point 3
public static double CalculateAngle0To180(double centerX, double centerY, double x2, double y2, double x3, double y3)
{
var angle = CalculateAngle0To360(centerX, centerY, x2, y2, x3, y3);
if (angle > 180.0)
{
angle = 360 - angle;
}
return angle;
}
}
I'm trying to create a fast 2D point inside polygon algorithm, for use in hit-testing (e.g. Polygon.contains(p:Point)). Suggestions for effective techniques would be appreciated.
For graphics, I'd rather not prefer integers. Many systems use integers for UI painting (pixels are ints after all), but macOS, for example, uses float for everything. macOS only knows points and a point can translate to one pixel, but depending on monitor resolution, it might translate to something else. On retina screens half a point (0.5/0.5) is pixel. Still, I never noticed that macOS UIs are significantly slower than other UIs. After all, 3D APIs (OpenGL or Direct3D) also work with floats and modern graphics libraries very often take advantage of GPU acceleration.
Now you said speed is your main concern, okay, let's go for speed. Before you run any sophisticated algorithm, first do a simple test. Create an axis aligned bounding box around your polygon. This is very easy, fast and can already save you a lot of calculations. How does that work? Iterate over all points of the polygon and find the min/max values of X and Y.
E.g. you have the points (9/1), (4/3), (2/7), (8/2), (3/6). This means Xmin is 2, Xmax is 9, Ymin is 1 and Ymax is 7. A point outside of the rectangle with the two edges (2/1) and (9/7) cannot be within the polygon.
// p is your point, p.x is the x coord, p.y is the y coord
if (p.x < Xmin || p.x > Xmax || p.y < Ymin || p.y > Ymax) {
// Definitely not within the polygon!
}
This is the first test to run for any point. As you can see, this test is ultra fast but it's also very coarse. To handle points that are within the bounding rectangle, we need a more sophisticated algorithm. There are a couple of ways how this can be calculated. Which method works also depends on whether the polygon can have holes or will always be solid. Here are examples of solid ones (one convex, one concave):
And here's one with a hole:
The green one has a hole in the middle!
The easiest algorithm, that can handle all three cases above and is still pretty fast is named ray casting. The idea of the algorithm is pretty simple: Draw a virtual ray from anywhere outside the polygon to your point and count how often it hits a side of the polygon. If the number of hits is even, it's outside of the polygon, if it's odd, it's inside.
The winding number algorithm would be an alternative, it is more accurate for points being very close to a polygon line but it's also much slower. Ray casting may fail for points too close to a polygon side because of limited floating point precision and rounding issues, but in reality that is hardly a problem, as if a point lies that close to a side, it's often visually not even possible for a viewer to recognize if it is already inside or still outside.
You still have the bounding box of above, remember? Just pick a point outside the bounding box and use it as starting point for your ray. E.g. the point (Xmin - e/p.y) is outside the polygon for sure.
But what is e? Well, e (actually epsilon) gives the bounding box some padding. As I said, ray tracing fails if we start too close to a polygon line. Since the bounding box might equal the polygon (if the polygon is an axis aligned rectangle, the bounding box is equal to the polygon itself!), we need some padding to make this safe, that's all. How big should you choose e? Not too big. It depends on the coordinate system scale you use for drawing. If your pixel step width is 1.0, then just choose 1.0 (yet 0.1 would have worked as well)
Now that we have the ray with its start and end coordinates, the problem shifts from "is the point within the polygon" to "how often does the ray intersects a polygon side". Therefore we can't just work with the polygon points as before, now we need the actual sides. A side is always defined by two points.
side 1: (X1/Y1)-(X2/Y2)
side 2: (X2/Y2)-(X3/Y3)
side 3: (X3/Y3)-(X4/Y4)
:
You need to test the ray against all sides. Consider the ray to be a vector and every side to be a vector. The ray has to hit each side exactly once or never at all. It can't hit the same side twice. Two lines in 2D space will always intersect exactly once, unless they are parallel, in which case they never intersect. However since vectors have a limited length, two vectors might not be parallel and still never intersect because they are too short to ever meet each other.
// Test the ray against all sides
int intersections = 0;
for (side = 0; side < numberOfSides; side++) {
// Test if current side intersects with ray.
// If yes, intersections++;
}
if ((intersections & 1) == 1) {
// Inside of polygon
} else {
// Outside of polygon
}
So far so well, but how do you test if two vectors intersect? Here's some C code (not tested), that should do the trick:
#define NO 0
#define YES 1
#define COLLINEAR 2
int areIntersecting(
float v1x1, float v1y1, float v1x2, float v1y2,
float v2x1, float v2y1, float v2x2, float v2y2
) {
float d1, d2;
float a1, a2, b1, b2, c1, c2;
// Convert vector 1 to a line (line 1) of infinite length.
// We want the line in linear equation standard form: A*x + B*y + C = 0
// See: http://en.wikipedia.org/wiki/Linear_equation
a1 = v1y2 - v1y1;
b1 = v1x1 - v1x2;
c1 = (v1x2 * v1y1) - (v1x1 * v1y2);
// Every point (x,y), that solves the equation above, is on the line,
// every point that does not solve it, is not. The equation will have a
// positive result if it is on one side of the line and a negative one
// if is on the other side of it. We insert (x1,y1) and (x2,y2) of vector
// 2 into the equation above.
d1 = (a1 * v2x1) + (b1 * v2y1) + c1;
d2 = (a1 * v2x2) + (b1 * v2y2) + c1;
// If d1 and d2 both have the same sign, they are both on the same side
// of our line 1 and in that case no intersection is possible. Careful,
// 0 is a special case, that's why we don't test ">=" and "<=",
// but "<" and ">".
if (d1 > 0 && d2 > 0) return NO;
if (d1 < 0 && d2 < 0) return NO;
// The fact that vector 2 intersected the infinite line 1 above doesn't
// mean it also intersects the vector 1. Vector 1 is only a subset of that
// infinite line 1, so it may have intersected that line before the vector
// started or after it ended. To know for sure, we have to repeat the
// the same test the other way round. We start by calculating the
// infinite line 2 in linear equation standard form.
a2 = v2y2 - v2y1;
b2 = v2x1 - v2x2;
c2 = (v2x2 * v2y1) - (v2x1 * v2y2);
// Calculate d1 and d2 again, this time using points of vector 1.
d1 = (a2 * v1x1) + (b2 * v1y1) + c2;
d2 = (a2 * v1x2) + (b2 * v1y2) + c2;
// Again, if both have the same sign (and neither one is 0),
// no intersection is possible.
if (d1 > 0 && d2 > 0) return NO;
if (d1 < 0 && d2 < 0) return NO;
// If we get here, only two possibilities are left. Either the two
// vectors intersect in exactly one point or they are collinear, which
// means they intersect in any number of points from zero to infinite.
if ((a1 * b2) - (a2 * b1) == 0.0f) return COLLINEAR;
// If they are not collinear, they must intersect in exactly one point.
return YES;
}
The input values are the two endpoints of vector 1 (v1x1/v1y1 and v1x2/v1y2) and vector 2 (v2x1/v2y1 and v2x2/v2y2). So you have 2 vectors, 4 points, 8 coordinates. YES and NO are clear. YES increases intersections, NO does nothing.
What about COLLINEAR? It means both vectors lie on the same infinite line, depending on position and length, they don't intersect at all or they intersect in an endless number of points. I'm not absolutely sure how to handle this case, I would not count it as intersection either way. Well, this case is rather rare in practice anyway because of floating point rounding errors; better code would probably not test for == 0.0f but instead for something like < epsilon, where epsilon is a rather small number.
If you need to test a larger number of points, you can certainly speed up the whole thing a bit by keeping the linear equation standard forms of the polygon sides in memory, so you don't have to recalculate these every time. This will save you two floating point multiplications and three floating point subtractions on every test in exchange for storing three floating point values per polygon side in memory. It's a typical memory vs computation time trade off.
Last but not least: If you may use 3D hardware to solve the problem, there is an interesting alternative. Just let the GPU do all the work for you. Create a painting surface that is off screen. Fill it completely with the color black. Now let OpenGL or Direct3D paint your polygon (or even all of your polygons if you just want to test if the point is within any of them, but you don't care for which one) and fill the polygon(s) with a different color, e.g. white. To check if a point is within the polygon, get the color of this point from the drawing surface. This is just a O(1) memory fetch.
Of course this method is only usable if your drawing surface doesn't have to be huge. If it cannot fit into the GPU memory, this method is slower than doing it on the CPU. If it would have to be huge and your GPU supports modern shaders, you can still use the GPU by implementing the ray casting shown above as a GPU shader, which absolutely is possible. For a larger number of polygons or a large number of points to test, this will pay off, consider some GPUs will be able to test 64 to 256 points in parallel. Note however that transferring data from CPU to GPU and back is always expensive, so for just testing a couple of points against a couple of simple polygons, where either the points or the polygons are dynamic and will change frequently, a GPU approach will rarely pay off.
I think the following piece of code is the best solution (taken from here):
int pnpoly(int nvert, float *vertx, float *verty, float testx, float testy)
{
int i, j, c = 0;
for (i = 0, j = nvert-1; i < nvert; j = i++) {
if ( ((verty[i]>testy) != (verty[j]>testy)) &&
(testx < (vertx[j]-vertx[i]) * (testy-verty[i]) / (verty[j]-verty[i]) + vertx[i]) )
c = !c;
}
return c;
}
Arguments
nvert: Number of vertices in the polygon. Whether to repeat the first vertex at the end has been discussed in the article referred above.
vertx, verty: Arrays containing the x- and y-coordinates of the polygon's vertices.
testx, testy: X- and y-coordinate of the test point.
It's both short and efficient and works both for convex and concave polygons. As suggested before, you should check the bounding rectangle first and treat polygon holes separately.
The idea behind this is pretty simple. The author describes it as follows:
I run a semi-infinite ray horizontally (increasing x, fixed y) out from the test point, and count how many edges it crosses. At each crossing, the ray switches between inside and outside. This is called the Jordan curve theorem.
The variable c is switching from 0 to 1 and 1 to 0 each time the horizontal ray crosses any edge. So basically it's keeping track of whether the number of edges crossed are even or odd. 0 means even and 1 means odd.
Here is a C# version of the answer given by nirg, which comes from this RPI professor. Note that use of the code from that RPI source requires attribution.
A bounding box check has been added at the top. However, as James Brown points out, the main code is almost as fast as the bounding box check itself, so the bounding box check can actually slow the overall operation, in the case that most of the points you are checking are inside the bounding box. So you could leave the bounding box check out, or an alternative would be to precompute the bounding boxes of your polygons if they don't change shape too often.
public bool IsPointInPolygon( Point p, Point[] polygon )
{
double minX = polygon[ 0 ].X;
double maxX = polygon[ 0 ].X;
double minY = polygon[ 0 ].Y;
double maxY = polygon[ 0 ].Y;
for ( int i = 1 ; i < polygon.Length ; i++ )
{
Point q = polygon[ i ];
minX = Math.Min( q.X, minX );
maxX = Math.Max( q.X, maxX );
minY = Math.Min( q.Y, minY );
maxY = Math.Max( q.Y, maxY );
}
if ( p.X < minX || p.X > maxX || p.Y < minY || p.Y > maxY )
{
return false;
}
// https://wrf.ecse.rpi.edu/Research/Short_Notes/pnpoly.html
bool inside = false;
for ( int i = 0, j = polygon.Length - 1 ; i < polygon.Length ; j = i++ )
{
if ( ( polygon[ i ].Y > p.Y ) != ( polygon[ j ].Y > p.Y ) &&
p.X < ( polygon[ j ].X - polygon[ i ].X ) * ( p.Y - polygon[ i ].Y ) / ( polygon[ j ].Y - polygon[ i ].Y ) + polygon[ i ].X )
{
inside = !inside;
}
}
return inside;
}
Here is a JavaScript variant of the answer by M. Katz based on Nirg's approach:
function pointIsInPoly(p, polygon) {
var isInside = false;
var minX = polygon[0].x, maxX = polygon[0].x;
var minY = polygon[0].y, maxY = polygon[0].y;
for (var n = 1; n < polygon.length; n++) {
var q = polygon[n];
minX = Math.min(q.x, minX);
maxX = Math.max(q.x, maxX);
minY = Math.min(q.y, minY);
maxY = Math.max(q.y, maxY);
}
if (p.x < minX || p.x > maxX || p.y < minY || p.y > maxY) {
return false;
}
var i = 0, j = polygon.length - 1;
for (i, j; i < polygon.length; j = i++) {
if ( (polygon[i].y > p.y) != (polygon[j].y > p.y) &&
p.x < (polygon[j].x - polygon[i].x) * (p.y - polygon[i].y) / (polygon[j].y - polygon[i].y) + polygon[i].x ) {
isInside = !isInside;
}
}
return isInside;
}
Compute the oriented sum of angles between the point p and each of the polygon apices. If the total oriented angle is 360 degrees, the point is inside. If the total is 0, the point is outside.
I like this method better because it is more robust and less dependent on numerical precision.
Methods that compute evenness of number of intersections are limited because you can 'hit' an apex during the computation of the number of intersections.
EDIT: By The Way, this method works with concave and convex polygons.
EDIT: I recently found a whole Wikipedia article on the topic.
This question is so interesting. I have another workable idea different from other answers to this post. The idea is to use the sum of angles to decide whether the target is inside or outside. Better known as winding number.
Let x be the target point. Let array [0, 1, .... n] be the all the points of the area. Connect the target point with every border point with a line. If the target point is inside of this area. The sum of all angles will be 360 degrees. If not the angles will be less than 360.
Refer to this image to get a basic understanding of the idea:
My algorithm assumes the clockwise is the positive direction. Here is a potential input:
[[-122.402015, 48.225216], [-117.032049, 48.999931], [-116.919132, 45.995175], [-124.079107, 46.267259], [-124.717175, 48.377557], [-122.92315, 47.047963], [-122.402015, 48.225216]]
The following is the python code that implements the idea:
def isInside(self, border, target):
degree = 0
for i in range(len(border) - 1):
a = border[i]
b = border[i + 1]
# calculate distance of vector
A = getDistance(a[0], a[1], b[0], b[1]);
B = getDistance(target[0], target[1], a[0], a[1])
C = getDistance(target[0], target[1], b[0], b[1])
# calculate direction of vector
ta_x = a[0] - target[0]
ta_y = a[1] - target[1]
tb_x = b[0] - target[0]
tb_y = b[1] - target[1]
cross = tb_y * ta_x - tb_x * ta_y
clockwise = cross < 0
# calculate sum of angles
if(clockwise):
degree = degree + math.degrees(math.acos((B * B + C * C - A * A) / (2.0 * B * C)))
else:
degree = degree - math.degrees(math.acos((B * B + C * C - A * A) / (2.0 * B * C)))
if(abs(round(degree) - 360) <= 3):
return True
return False
The Eric Haines article cited by bobobobo is really excellent. Particularly interesting are the tables comparing performance of the algorithms; the angle summation method is really bad compared to the others. Also interesting is that optimisations like using a lookup grid to further subdivide the polygon into "in" and "out" sectors can make the test incredibly fast even on polygons with > 1000 sides.
Anyway, it's early days but my vote goes to the "crossings" method, which is pretty much what Mecki describes I think. However I found it most succintly described and codified by David Bourke. I love that there is no real trigonometry required, and it works for convex and concave, and it performs reasonably well as the number of sides increases.
By the way, here's one of the performance tables from the Eric Haines' article for interest, testing on random polygons.
number of edges per polygon
3 4 10 100 1000
MacMartin 2.9 3.2 5.9 50.6 485
Crossings 3.1 3.4 6.8 60.0 624
Triangle Fan+edge sort 1.1 1.8 6.5 77.6 787
Triangle Fan 1.2 2.1 7.3 85.4 865
Barycentric 2.1 3.8 13.8 160.7 1665
Angle Summation 56.2 70.4 153.6 1403.8 14693
Grid (100x100) 1.5 1.5 1.6 2.1 9.8
Grid (20x20) 1.7 1.7 1.9 5.7 42.2
Bins (100) 1.8 1.9 2.7 15.1 117
Bins (20) 2.1 2.2 3.7 26.3 278
Really like the solution posted by Nirg and edited by bobobobo. I just made it javascript friendly and a little more legible for my use:
function insidePoly(poly, pointx, pointy) {
var i, j;
var inside = false;
for (i = 0, j = poly.length - 1; i < poly.length; j = i++) {
if(((poly[i].y > pointy) != (poly[j].y > pointy)) && (pointx < (poly[j].x-poly[i].x) * (pointy-poly[i].y) / (poly[j].y-poly[i].y) + poly[i].x) ) inside = !inside;
}
return inside;
}
Swift version of the answer by nirg:
extension CGPoint {
func isInsidePolygon(vertices: [CGPoint]) -> Bool {
guard !vertices.isEmpty else { return false }
var j = vertices.last!, c = false
for i in vertices {
let a = (i.y > y) != (j.y > y)
let b = (x < (j.x - i.x) * (y - i.y) / (j.y - i.y) + i.x)
if a && b { c = !c }
j = i
}
return c
}
}
Most of the answers in this question are not handling all corner cases well. Some subtle corner cases like below:
This is a javascript version with all corner cases well handled.
/** Get relationship between a point and a polygon using ray-casting algorithm
* #param {{x:number, y:number}} P: point to check
* #param {{x:number, y:number}[]} polygon: the polygon
* #returns -1: outside, 0: on edge, 1: inside
*/
function relationPP(P, polygon) {
const between = (p, a, b) => p >= a && p <= b || p <= a && p >= b
let inside = false
for (let i = polygon.length-1, j = 0; j < polygon.length; i = j, j++) {
const A = polygon[i]
const B = polygon[j]
// corner cases
if (P.x == A.x && P.y == A.y || P.x == B.x && P.y == B.y) return 0
if (A.y == B.y && P.y == A.y && between(P.x, A.x, B.x)) return 0
if (between(P.y, A.y, B.y)) { // if P inside the vertical range
// filter out "ray pass vertex" problem by treating the line a little lower
if (P.y == A.y && B.y >= A.y || P.y == B.y && A.y >= B.y) continue
// calc cross product `PA X PB`, P lays on left side of AB if c > 0
const c = (A.x - P.x) * (B.y - P.y) - (B.x - P.x) * (A.y - P.y)
if (c == 0) return 0
if ((A.y < B.y) == (c > 0)) inside = !inside
}
}
return inside? 1 : -1
}
I did some work on this back when I was a researcher under Michael Stonebraker - you know, the professor who came up with Ingres, PostgreSQL, etc.
We realized that the fastest way was to first do a bounding box because it's SUPER fast. If it's outside the bounding box, it's outside. Otherwise, you do the harder work...
If you want a great algorithm, look to the open source project PostgreSQL source code for the geo work...
I want to point out, we never got any insight into right vs left handedness (also expressible as an "inside" vs "outside" problem...
UPDATE
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.
In addition, my work used separate functions for "on the line" tests.
...Since someone asked: we figured out that bounding box tests were best when the number of verticies went beyond some number - do a very quick test before doing the longer test if necessary... A bounding box is created by simply taking the largest x, smallest x, largest y and smallest y and putting them together to make four points of a box...
Another tip for those that follow: we did all our more sophisticated and "light-dimming" computing in a grid space all in positive points on a plane and then re-projected back into "real" longitude/latitude, thus avoiding possible errors of wrapping around when one crossed line 180 of longitude and when handling polar regions. Worked great!
The trivial solution would be to divide the polygon to triangles and hit test the triangles as explained here
If your polygon is CONVEX there might be a better approach though. Look at the polygon as a collection of infinite lines. Each line dividing space into two. for every point it's easy to say if its on the one side or the other side of the line. If a point is on the same side of all lines then it is inside the polygon.
David Segond's answer is pretty much the standard general answer, and Richard T's is the most common optimization, though therre are some others. Other strong optimizations are based on less general solutions. For example if you are going to check the same polygon with lots of points, triangulating the polygon can speed things up hugely as there are a number of very fast TIN searching algorithms. Another is if the polygon and points are on a limited plane at low resolution, say a screen display, you can paint the polygon onto a memory mapped display buffer in a given colour, and check the color of a given pixel to see if it lies in the polygons.
Like many optimizations, these are based on specific rather than general cases, and yield beneifits based on amortized time rather than single usage.
Working in this field, i found Joeseph O'Rourkes 'Computation Geometry in C' ISBN 0-521-44034-3 to be a great help.
Java Version:
public class Geocode {
private float latitude;
private float longitude;
public Geocode() {
}
public Geocode(float latitude, float longitude) {
this.latitude = latitude;
this.longitude = longitude;
}
public float getLatitude() {
return latitude;
}
public void setLatitude(float latitude) {
this.latitude = latitude;
}
public float getLongitude() {
return longitude;
}
public void setLongitude(float longitude) {
this.longitude = longitude;
}
}
public class GeoPolygon {
private ArrayList<Geocode> points;
public GeoPolygon() {
this.points = new ArrayList<Geocode>();
}
public GeoPolygon(ArrayList<Geocode> points) {
this.points = points;
}
public GeoPolygon add(Geocode geo) {
points.add(geo);
return this;
}
public boolean inside(Geocode geo) {
int i, j;
boolean c = false;
for (i = 0, j = points.size() - 1; i < points.size(); j = i++) {
if (((points.get(i).getLongitude() > geo.getLongitude()) != (points.get(j).getLongitude() > geo.getLongitude())) &&
(geo.getLatitude() < (points.get(j).getLatitude() - points.get(i).getLatitude()) * (geo.getLongitude() - points.get(i).getLongitude()) / (points.get(j).getLongitude() - points.get(i).getLongitude()) + points.get(i).getLatitude()))
c = !c;
}
return c;
}
}
I realize this is old, but here is a ray casting algorithm implemented in Cocoa, in case anyone is interested. Not sure it is the most efficient way to do things, but it may help someone out.
- (BOOL)shape:(NSBezierPath *)path containsPoint:(NSPoint)point
{
NSBezierPath *currentPath = [path bezierPathByFlatteningPath];
BOOL result;
float aggregateX = 0; //I use these to calculate the centroid of the shape
float aggregateY = 0;
NSPoint firstPoint[1];
[currentPath elementAtIndex:0 associatedPoints:firstPoint];
float olderX = firstPoint[0].x;
float olderY = firstPoint[0].y;
NSPoint interPoint;
int noOfIntersections = 0;
for (int n = 0; n < [currentPath elementCount]; n++) {
NSPoint points[1];
[currentPath elementAtIndex:n associatedPoints:points];
aggregateX += points[0].x;
aggregateY += points[0].y;
}
for (int n = 0; n < [currentPath elementCount]; n++) {
NSPoint points[1];
[currentPath elementAtIndex:n associatedPoints:points];
//line equations in Ax + By = C form
float _A_FOO = (aggregateY/[currentPath elementCount]) - point.y;
float _B_FOO = point.x - (aggregateX/[currentPath elementCount]);
float _C_FOO = (_A_FOO * point.x) + (_B_FOO * point.y);
float _A_BAR = olderY - points[0].y;
float _B_BAR = points[0].x - olderX;
float _C_BAR = (_A_BAR * olderX) + (_B_BAR * olderY);
float det = (_A_FOO * _B_BAR) - (_A_BAR * _B_FOO);
if (det != 0) {
//intersection points with the edges
float xIntersectionPoint = ((_B_BAR * _C_FOO) - (_B_FOO * _C_BAR)) / det;
float yIntersectionPoint = ((_A_FOO * _C_BAR) - (_A_BAR * _C_FOO)) / det;
interPoint = NSMakePoint(xIntersectionPoint, yIntersectionPoint);
if (olderX <= points[0].x) {
//doesn't matter in which direction the ray goes, so I send it right-ward.
if ((interPoint.x >= olderX && interPoint.x <= points[0].x) && (interPoint.x > point.x)) {
noOfIntersections++;
}
} else {
if ((interPoint.x >= points[0].x && interPoint.x <= olderX) && (interPoint.x > point.x)) {
noOfIntersections++;
}
}
}
olderX = points[0].x;
olderY = points[0].y;
}
if (noOfIntersections % 2 == 0) {
result = FALSE;
} else {
result = TRUE;
}
return result;
}
Obj-C version of nirg's answer with sample method for testing points. Nirg's answer worked well for me.
- (BOOL)isPointInPolygon:(NSArray *)vertices point:(CGPoint)test {
NSUInteger nvert = [vertices count];
NSInteger i, j, c = 0;
CGPoint verti, vertj;
for (i = 0, j = nvert-1; i < nvert; j = i++) {
verti = [(NSValue *)[vertices objectAtIndex:i] CGPointValue];
vertj = [(NSValue *)[vertices objectAtIndex:j] CGPointValue];
if (( (verti.y > test.y) != (vertj.y > test.y) ) &&
( test.x < ( vertj.x - verti.x ) * ( test.y - verti.y ) / ( vertj.y - verti.y ) + verti.x) )
c = !c;
}
return (c ? YES : NO);
}
- (void)testPoint {
NSArray *polygonVertices = [NSArray arrayWithObjects:
[NSValue valueWithCGPoint:CGPointMake(13.5, 41.5)],
[NSValue valueWithCGPoint:CGPointMake(42.5, 56.5)],
[NSValue valueWithCGPoint:CGPointMake(39.5, 69.5)],
[NSValue valueWithCGPoint:CGPointMake(42.5, 84.5)],
[NSValue valueWithCGPoint:CGPointMake(13.5, 100.0)],
[NSValue valueWithCGPoint:CGPointMake(6.0, 70.5)],
nil
];
CGPoint tappedPoint = CGPointMake(23.0, 70.0);
if ([self isPointInPolygon:polygonVertices point:tappedPoint]) {
NSLog(#"YES");
} else {
NSLog(#"NO");
}
}
There is nothing more beutiful than an inductive definition of a problem. For the sake of completeness here you have a version in prolog which might also clarify the thoughs behind ray casting:
Based on the simulation of simplicity algorithm in http://www.ecse.rpi.edu/Homepages/wrf/Research/Short_Notes/pnpoly.html
Some helper predicates:
exor(A,B):- \+A,B;A,\+B.
in_range(Coordinate,CA,CB) :- exor((CA>Coordinate),(CB>Coordinate)).
inside(false).
inside(_,[_|[]]).
inside(X:Y, [X1:Y1,X2:Y2|R]) :- in_range(Y,Y1,Y2), X > ( ((X2-X1)*(Y-Y1))/(Y2-Y1) + X1),toggle_ray, inside(X:Y, [X2:Y2|R]); inside(X:Y, [X2:Y2|R]).
get_line(_,_,[]).
get_line([XA:YA,XB:YB],[X1:Y1,X2:Y2|R]):- [XA:YA,XB:YB]=[X1:Y1,X2:Y2]; get_line([XA:YA,XB:YB],[X2:Y2|R]).
The equation of a line given 2 points A and B (Line(A,B)) is:
(YB-YA)
Y - YA = ------- * (X - XA)
(XB-YB)
It is important that the direction of rotation for the line is
setted to clock-wise for boundaries and anti-clock-wise for holes.
We are going to check whether the point (X,Y), i.e the tested point is at the left
half-plane of our line (it is a matter of taste, it could also be
the right side, but also the direction of boundaries lines has to be changed in
that case), this is to project the ray from the point to the right (or left)
and acknowledge the intersection with the line. We have chosen to project
the ray in the horizontal direction (again it is a matter of taste,
it could also be done in vertical with similar restrictions), so we have:
(XB-XA)
X < ------- * (Y - YA) + XA
(YB-YA)
Now we need to know if the point is at the left (or right) side of
the line segment only, not the entire plane, so we need to
restrict the search only to this segment, but this is easy since
to be inside the segment only one point in the line can be higher
than Y in the vertical axis. As this is a stronger restriction it
needs to be the first to check, so we take first only those lines
meeting this requirement and then check its possition. By the Jordan
Curve theorem any ray projected to a polygon must intersect at an
even number of lines. So we are done, we will throw the ray to the
right and then everytime it intersects a line, toggle its state.
However in our implementation we are goint to check the lenght of
the bag of solutions meeting the given restrictions and decide the
innership upon it. for each line in the polygon this have to be done.
is_left_half_plane(_,[],[],_).
is_left_half_plane(X:Y,[XA:YA,XB:YB], [[X1:Y1,X2:Y2]|R], Test) :- [XA:YA, XB:YB] = [X1:Y1, X2:Y2], call(Test, X , (((XB - XA) * (Y - YA)) / (YB - YA) + XA));
is_left_half_plane(X:Y, [XA:YA, XB:YB], R, Test).
in_y_range_at_poly(Y,[XA:YA,XB:YB],Polygon) :- get_line([XA:YA,XB:YB],Polygon), in_range(Y,YA,YB).
all_in_range(Coordinate,Polygon,Lines) :- aggregate(bag(Line), in_y_range_at_poly(Coordinate,Line,Polygon), Lines).
traverses_ray(X:Y, Lines, Count) :- aggregate(bag(Line), is_left_half_plane(X:Y, Line, Lines, <), IntersectingLines), length(IntersectingLines, Count).
% This is the entry point predicate
inside_poly(X:Y,Polygon,Answer) :- all_in_range(Y,Polygon,Lines), traverses_ray(X:Y, Lines, Count), (1 is mod(Count,2)->Answer=inside;Answer=outside).
I've made a Python implementation of nirg's c++ code:
Inputs
bounding_points: nodes that make up the polygon.
bounding_box_positions: candidate points to filter. (In my implementation created from the bounding box.
(The inputs are lists of tuples in the format: [(xcord, ycord), ...])
Returns
All the points that are inside the polygon.
def polygon_ray_casting(self, bounding_points, bounding_box_positions):
# Arrays containing the x- and y-coordinates of the polygon's vertices.
vertx = [point[0] for point in bounding_points]
verty = [point[1] for point in bounding_points]
# Number of vertices in the polygon
nvert = len(bounding_points)
# Points that are inside
points_inside = []
# For every candidate position within the bounding box
for idx, pos in enumerate(bounding_box_positions):
testx, testy = (pos[0], pos[1])
c = 0
for i in range(0, nvert):
j = i - 1 if i != 0 else nvert - 1
if( ((verty[i] > testy ) != (verty[j] > testy)) and
(testx < (vertx[j] - vertx[i]) * (testy - verty[i]) / (verty[j] - verty[i]) + vertx[i]) ):
c += 1
# If odd, that means that we are inside the polygon
if c % 2 == 1:
points_inside.append(pos)
return points_inside
Again, the idea is taken from here
C# version of nirg's answer is here: I'll just share the code. It may save someone some time.
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;
}
VBA VERSION:
Note: Remember that if your polygon is an area within a map that Latitude/Longitude are Y/X values as opposed to X/Y (Latitude = Y, Longitude = X) due to from what I understand are historical implications from way back when Longitude was not a measurement.
CLASS MODULE: 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
MODULE:
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
.Net port:
static void Main(string[] args)
{
Console.Write("Hola");
List<double> vertx = new List<double>();
List<double> verty = new List<double>();
int i, j, c = 0;
vertx.Add(1);
vertx.Add(2);
vertx.Add(1);
vertx.Add(4);
vertx.Add(4);
vertx.Add(1);
verty.Add(1);
verty.Add(2);
verty.Add(4);
verty.Add(4);
verty.Add(1);
verty.Add(1);
int nvert = 6; //Vértices del poligono
double testx = 2;
double testy = 5;
for (i = 0, j = nvert - 1; i < nvert; j = i++)
{
if (((verty[i] > testy) != (verty[j] > testy)) &&
(testx < (vertx[j] - vertx[i]) * (testy - verty[i]) / (verty[j] - verty[i]) + vertx[i]))
c = 1;
}
}
Surprised nobody brought this up earlier, but for the pragmatists requiring a database: MongoDB has excellent support for Geo queries including this one.
What you are looking for is:
db.neighborhoods.findOne({ geometry: { $geoIntersects: { $geometry: {
type: "Point", coordinates: [ "longitude", "latitude" ] } } }
})
Neighborhoods is the collection that stores one or more polygons in standard GeoJson format. If the query returns null it is not intersected otherwise it is.
Very well documented here:
https://docs.mongodb.com/manual/tutorial/geospatial-tutorial/
The performance for more than 6,000 points classified in a 330 irregular polygon grid was less than one minute with no optimization at all and including the time to update documents with their respective polygon.
Heres a point in polygon test in C that isn't using ray-casting. And it can work for overlapping areas (self intersections), see the use_holes argument.
/* math lib (defined below) */
static float dot_v2v2(const float a[2], const float b[2]);
static float angle_signed_v2v2(const float v1[2], const float v2[2]);
static void copy_v2_v2(float r[2], const float a[2]);
/* intersection function */
bool isect_point_poly_v2(const float pt[2], const float verts[][2], const unsigned int nr,
const bool use_holes)
{
/* we do the angle rule, define that all added angles should be about zero or (2 * PI) */
float angletot = 0.0;
float fp1[2], fp2[2];
unsigned int i;
const float *p1, *p2;
p1 = verts[nr - 1];
/* first vector */
fp1[0] = p1[0] - pt[0];
fp1[1] = p1[1] - pt[1];
for (i = 0; i < nr; i++) {
p2 = verts[i];
/* second vector */
fp2[0] = p2[0] - pt[0];
fp2[1] = p2[1] - pt[1];
/* dot and angle and cross */
angletot += angle_signed_v2v2(fp1, fp2);
/* circulate */
copy_v2_v2(fp1, fp2);
p1 = p2;
}
angletot = fabsf(angletot);
if (use_holes) {
const float nested = floorf((angletot / (float)(M_PI * 2.0)) + 0.00001f);
angletot -= nested * (float)(M_PI * 2.0);
return (angletot > 4.0f) != ((int)nested % 2);
}
else {
return (angletot > 4.0f);
}
}
/* math lib */
static float dot_v2v2(const float a[2], const float b[2])
{
return a[0] * b[0] + a[1] * b[1];
}
static float angle_signed_v2v2(const float v1[2], const float v2[2])
{
const float perp_dot = (v1[1] * v2[0]) - (v1[0] * v2[1]);
return atan2f(perp_dot, dot_v2v2(v1, v2));
}
static void copy_v2_v2(float r[2], const float a[2])
{
r[0] = a[0];
r[1] = a[1];
}
Note: this is one of the less optimal methods since it includes a lot of calls to atan2f, but it may be of interest to developers reading this thread (in my tests its ~23x slower then using the line intersection method).
If you're using Google Map SDK and want to check if a point is inside a polygon, you can try to use GMSGeometryContainsLocation. It works great!! Here is how that works,
if GMSGeometryContainsLocation(point, polygon, true) {
print("Inside this polygon.")
} else {
print("outside this polygon")
}
Here is the reference: https://developers.google.com/maps/documentation/ios-sdk/reference/group___geometry_utils#gaba958d3776d49213404af249419d0ffd
This is a presumably slightly less optimized version of the C code from here which was sourced from this page.
My C++ version uses a std::vector<std::pair<double, double>> and two doubles as an x and y. The logic should be exactly the same as the original C code, but I find mine easier to read. I can't speak for the performance.
bool point_in_poly(std::vector<std::pair<double, double>>& verts, double point_x, double point_y)
{
bool in_poly = false;
auto num_verts = verts.size();
for (int i = 0, j = num_verts - 1; i < num_verts; j = i++) {
double x1 = verts[i].first;
double y1 = verts[i].second;
double x2 = verts[j].first;
double y2 = verts[j].second;
if (((y1 > point_y) != (y2 > point_y)) &&
(point_x < (x2 - x1) * (point_y - y1) / (y2 - y1) + x1))
in_poly = !in_poly;
}
return in_poly;
}
The original C code is
int pnpoly(int nvert, float *vertx, float *verty, float testx, float testy)
{
int i, j, c = 0;
for (i = 0, j = nvert-1; i < nvert; j = i++) {
if ( ((verty[i]>testy) != (verty[j]>testy)) &&
(testx < (vertx[j]-vertx[i]) * (testy-verty[i]) / (verty[j]-verty[i]) + vertx[i]) )
c = !c;
}
return c;
}
Yet another numpyic implementation which I believe is the most concise one out of all the answers so far.
For example, let's say we have a polygon with polygon hollows that looks like this:
The 2D coordinates for the vertices of the large polygon are
[[139, 483], [227, 792], [482, 849], [523, 670], [352, 330]]
The coordinates for the vertices of the square hollow are
[[248, 518], [336, 510], [341, 614], [250, 620]]
The coordinates for the vertices of the triangle hollow are
[[416, 531], [505, 517], [495, 616]]
Say we want to test two points [296, 557] and [422, 730] if they are within the red area (excluding the edges). If we locate the two points, it will look like this:
Obviously, [296, 557] is not inside the read area, whereas [422, 730] is.
My solution is based on the winding number algorithm. Below is my 4-line python code using only numpy:
def detect(points, *polygons):
import numpy as np
endpoint1 = np.r_[tuple(np.roll(p, 1, 0) for p in polygons)][:, None] - points
endpoint2 = np.r_[polygons][:, None] - points
p1, p2 = np.cross(endpoint1, endpoint2), np.einsum('...i,...i', endpoint1, endpoint2)
return ~((p1.sum(0) < 0) ^ (abs(np.arctan2(p1, p2).sum(0)) > np.pi) | ((p1 == 0) & (p2 <= 0)).any(0))
To test the implementation:
points = [[296, 557], [422, 730]]
polygon1 = [[139, 483], [227, 792], [482, 849], [523, 670], [352, 330]]
polygon2 = [[248, 518], [336, 510], [341, 614], [250, 620]]
polygon3 = [[416, 531], [505, 517], [495, 616]]
print(detect(points, polygon1, polygon2, polygon3))
Output:
[False True]
For Detecting hit on Polygon we need to test two things:
If Point is inside polygon area. (can be accomplished by Ray-Casting Algorithm)
If Point is on the polygon border(can be accomplished by same algorithm which is used for point detection on polyline(line)).
To deal with the following special cases in Ray casting algorithm:
The ray overlaps one of the polygon's side.
The point is inside of the polygon and the ray passes through a vertex of the polygon.
The point is outside of the polygon and the ray just touches one of the polygon's angle.
Check Determining Whether A Point Is Inside A Complex Polygon. The article provides an easy way to resolve them so there will be no special treatment required for the above cases.
You can do this by checking if the area formed by connecting the desired point to the vertices of your polygon matches the area of the polygon itself.
Or you could check if the sum of the inner angles from your point to each pair of two consecutive polygon vertices to your check point sums to 360, but I have the feeling that the first option is quicker because it doesn't involve divisions nor calculations of inverse of trigonometric functions.
I don't know what happens if your polygon has a hole inside it but it seems to me that the main idea can be adapted to this situation
You can as well post the question in a math community. I bet they have one million ways of doing that
If you are looking for a java-script library there's a javascript google maps v3 extension for the Polygon class to detect whether or not a point resides within it.
var polygon = new google.maps.Polygon([], "#000000", 1, 1, "#336699", 0.3);
var isWithinPolygon = polygon.containsLatLng(40, -90);
Google Extention Github