Changeset - 40cc3b625eb2
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Laman - 6 years ago 2019-01-22 11:18:50

work on polar hough
2 files changed with 86 insertions and 90 deletions:
exp/stone_detect.py
78
90
src/analyzer/epoint.py
8
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exp/stone_detect.py
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import sys
 
sys.path.append("../src")
 

			




	
	
	
		
 
import os

			




	
	
	
		
 
import math

			




	
	
	
		
 
import random

			




	
	
	
		
 
import itertools

			




	
	
	
		
 

			




	
	
	
		
 
import cv2 as cv

			




	
	
	
		
 
import numpy as np

			




	
	
	
		
 
import scipy.cluster

			




	
	
	
		
 
import scipy.ndimage

			




	
	
	
		
 
import scipy.signal

			




	
	
	
		
 

			




	
	
	
		
 
from annotations import DataFile,computeBoundingBox

			




	
	
	
		
 
from hough import show

			




	
	
	
		
 
from analyzer.epoint import EPoint,homogenize

			




	
	
	
		
 
from analyzer.grid import transformPoint

			




	
	
	
		
 

			




	
	
	
		
 
random.seed(361)

			




	
	
	
		
 

			




	
	
	
		
 

			




	
	
	
		
 
class Line():

			




	
	
	
		
 
	def __init__(self,a,b):

			




	
	
	
		
 
		self.a=a

			




	
	
	
		
 
		self.b=b

			




	
	
	
		
 
		self.points={a,b}

			




	
	
	
		
 

			




	
	
	
		
 
	def getSortedPoints(self):

			




	
	
	
		
 
		return tuple(sorted(self.points))

			




	
	
	
		
 

			




	
	
	
		
 
	def computeAngle(self,line):

			




	
	
	
		
 
		ab=self.a-self.b

			




	
	
	
		
 
		cd=line.a-line.b

			




	
	
	
		
 
		alpha=math.atan(ab.y/ab.x)

			




	
	
	
		
 
		gamma=math.atan(cd.y/cd.x)

			




	
	
	
		
 
		fi=max(alpha,gamma)-min(alpha,gamma)

			




	
	
	
		
 
		return min(fi,math.pi-fi)

			




	
	
	
		
 

			




	
	
	
		
 
	def intersect(self,line):

			




	
	
	
		
 
		p=self.toProjective()

			




	
	
	
		
 
		q=line.toProjective()

			




	
	
	
		
 
		return EPoint.fromProjective(np.cross(p,q))

			




	
	
	
		
 

			




	
	
	
		
 
	def toProjective(self):

			




	
	
	
		
 
		return homogenize(np.cross(self.a.toProjective(),self.b.toProjective()))

			




	
	
	
		
 

			




	
	
	
		
 
	def transform(self,matrix):

			




	
	
	
		
 
		a=EPoint.fromProjective(transformPoint(self.a.toProjective(),matrix))

			




	
	
	
		
 
		b=EPoint.fromProjective(transformPoint(self.b.toProjective(),matrix))

			




	
	
	
		
 
		if a is None or b is None: return None

			




	
	
	
		
 
		return Line(a,b)

			




	
	
	
		
 

			




	
	
	
		
 
	def __str__(self): return "({0},{1})".format(self.a,self.b)

			




	
	
	
		
 
	def __repr__(self): return "Line({0},{1})".format(repr(self.a),repr(self.b))

			




	
	
	
		
 

			




	
	
	
		
 

			




	
	
	
		
 
class PolarHough:

			




	
	
	
		
 
	# http://www.cis.pku.edu.cn/vision/vpdetection/LiPYZ10isvc.pdf

			




	
	
	
		
 
	def __init__(self,anglePrecision,lengthPrecision):

			




	
	
	
		
 
		self._anglePrecision=anglePrecision

			




	
	
	
		
 
		self._lengthPrecision=lengthPrecision

			




	
	
	
		
 
		self._maxLength=4096

			




	
	
	
		
 
		n=math.ceil(2*math.pi/anglePrecision)

			




	
	
	
		
 
		self._acc=[[] for i in range(n)]

			




	
	
	
		
 

			




	
	
	
		
 
	def put(self,item):

			




	
	
	
		
 
		k=int(item[0]//self._anglePrecision)

			




	
	
	
		
 
		self._acc[k].append(item)

			




	
	
	
		
 

			




	
	
	
		
 
	def extract(self,count):

			




	
	
	
		
 
		vanishingPoints=[]

			




	
	
	
		
 
		angles=self._extractAngles(count)

			




	
	
	
		
 
		angles=[alpha for (alpha,prominence) in angles]

			




	
	
	
		
 
		bins=self._mapAngles(angles)

			




	
	
	
		
 
		for (alpha,bin) in zip(angles,bins):

			




	
	
	
		
 
			(length,sampleCount)=self._extractLength([dist for (beta,dist) in bin])

			




	
	
	
		
 
			vanishingPoints.append((alpha,length))

			




	
	
	
		
 
		return vanishingPoints

			




	
	
	
		
 

			




	
	
	
		
 
	def _extractAngles(self,k):

			




	
	
	
		
 
		lens=np.array(list(map(len,self._acc)))

			




	
	
	
		
 
		print(lens)

			




	
	
	
		
 
		(peakKeys,info)=scipy.signal.find_peaks(lens,prominence=0)

			




	
	
	
		
 
		res=sorted(zip(info["prominences"],peakKeys),reverse=True)[:k]

			




	
	
	
		
 
		res=[(key*self._anglePrecision,prominence) for (prominence,key) in res]

			




	
	
	
		
 
		print("(angle, prominence):",res,"...",[alpha/self._anglePrecision for (alpha,_) in res])

			




	
	
	
		
 
		return res

			




	
	
	
		
 

			




	
	
	
		
 
	def _mapAngles(self,angles):

			




	
	
	
		
 
		res=[[] for alpha in angles]

			




	
	
	
		
 
		for (i,bin) in enumerate(self._acc):

			




	
	
	
		
 
			beta=i*self._anglePrecision

			




	
	
	
		
 
			key=min(zip(map(lambda alpha: angleDiff(alpha,beta), angles), itertools.count()))[1]

			




	
	
	
		
 
			res[key].extend(bin)

			




	
	
	
		
 
		return res

			




	
	
	
		
 

			




	
	
	
		
 
	def _extractLength(self,arr):

			




	
	
	
		
 
		acc=np.zeros(self._maxLength+1,dtype=np.int32)

			




	
	
	
		
 
		for dist in arr:

			




	
	
	
		
 
			dist=min(dist,self._maxLength)

			




	
	
	
		
 
			acc[int(dist//self._lengthPrecision)]+=1

			




	
	
	
		
 
		res=acc.argmax()

			




	
	
	
		
 
		print("(length, count):",(res*self._lengthPrecision,acc[res]))

			




	
	
	
		
 
		return (res*self._lengthPrecision,acc[res])

			




	
	
	
		
 

			




	
	
	
		
 

			




	
	
	
		
 
def angleDiff(alpha,beta):

			




	
	
	
		
 
	diff=abs(alpha-beta)

			




	
	
	
		
 
	if diff>math.pi: diff=2*math.pi-diff

			




	
	
	
		
 
	return diff

			




	
	
	
		
 

			




	
	
	
		
 

			




	
	
	
		
 
def kmeans(img):

			




	
	
	
		
 
	arr=np.reshape(img,(-1,3)).astype(np.float)

			




	
	
	
		
 
	colors=np.array([[0,0,0],[255,255,255],[193,165,116]],np.float)

			




	
	
	
		
 
	print(colors)

			




	
	
	
		
 
	(centers,distortion)=scipy.cluster.vq.kmeans(arr,colors)

			




	
	
	
		
 
	print("k-means centers:",centers)

			




	
	
	
		
 
	return centers

			




	
	
	
		
 

			




	
	
	
		
 

			




	
	
	
		
 
def quantize(img,centers):

			




	
	
	
		
 
	origShape=img.shape

			




	
	
	
		
 
	data=np.reshape(img,(-1,3))

			




	
	
	
		
 
	(keys,dists)=scipy.cluster.vq.vq(data,centers)

			




	
	
	
		
 
	pixels=np.array([centers[k] for k in keys],dtype=np.uint8).reshape(origShape)

			




	
	
	
		
 
	return pixels

			




	
	
	
		
 

			




	
	
	
		
 

			




	
	
	
		
 
def filterStones(contours,bwImg,stoneDims):

			




	
	
	
		
 
	contourImg=cv.cvtColor(bwImg,cv.COLOR_GRAY2BGR)

			




	
	
	
		
 
	res=[]

			




	
	
	
		
 
	for (i,c) in enumerate(contours):

			




	
	
	
		
 
		keep=True

			




	
	
	
		
 
		moments=cv.moments(c)

			




	
	
	
		
 
		center=(moments["m10"]/(moments["m00"] or 1), moments["m01"]/(moments["m00"] or 1))

			




	
	
		
 
@@ -169,164 +227,94 @@ def groupParallels(lines,tolerance,w):

			




	
	
	
		
 
			continue

			




	
	
	
		
 

			




	
	
	
		
 

			




	
	
	
		
 
if __name__=="__main__":

			




	
	
	
		
 
	filepath=sys.argv[1]

			




	
	
	
		
 
	annotations=DataFile(sys.argv[2])

			




	
	
	
		
 
	filename=os.path.basename(filepath)

			




	
	
	
		
 
	corners=annotations[filename][0]

			




	
	
	
		
 
	(x1,y1,x2,y2)=computeBoundingBox(corners)

			




	
	
	
		
 
	(w,h)=(x2-x1,y2-y1)

			




	
	
	
		
 
	img=cv.imread(filepath)

			




	
	
	
		
 
	(x3,x4,y3,y4)=(x1+w//4,x1+3*w//4,y1+h//4,y1+3*h//4)

			




	
	
	
		
 
	print("x3,x4,y3,y4:",x3,x4,y3,y4)

			




	
	
	
		
 
	rect=img[y3:y4,x3:x4,:]

			




	
	
	
		
 
	centers=kmeans(rect)

			




	
	
	
		
 
	print("x1,x2,y1,y2:",(x1,x2,y1,y2))

			




	
	
	
		
 
	img[y1:y2,x1:x2,:]=quantize(img[y1:y2,x1:x2,:],centers)

			




	
	
	
		
 
	print("image quantized")

			




	
	
	
		
 

			




	
	
	
		
 
	rect=img[y1:y2,x1:x2]

			




	
	
	
		
 
	unit=np.array([1,1,1],dtype=np.uint8)

			




	
	
	
		
 
	kernel=np.ones((3,3),np.uint8)

			




	
	
	
		
 
	maskB=cv.inRange(rect,centers[0]-unit,centers[0]+unit)

			




	
	
	
		
 
	maskB=cv.morphologyEx(maskB,cv.MORPH_OPEN,kernel,iterations=1)

			




	
	
	
		
 
	maskB=cv.erode(maskB,kernel,iterations=2)

			




	
	
	
		
 
	maskB=cv.erode(maskB,kernel,iterations=4)

			




	
	
	
		
 
	maskW=cv.inRange(rect,centers[1]-unit,centers[1]+unit)

			




	
	
	
		
 
	maskW=cv.erode(maskW,kernel,iterations=2)

			




	
	
	
		
 
	maskW=cv.erode(maskW,kernel,iterations=3)

			




	
	
	
		
 

			




	
	
	
		
 
	show(img,filename)

			




	
	
	
		
 
	show(maskB,filename)

			




	
	
	
		
 
	show(maskW,filename)

			




	
	
	
		
 
	stones=cv.bitwise_or(maskB,maskW)

			




	
	
	
		
 
	show(stones)

			




	
	
	
		
 

			




	
	
	
		
 
	stoneDims=(w/19,h/19)

			




	
	
	
		
 
	print("stone dims:",tuple(x/2 for x in stoneDims),"-",stoneDims)

			




	
	
	
		
 

			




	
	
	
		
 
	(contours,hierarchy)=cv.findContours(stones,cv.RETR_LIST,cv.CHAIN_APPROX_SIMPLE)

			




	
	
	
		
 
	stoneLocs=filterStones(contours,stones,stoneDims)

			




	
	
	
		
 

			




	
	
	
		
 
	linesImg=cv.cvtColor(np.zeros((h,w),np.uint8),cv.COLOR_GRAY2BGR)

			




	
	
	
		
 
	cv.drawContours(linesImg,[c for (point,c) in stoneLocs],-1,(255,255,255),-1)

			




	
	
	
		
 
	for (p,c) in stoneLocs:

			




	
	
	
		
 
		cv.drawMarker(linesImg,(int(p.x),int(p.y)),(255,0,0),cv.MARKER_CROSS)

			




	
	
	
		
 
	matsImg=np.copy(linesImg)

			




	
	
	
		
 

			




	
	
	
		
 
	lineDict=dict()

			




	
	
	
		
 
	minCount=min(max(math.sqrt(len(stoneLocs))-4,3),7)

			




	
	
	
		
 
	print("min count:",minCount)

			




	
	
	
		
 
	for line in groupLines([point for (point,contour) in stoneLocs],minCount,2):

			




	
	
	
		
 
		key=line.getSortedPoints()

			




	
	
	
		
 
		if key in lineDict: # we already have a line with the same incident points

			




	
	
	
		
 
			continue

			




	
	
	
		
 
		lineDict[line.getSortedPoints()]=line

			




	
	
	
		
 
		obsolete=set()

			




	
	
	
		
 
		for ab in lineDict.values():

			




	
	
	
		
 
			if ab is line: continue

			




	
	
	
		
 
			if line.points<ab.points: # == impossible

			




	
	
	
		
 
				del lineDict[key]

			




	
	
	
		
 
				break

			




	
	
	
		
 
			if ab.points<line.points:

			




	
	
	
		
 
				obsolete.add(ab.getSortedPoints())

			




	
	
	
		
 
		for key in obsolete: del lineDict[key]

			




	
	
	
		
 

			




	
	
	
		
 
	print("valid lines:",len(lineDict))

			




	
	
	
		
 
	lines=sorted(lineDict.values(), key=lambda ab: len(ab.points), reverse=True)

			




	
	
	
		
 
	res=[]

			




	
	
	
		
 
	for line in lines:

			




	
	
	
		
 
		v=line.b-line.a

			




	
	
	
		
 
		alpha=math.atan(v.y/v.x)

			




	
	
	
		
 
		res.append((round(math.pi/2-alpha if alpha>0 else math.pi/2+alpha,3),repr(line)))

			




	
	
	
		
 
		(xa,ya)=line.a

			




	
	
	
		
 
		(xb,yb)=line.b

			




	
	
	
		
 
		cv.line(linesImg,(int(xa),int(ya)),(int(xb),int(yb)),(255,255,0),1)

			




	
	
	
		
 
	res.sort()

			




	
	
	
		
 
	# for row in res: print(row)

			




	
	
	
		
 
	# linePack=[

			




	
	
	
		
 
	# 	Line(EPoint(174.457,166.127),EPoint(174.96,27.253)),

			




	
	
	
		
 
	# 	Line(EPoint(191.333,38.075),EPoint(191.485,71.227)),

			




	
	
	
		
 
	# 	Line(EPoint(210.117,167.092),EPoint(205.0,50.0)),

			




	
	
	
		
 
	# 	Line(EPoint(127.7,25.6),EPoint(120.172,179.405)),

			




	
	
	
		
 
	# 	Line(EPoint(127.809,58.481),EPoint(124.324,127.427)),

			




	
	
	
		
 
	# 	Line(EPoint(85.964,191.64),EPoint(97.68,14.477)),

			




	
	
	
		
 
	# 	Line(EPoint(56.447,124.662),EPoint(54.889,137.918))

			




	
	
	
		
 
	# ]

			




	
	
	
		
 
	# linePack=[

			




	
	
	
		
 
	# 	Line(EPoint(56.447,124.662),EPoint(139.695,128.104)),

			




	
	
	
		
 
	# 	Line(EPoint(288.267,74.433),EPoint(191.485,71.227)),

			




	
	
	
		
 
	# 	Line(EPoint(252.926,29.71),EPoint(174.96,27.253)),

			




	
	
	
		
 
	# 	Line(EPoint(274.412,120.07),EPoint(41.065,112.759)),

			




	
	
	
		
 
	# 	Line(EPoint(289.674,108.019),EPoint(26.17,100.538)),

			




	
	
	
		
 
	# 	Line(EPoint(240.702,107.818),EPoint(41.065,112.759)),

			




	
	
	
		
 
	# 	Line(EPoint(174.457,166.127),EPoint(88.192,164.5))

			




	
	
	
		
 
	# ]

			




	
	
	
		
 
	# for (i,p) in enumerate(linePack):

			




	
	
	
		
 
	# 	for q in linePack[i+1:]:

			




	
	
	
		
 
	# 		print(p.intersect(q))

			




	
	
	
		
 
	# show(linesImg)

			




	
	
	
		
 

			




	
	
	
		
 
	# parallels=[]

			




	
	
	
		
 
	# for (i,ab) in enumerate(lines):

			




	
	
	
		
 
	# 	for cd in lines[i+1:]:

			




	
	
	
		
 
	# 		if ab.computeAngle(cd)>math.pi/4:

			




	
	
	
		
 
	# 			continue

			




	
	
	
		
 
	# 		parallels.append((ab,cd))

			




	
	
	
		
 
	# print("parallel lines candidate pairs:",len(parallels))

			




	
	
	
		
 

			




	
	
	
		
 
	# parallels=list(groupParallels(lines,50,w))

			




	
	
	
		
 
	# print("parallel triples:",len(parallels))

			




	
	
	
		
 
	# for (p,q,r) in parallels:

			




	
	
	
		
 
	# 	print(p,q,r)

			




	
	
	
		
 
	# 	print(p.intersect(q))

			




	
	
	
		
 
	# 	print(p.intersect(r))

			




	
	
	
		
 
	# 	print(q.intersect(r))

			




	
	
	
		
 
	# 	print()

			




	
	
	
		
 
	# 	matsImg_=np.copy(matsImg)

			




	
	
	
		
 
	# 	for pi in (p,q,r):

			




	
	
	
		
 
	# 		cv.line(matsImg_,(int(pi.a.x),int(pi.a.y)),(int(pi.b.x),int(pi.b.y)),(0,255,255),1)

			




	
	
	
		
 
	# 	show(matsImg_)

			




	
	
	
		
 
	show(linesImg)

			




	
	
	
		
 

			




	
	
	
		
 
	# p=Line(corners[0],corners[1])

			




	
	
	
		
 
	# q=Line(corners[2],corners[3])

			




	
	
	
		
 
	# r=Line(corners[1],corners[2])

			




	
	
	
		
 
	# s=Line(corners[3],corners[0])

			




	
	
	
		
 
	#

			




	
	
	
		
 
	# matrix=computeRectiMatrix(p,q,r,s)

			




	
	
	
		
 
	# print(matrix)

			




	
	
	
		
 
	# for (point,contour) in stoneLocs:

			




	
	
	
		
 
	# 	point_=EPoint.fromProjective(transformPoint(point.toProjective(),matrix))

			




	
	
	
		
 
	# 	# print(point,"->",point_)

			




	
	
	
		
 
	# 	cv.line(matsImg,(int(point.x),int(point.y)),(int(point_.x),int(point_.y)),(0,255,255),1)

			




	
	
	
		
 
	# points1=np.float32([[p.x-x1,p.y-y1] for p in corners])

			




	
	
	
		
 
	# points2=np.float32([[0,0],[0,400],[400,400],[400,0]])

			




	
	
	
		
 
	# print(points1)

			




	
	
	
		
 
	# print(points2)

			




	
	
	
		
 
	# mat=cv.getPerspectiveTransform(points1,points2)

			




	
	
	
		
 
	# print(mat)

			




	
	
	
		
 
	# show(rect)

			




	
	
	
		
 
	# warped=cv.warpPerspective(rect,matrix,(400,400))

			




	
	
	
		
 
	# show(warped)

			




	
	
	
		
 

			




	
	
	
		
 
	# mats=[]

			




	
	
	
		
 
	# for (i,(p,q)) in enumerate(parallels):

			




	
	
	
		
 
	# 	for (r,s) in parallels[i+1:]:

			




	
	
	
		
 
	# 		if p is r or p is s or q is r or q is s:

			




	
	
	
		
 
	# 			continue

			




	
	
	
		
 
	# 		matrix=computeRectiMatrix(p,q,r,s)

			




	
	
	
		
 
	# 		score=scoreMatrix(matrix,p,r,lines)

			




	
	
	
		
 
	# 		mats.append((score,p,q,r,s,matrix))

			




	
	
	
		
 
	# mats.sort(key=lambda x: x[0])

			




	
	
	
		
 
	# for (score,p,q,r,s,matrix) in mats[:4]:

			




	
	
	
		
 
	# 	print(score,p,q,r,s,matrix)

			




	
	
	
		
 
	# 	matsImg_=np.copy(matsImg)

			




	
	
	
		
 
	# 	for ab in (p,q,r,s):

			




	
	
	
		
 
	# 		((xa,ya),(xb,yb))=(ab.a,ab.b)

			




	
	
	
		
 
	# 		cv.line(matsImg_,(int(xa),int(ya)),(int(xb),int(yb)),(255,255,0),1)

			




	
	
	
		
 
	# 	show(matsImg_)

			




	
	
	
		
 
	# for (score,p,q,r,s,matrix) in mats[-4:]:

			




	
	
	
		
 
	# 	print(score,p,q,r,s,matrix)

			




	
	
	
		
 
	# 	matsImg_=np.copy(matsImg)

			




	
	
	
		
 
	# 	for ab in (p,q,r,s):

			




	
	
	
		
 
	# 		((xa,ya),(xb,yb))=(ab.a,ab.b)

			




	
	
	
		
 
	# 		cv.line(matsImg_,(int(xa),int(ya)),(int(xb),int(yb)),(0,0,255),1)

			




	
	
	
		
 
	# 	show(matsImg_)

			




	
	
	
		
 
	imgSize=img.shape[:2]

			




	
	
	
		
 
	print("image size:",imgSize)

			




	
	
	
		
 
	imgCenter=EPoint(imgSize[1]/2,imgSize[0]/2)-EPoint(x1,y1)

			




	
	
	
		
 
	polarHough=PolarHough(math.pi/180,10)

			




	
	
	
		
 
	for (i,ab) in enumerate(lines):

			




	
	
	
		
 
		for cd in lines[i+1:]:

			




	
	
	
		
 
			point=ab.intersect(cd)

			




	
	
	
		
 
			if 0<=point.x<=w and 0<=point.y<=h: continue

			




	
	
	
		
 
			print(point,"->",point.toPolar(imgCenter))

			




	
	
	
		
 
			polarHough.put(point.toPolar(imgCenter))

			




	
	
	
		
 
	vanish=[EPoint.fromPolar(p,imgCenter) for p in polarHough.extract(2)]

			




	
	
	
		
 
	print(vanish)

			




	
	
	
		
 
	(a,b,c,d)=corners

			




	
	
	
		
 
	(p,q,r,s)=(Line(a,b),Line(b,c),Line(c,d),Line(d,a))

			




	
	
	
		
 
	v1=p.intersect(r)

			




	
	
	
		
 
	v2=q.intersect(s)

			




	
	
	
		
 
	print("true vanishing points:",v1,"~",v1.toPolar(imgCenter),"and",v2,"~",v2.toPolar(imgCenter))

			




        

    


    
    

    

        

                

                    src/analyzer/epoint.py
                

                

                  
                      
                        

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@@ -3,54 +3,62 @@

			




	
	
	
		
 

			




	
	
	
		
 
def homogenize(v):

			




	
	
	
		
 
	for k in v:

			




	
	
	
		
 
		if k!=0: return v/k

			




	
	
	
		
 
	return v

			




	
	
	
		
 

			




	
	
	
		
 

			




	
	
	
		
 
## Euclidean 2D plane point: (x,y).

			




	
	
	
		
 
class EPoint:

			




	
	
	
		
 
	def __init__(self,x,y):

			




	
	
	
		
 
		self._x=x

			




	
	
	
		
 
		self._y=y

			




	
	
	
		
 

			




	
	
	
		
 
	@property

			




	
	
	
		
 
	def x(self): return self._x

			




	
	
	
		
 
	

			




	
	
	
		
 
	@property

			




	
	
	
		
 
	def y(self): return self._y

			




	
	
	
		
 

			




	
	
	
		
 
	@staticmethod

			




	
	
	
		
 
	def fromProjective(point):

			




	
	
	
		
 
		if point.item(0)==0: return None

			




	
	
	
		
 
		return EPoint(point.item(1)/point.item(0),point.item(2)/point.item(0))

			




	
	
	
		
 

			




	
	
	
		
 
	@staticmethod

			




	
	
	
		
 
	def fromPolar(point,center):

			




	
	
	
		
 
		(alpha,length)=point

			




	
	
	
		
 
		x=math.cos(alpha)

			




	
	
	
		
 
		y=math.sin(alpha)

			




	
	
	
		
 
		return length*EPoint(x,y)+center

			




	
	
	
		
 

			




	
	
	
		
 
	def toProjective(self):

			




	
	
	
		
 
		return (1,self._x,self._y)

			




	
	
	
		
 

			




	
	
	
		
 
	def toPolar(self,center):

			




	
	
	
		
 
		v=self-center

			




	
	
	
		
 
		alpha=math.atan2(v.y,v.x)

			




	
	
	
		
 
		if alpha<0: alpha+=2*math.pi

			




	
	
	
		
 
		k=self.dist(center)

			




	
	
	
		
 
		return (alpha,k)

			




	
	
	
		
 

			




	
	
	
		
 
	def dist(self,a):

			




	
	
	
		
 
		return math.sqrt((self._x-a._x)**2+(self._y-a._y)**2)

			




	
	
	
		
 

			




	
	
	
		
 
	def __add__(self,a):

			




	
	
	
		
 
		return EPoint(self._x+a._x,self._y+a._y)

			




	
	
	
		
 

			




	
	
	
		
 
	def __sub__(self,a):

			




	
	
	
		
 
		return EPoint(self._x-a._x,self._y-a._y)

			




	
	
	
		
 

			




	
	
	
		
 
	def __mul__(self,k):

			




	
	
	
		
 
		return EPoint(self._x*k,self._y*k)

			




	
	
	
		
 

			




	
	
	
		
 
	def __rmul__(self,k):

			




	
	
	
		
 
		return self*k

			




	
	
	
		
 

			




	
	
	
		
 
	def __truediv__(self,k):

			




	
	
	
		
 
		return EPoint(self._x/k,self._y/k)

			




	
	
	
		
 

			




	
	
	
		
 
	def __floordiv__(self,k):

			




	
	
	
		
 
		return EPoint(self._x//k,self._y//k)

			




	
	
	
		
 

			




        

    



    


    



    



      

      





    
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