from epoint import *

class Corners:

  def __init__(self):

    self.corners=[]

  def add(self,x,y):

    a=EPoint(x,y)

    # for i,c in enumerate(self.corners): # move an improperly placed point

      # if a.dist(c)<20:

        # self.corners[i]=a

        # return

    if len(self.corners)<4: # add a new corner

      self.corners.append(a)

    if len(self.corners)<4: 

      return

    index,minDist=0,float('inf') # replace the corner closest to the clicked point

    for i,c in enumerate(self.corners):

      if a.dist(c)<minDist:

        index,minDist=i,a.dist(c)

    self.corners[index]=a

    self._canonizeOrder()

  ## Computes twice the area of a triangle formed by points a,b,c.

  #  

  #  @return positive value for points oriented counter-clockwise, negative for clockwise, zero for degenerate cases.

  def _doubleTriangleArea(a,b,c):

    return (a.x-b.x)*(c.y-a.y)-(c.x-a.x)*(a.y-b.y)

  def _slope(a,b):

    if(b.x==a.x): return float("inf")

    return (b.y-a.y)/(b.x-a.x)

  ## Order the corners (0,1,2,3) so they make a quadrangle with vertices KLMN in counter-clockwise order, K being in the upper left.

  #  

  #  For four points ABCD, there are 24 possible permutations corresponding to the desired KLMN.

  #  When we relax the condition of K being the upper left one, we get six groups of four equivalent permuations. KLMN ~ LMNK ~ MNKL ~ NKLM.

  #  

  #  xxxx -> KLMN | ABC | ABD | ACD | BCD | index | swap

  #  ------------ | :-: | :-: | :-: | :-: | ----: | ----

  #  A BCD        |  +  |  +  |  +  |  +  |    15 | 0

  #  A BDC        |  +  |  +  |  -  |  -  |    12 | CD

  #  A CBD        |  -  |  +  |  +  |  -  |     6 | BC

  #  A CDB        |  -  |  -  |  +  |  +  |     3 | AB

  #  A DBC        |  +  |  -  |  -  |  +  |     9 | AD

  #  A DCB        |  -  |  -  |  -  |  -  |     0 | BD

  #  

  #  For every non-degenerate quadrangle, there must be 1-3 edges going right-left (from a higher to a lower x coordinate).

  #  From these pick the one with the lowest slope (dy/dx) and declare its ending point the upper left corner. For the same slope pick the one further left.

  #  

  #  @return True for a convex quadrangle, False for concave or degenerate cases.

  def _canonizeOrder(self):

    if len(self.corners)!=4: return False # erroneus call

    a,b,c,d=(x for x in self.corners)

    abc=Corners._doubleTriangleArea(a,b,c)

    abd=Corners._doubleTriangleArea(a,b,d)

    acd=Corners._doubleTriangleArea(a,c,d)

    bcd=Corners._doubleTriangleArea(b,c,d)

    if any(x==0 for x in (abc,abd,acd,bcd)): return False # collinear degenerate

    swaps=[(1,3),(0,1),(1,2),(0,3),(2,3),(0,0)]

    index=(8 if abc>0 else 0)+(4 if abd>0 else 0)+(2 if acd>0 else 0)+(1 if bcd>0 else 0)

    if index%3!=0: return False # concave degenerate

    swap=swaps[index//3]

    self.corners[swap[0]], self.corners[swap[1]] = self.corners[swap[1]], self.corners[swap[0]] # counter-clockwise order

    kIndex=None

    lowestSlope=float("inf")

    for i,corner in enumerate(self.corners): # find the NK edge: going right-left with the lowest slope, secondarily the one going down

      ii=(i+1)%4

      slope=abs(Corners._slope(corner,self.corners[ii]))

      if corner.x>self.corners[ii].x and (slope<lowestSlope or (slope==lowestSlope and corner.y<self.corners[ii].y)):

        kIndex=ii

        lowestSlope=slope

    self.corners=self.corners[kIndex:]+self.corners[:kIndex] # rotate the upper left corner to the first place

    return True # success

# import itertools

# points=[(10,8),(8,10),(12,10),(10,12)]

# for perm in itertools.permutations(points):

  # corn=Corners()

  # for p in perm: corn.add(p[0],p[1])

  # print(corn._canonizeOrder())

  # print(corn.corners)

import math

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

class EPoint:

  def __init__(self,x,y):

    self.x=x

    self.y=y

  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 __truediv__(self,k):

    return EPoint(self.x/k,self.y/k)

  def __str__(self): return "({0},{1})".format(self.x,self.y)

  def __repr__(self): return "EPoint({0},{1})".format(self.x,self.y)

from epoint import *

  ## Stores a grid corner located at x,y coordinates.

  ## Redraws the current image and its overlay.

  ## Marks a point at the image with a green cross. Used for corners.

## Handles and presents the game state as detected by the program.