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#!/usr/bin/env python3
# Generate Apollonian Gaskets -- the math part.
# Copyright (c) 2014 Ludger Sandig
# This file is part of apollon.
# Apollon is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
# Apollon is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# GNU General Public License for more details.
# You should have received a copy of the GNU General Public License
# along with Apollon. If not, see <>.
from cmath import *
import random
class Circle(object):
A circle represented by center point as complex number and radius.
def __init__ ( self, mx, my, r ):
@param mx: x center coordinate
@type mx: int or float
@param my: y center coordinate
@type my: int or float
@param r: radius
@type r: int or float
self.r = r
self.m = (mx +my*1j)
def __repr__ ( self ):
Pretty printing
return "Circle( self, %s, %s, %s )" % (self.m.real, self.m.imag, self.r)
def __str__ ( self ):
Pretty printing
return "Circle x:%.3f y:%.3f r:%.3f [cur:%.3f]" % (self.m.real, self.m.imag, self.r.real, self.curvature().real)
def curvature (self):
Get circle's curvature.
@rtype: float
@return: Curvature of the circle.
return 1/self.r
def outerTangentCircle( circle1, circle2, circle3 ):
Takes three externally tangent circles and calculates the fourth one enclosing them.
@param circle1: first circle
@param circle2: second circle
@param circle3: third circle
@type circle1: L{Circle}
@type circle2: L{Circle}
@type circle3: L{Circle}
@return: The enclosing circle
@rtype: L{Circle}
cur1 = circle1.curvature()
cur2 = circle2.curvature()
cur3 = circle3.curvature()
m1 = circle1.m
m2 = circle2.m
m3 = circle3.m
cur4 = -2 * sqrt( cur1*cur2 + cur2*cur3 + cur1 * cur3 ) + cur1 + cur2 + cur3
m4 = ( -2 * sqrt( cur1*m1*cur2*m2 + cur2*m2*cur3*m3 + cur1*m1*cur3*m3 ) + cur1*m1 + cur2*m2 + cur3*m3 ) / cur4
circle4 = Circle( m4.real, m4.imag, 1/cur4 )
return circle4
def tangentCirclesFromRadii( r2, r3, r4 ):
Takes three radii and calculates the corresponding externally
tangent circles as well as a fourth one enclosing them. The enclosing
circle is the first one.
@param r2, r3, r4: Radii of the circles to calculate
@type r2: int or float
@type r3: int or float
@type r4: int or float
@return: The four circles, where the first one is the enclosing one.
@rtype: (L{Circle}, L{Circle}, L{Circle}, L{Circle})
circle2 = Circle( 0, 0, r2 )
circle3 = Circle( r2 + r3, 0, r3 )
m4x = (r2*r2 + r2*r4 + r2*r3 - r3*r4) / (r2 + r3)
m4y = sqrt( (r2 + r4) * (r2 + r4) - m4x*m4x )
circle4 = Circle( m4x, m4y, r4 )
circle1 = outerTangentCircle( circle2, circle3, circle4 )
return ( circle1, circle2, circle3, circle4 )
def secondSolution( fixed, c1, c2, c3 ):
If given four tangent circles, calculate the other one that is tangent
to the last three.
@param fixed: The fixed circle touches the other three, but not
the one to be calculated.
@param c1, c2, c3: Three circles to which the other tangent circle
is to be calculated.
@type fixed: L{Circle}
@type c1: L{Circle}
@type c2: L{Circle}
@type c3: L{Circle}
@return: The circle.
@rtype: L{Circle}
curf = fixed.curvature()
cur1 = c1.curvature()
cur2 = c2.curvature()
cur3 = c3.curvature()
curn = 2 * (cur1 + cur2 + cur3) - curf
mn = (2 * (cur1*c1.m + cur2*c2.m + cur3*c3.m) - curf*fixed.m ) / curn
return Circle( mn.real, mn.imag, 1/curn )
class ApollonianGasket(object):
Container for an Apollonian Gasket.
def __init__(self, c1, c2, c3):
Creates a basic apollonian Gasket with four circles.
@param c1, c2, c3: The curvatures of the three inner circles of the
starting set (i.e. depth 0 of the recursion). The fourth,
enclosing circle will be calculated from them.
@type c1: int or float
@type c2: int or float
@type c3: int or float
self.start = tangentCirclesFromRadii( 1/c1, 1/c2, 1/c3 )
self.genCircles = list(self.start)
def recurse(self, circles, depth, maxDepth):
"""Recursively calculate the smaller circles of the AG up to the
given depth. Note that for depth n we get 2*3^{n+1} circles.
@param maxDepth: Maximal depth of the recursion.
@type maxDepth: int
@param circles: 4-Tuple of circles for which the second
solutions are calculated
@type circles: (L{Circle}, L{Circle}, L{Circle}, L{Circle})
@param depth: Current depth
@type depth: int
if( depth == maxDepth ):
(c1, c2, c3, c4) = circles
if( depth == 0 ):
# First recursive step, this is the only time we need to
# calculate 4 new circles.
del self.genCircles[4:]
cspecial = secondSolution( c1, c2, c3, c4 )
self.genCircles.append( cspecial )
self.recurse( (cspecial, c2, c3, c4), 1, maxDepth )
cn2 = secondSolution( c2, c1, c3, c4 )
self.genCircles.append( cn2 )
cn3 = secondSolution( c3, c1, c2, c4 )
self.genCircles.append( cn3 )
cn4 = secondSolution( c4, c1, c2, c3 )
self.genCircles.append( cn4 )
self.recurse( (cn2, c1, c3, c4), depth+1, maxDepth )
self.recurse( (cn3, c1, c2, c4), depth+1, maxDepth )
self.recurse( (cn4, c1, c2, c3), depth+1, maxDepth )
def generate(self, depth):
Wrapper for the recurse function. Generate the AG,
@param depth: Recursion depth of the Gasket
@type depth: int
self.recurse(self.start, 0, depth)