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