Packing circles in a circle


The following code attempts to pack a predefined number of smaller circles (of random radii between two given limits) into a larger one.

The algorithm simply selects a random set of radii, sorts them into decreasing order (so the larger circles get placed first), and keeps track of where each circle has been placed. In attempting to place the next circle, it selects a uniformly random point within the containing circle as its centre and checks that it will not overlap any other. If it will, another point is selected (up to some sensible limit of attempts, after which we give up trying to place the circle and move onto the next, smaller one).

enter image description here

The image dimensions, colour palette, outer circle size, inner circle size range and number of inner circles can be customized in the constructor to the Circles class.

import numpy as np

class Circle:
    """A little class representing an SVG circle."""

    def __init__(self, cx, cy, r, icolour=None):
        """Initialize the circle with its centre, (cx,cy) and radius, r.

        icolour is the index of the circle's colour.

        """,, self.r = cx, cy, r
        self.icolour = icolour

    def overlap_with(self, cx, cy, r):
        """Does the circle overlap with another of radius r at (cx, cy)?"""

        d = np.hypot(,
        return d < r + self.r

    def draw_circle(self, fo):
        """Write the circle's SVG to the output stream, fo."""

        print('<circle cx="{}" cy="{}" r="{}" class="c{}"/>'
            .format(,, self.r, self.icolour), file=fo)

class Circles:
    """A class for drawing circles-inside-a-circle."""

    def __init__(self, width=600, height=600, R=250, n=800, rho_min=0.005,
                 rho_max=0.05, colours=None):
        """Initialize the Circles object.

        width, height are the SVG canvas dimensions
        R is the radius of the large circle within which the small circles are
        to fit.
        n is the maximum number of circles to pack inside the large circle.
        rho_min is rmin/R, giving the minimum packing circle radius.
        rho_max is rmax/R, giving the maximum packing circle radius.
        colours is a list of SVG fill colour specifiers to be referenced by
            the class identifiers c<i>. If None, a default palette is set.


        self.width, self.height = width, height
        self.R, self.n = R, n
        # The centre of the canvas
        self.CX, self.CY = self.width // 2, self.height // 2
        self.rmin, self.rmax = R * rho_min, R * rho_max
        self.colours = colours or ['#993300', '#a5c916', '#00AA66', '#FF9900']

    def preamble(self):
        """The usual SVG preamble, including the image size."""

        print('<?xml version="1.0" encoding="utf-8"?>\n'

        '<svg xmlns=""\n' + ' '*5 +
          'xmlns:xlink="" width="{}" height="{}" >'
                .format(self.width, self.height),

    def defs_decorator(func):
        """For convenience, wrap the CSS styles with the needed SVG tags."""

        def wrapper(self):
            <style type="text/css"><![CDATA[""",


        return wrapper

    def svg_styles(self):
        """Set the SVG styles: circles are coloured with no border."""

        print('circle {stroke: none;}',
        for i, c in enumerate(self.colours):
            print('.c{} {{fill: {};}}'.format(i, c),

    def make_svg(self, filename, *args, **kwargs):
        """Create the image as an SVG file with name filename."""

        ncolours = len(self.colours)
        with open(filename, 'w') as
            for circle in self.circles:

    def _place_circle(self, r):
        # The guard number: if we don't place a circle within this number
        # of trials, we give up.
        guard = 500
        while guard:
            # Pick a random position, uniformly on the larger circle's interior
            cr, cphi = ( self.R * np.sqrt(np.random.random()),
                         2*np.pi * np.random.random() )
            cx, cy = cr * np.cos(cphi), cr * np.sin(cphi)
            if cr+r < self.R:
            # The circle fits inside the larger circle.
                if not any(circle.overlap_with(self.CX+cx, self.CY+cy, r)
                                    for circle in self.circles):
                    # The circle doesn't overlap any other circle: place it.
                    circle = Circle(cx+self.CX, cy+self.CY, r,
            guard -= 1
        # Warn that we reached the guard number of attempts and gave up for
        # for this circle.
        print('guard reached.')

    def make_circles(self):
        """Place the little circles inside the big one."""

        # First choose a set of n random radii and sort them. We use
        # random.random() * random.random() to favour small circles.
        self.circles = []
        r = self.rmin + (self.rmax - self.rmin) * np.random.random(
                                self.n) * np.random.random(self.n)
        # Do our best to place the circles, larger ones first.
        for i in range(self.n):

circles = Circles(n=2000)

This project was inspired by Ken Goulding's block in JavaScript.

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