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Deskribapena

English: Pythagoras tree

Français : Arbre de Pythagore

Русский: Дерево Пифагора

Data

2010ko martxoaren 1ea

Jatorria

Norberak egina

Egilea

Guillaume Jacquenot Gjacquenot

SVG genesis

SVG irudi honen iturburu kodea baliozkoa.

Fitxategi hau (bektore-irudia) hau MATLAB-ekin sortu da .

Iturburu kode

MATLAB code

function M = Pythagor_tree(m,n,Colormap)
% function M = Pythagor_tree(m,n,Colormap)
% Compute Pythagoras_tree
% The Pythagoras Tree is a plane fractal constructed from squares.
% It is named after Pythagoras  because each triple of touching squares 
% encloses a right triangle, in a configuration traditionally used to
% depict the Pythagorean theorem.
% http://en.wikipedia.org/wiki/Pythagoras_tree
%
% Input : 
%       - m ( double m> 0) is the relative length of one of the side
%         right-angled triangle. The second side of the right-angle is 
%         taken to be one.
%         To have a symmetric tree, m has to be 1.
%       - n ( integer ) is the level of recursion.
%         The number of elements of tree is equal to 2^(n+1)-1.
%         A reasonnable number for n is 10.
%       - Colormap: String used to generate color of the different levels
%         of the tree.
%       All these arguments are optional: the function can run with
%       argument.
% Output : 
%       - Matrix M: Pyhagoras tree is stored in a matrix M.
%         This matrix has 5 columns.
%         Each row corresponds to the coordinate of each square of the tree
%         The two first columns give the bottom-left position of each
%         square. The third column corresponds to the orientation angle of
%         each square. The fourth column gives the size of each square. The
%         fifth column specifies the level of recursion of each square.
%         The first row corresponds to the root of the tree. It is always
%         the same
%         M(1,:) = [0 -1 0 1 1];
%         The leaf located at row i will give 2 leaves located at 2*i and
%         2*i+1.
%       - A svg file giving a vectorial display of the tree. The name of
%         file is generated from the parameter m,n,Colormap. The file is
%         stored in the current folder.
%
% 2010 02 29
% Guillaume Jacquenot
% guillaume dot jacquenot at gmail dot com

%% Check inputs
narg = nargin;
if narg <= 2
    % Colormap = 'jet';
    Colormap = 'summer';
    if narg <= 1
        n = 12; % Recursion level    
        if nargin == 0
            m = 0.8;
        end
    end
end
if m <= 0
	error([mfilename ':e0'],'Length of m has to be greater than zero');
end
if rem(n,1)~=0
	error([mfilename ':e0'],'The number of level has to be integer');
end
if ~iscolormap(Colormap)
	error([mfilename ':e1'],'Input colormap is not valid');
end
%% Compute constants
d      = sqrt(1+m^2);                  % 
c1     = 1/d;                          % Normalized length 1
c2     = m/d;                          % Normalized length 2
T      = [0 1/(1+m^2);1 1+m/(1+m^2)];  % Translation pattern  
alpha1 = atan2(m,1);                   % Defines the first rotation angle
alpha2 = alpha1-pi/2;                  % Defines the second rotation angle
pi2    = 2*pi;                         % Defines pi2
nEle   = 2^(n+1)-1;                    % Number of elements (square)
M      = zeros(nEle,5);                % Matrice containing the tree
M(1,:) = [0 -1 0 1 1];                 % Initialization of the tree

%% Compute the level of each square contained in the resulting matrix
Offset = 0;
for i = 0:n
    tmp = 2^i;
    M(Offset+(1:tmp),5) = i;
    Offset = Offset + tmp;
end
%% Compute the position and size of each square wrt its parent
for i = 2:2:(nEle-1)
    j          = i/2;
    mT         = M(j,4) * mat_rot(M(j,3)) * T;
    Tx         = mT(1,:) + M(j,1);
    Ty         = mT(2,:) + M(j,2);    
    theta1     = rem(M(j,3)+alpha1,pi2);
    theta2     = rem(M(j,3)+alpha2,pi2);
    M(i  ,1:4) = [Tx(1) Ty(1) theta1 M(j,4)*c1];
    M(i+1,1:4) = [Tx(2) Ty(2) theta2 M(j,4)*c2];
end
%% Display the tree
Pythagor_tree_plot(M,n);

%% Write results to an SVG file
Pythagor_tree_write2svg(m,n,Colormap,M);

function Pythagor_tree_write2svg(m,n,Colormap,M)
% Determine the bounding box of the tree with an offset
% Display_metadata = false;
Display_metadata = true;

nEle    = size(M,1);
r2      = sqrt(2);
LOffset = M(nEle,4) + 0.1;
min_x   = min(M(:,1)-r2*M(:,4)) - LOffset;
max_x   = max(M(:,1)+r2*M(:,4)) + LOffset;
min_y   = min(M(:,2)          ) - LOffset;  % -r2*M(:,4)
max_y   = max(M(:,2)+r2*M(:,4)) + LOffset;

% Compute the color of tree
ColorM = zeros(n+1,3);
eval(['ColorM = flipud(' Colormap '(n+1));']);
co   = 100;
Wfig = ceil(co*(max_x-min_x));
Hfig = ceil(co*(max_y-min_y));
filename = ['Pythagoras_tree_1_' strrep(num2str(m),'.','_') '_'...
             num2str(n) '_' Colormap '.svg'];
fid  = fopen(filename, 'wt');
fprintf(fid,'<?xml version="1.0" encoding="UTF-8" standalone="no"?>\n');
if ~Display_metadata
    fprintf(fid,'<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN"\n'); 
    fprintf(fid,'  "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">\n');
end
fprintf(fid,'<svg width="%d" height="%d" version="1.1"\n',Wfig,Hfig); % 
% fprintf(fid,['<svg width="12cm" height="4cm" version="1.1"\n']); % Wfig,

% fprintf(fid,['<svg width="15cm" height="10cm" '...
%              'viewBox="0 0 %d %d" version="1.1"\n'],...
%              Wfig,Hfig);
if Display_metadata
    fprintf(fid,'\txmlns:dc="http://purl.org/dc/elements/1.1/"\n');
    fprintf(fid,'\txmlns:cc="http://creativecommons.org/ns#"\n');
    fprintf(fid,['\txmlns:rdf="http://www.w3.org/1999/02/22'...
                 '-rdf-syntax-ns#"\n']);
end
fprintf(fid,'\txmlns:svg="http://www.w3.org/2000/svg"\n');
fprintf(fid,'\txmlns="http://www.w3.org/2000/svg"\n');
fprintf(fid,'\txmlns:xlink="http://www.w3.org/1999/xlink">\n');

if Display_metadata
    fprintf(fid,'\t<title>Pythagoras tree</title>\n');
    fprintf(fid,'\t<metadata>\n');
    fprintf(fid,'\t\t<rdf:RDF>\n');
    fprintf(fid,'\t\t\t<cc:Work\n');
    fprintf(fid,'\t\t\t\trdf:about="">\n');
    fprintf(fid,'\t\t\t\t<dc:format>image/svg+xml</dc:format>\n');
    fprintf(fid,'\t\t\t\t<dc:type\n');
    fprintf(fid,'\t\t\t\t\trdf:resource="http://purl.org/dc/dcmitype/StillImage" />\n');
    fprintf(fid,'\t\t\t\t<dc:title>Pythagoras tree</dc:title>\n');
    fprintf(fid,'\t\t\t\t<dc:creator>\n');
    fprintf(fid,'\t\t\t\t\t<cc:Agent>\n');
    fprintf(fid,'\t\t\t\t\t\t<dc:title>Guillaume Jacquenot</dc:title>\n');
    fprintf(fid,'\t\t\t\t\t</cc:Agent>\n');
    fprintf(fid,'\t\t\t\t</dc:creator>\n');
    fprintf(fid,'\t\t\t\t<cc:license\n');
    fprintf(fid,'\t\t\t\t\t\trdf:resource="http://creativecommons.org/licenses/by-nc-sa/3.0/" />\n');
    fprintf(fid,'\t\t\t</cc:Work>\n');
    fprintf(fid,'\t\t\t<cc:License\n');
    fprintf(fid,'\t\t\t\trdf:about="http://creativecommons.org/licenses/by-nc-sa/3.0/">\n');
    fprintf(fid,'\t\t\t\t<cc:permits\n');
    fprintf(fid,'\t\t\t\t\trdf:resource="http://creativecommons.org/ns#Reproduction" />\n');
    fprintf(fid,'\t\t\t\t<cc:permits\n');
    fprintf(fid,'\t\t\t\t\trdf:resource="http://creativecommons.org/ns#Reproduction" />\n');
    fprintf(fid,'\t\t\t\t<cc:permits\n');
    fprintf(fid,'\t\t\t\t\trdf:resource="http://creativecommons.org/ns#Distribution" />\n');
    fprintf(fid,'\t\t\t\t<cc:requires\n');
    fprintf(fid,'\t\t\t\t\trdf:resource="http://creativecommons.org/ns#Notice" />\n');
    fprintf(fid,'\t\t\t\t<cc:requires\n');
    fprintf(fid,'\t\t\t\t\trdf:resource="http://creativecommons.org/ns#Attribution" />\n');
    fprintf(fid,'\t\t\t\t<cc:prohibits\n');
    fprintf(fid,'\t\t\t\t\trdf:resource="http://creativecommons.org/ns#CommercialUse" />\n');
    fprintf(fid,'\t\t\t\t<cc:permits\n');
    fprintf(fid,'\t\t\t\t\trdf:resource="http://creativecommons.org/ns#DerivativeWorks" />\n');
    fprintf(fid,'\t\t\t\t<cc:requires\n');
    fprintf(fid,'\t\t\t\t\trdf:resource="http://creativecommons.org/ns#ShareAlike" />\n');
    fprintf(fid,'\t\t\t</cc:License>\n');
    fprintf(fid,'\t\t</rdf:RDF>\n');
    fprintf(fid,'\t</metadata>\n'); 
end
fprintf(fid,'\t<defs>\n');
fprintf(fid,'\t\t<rect width="%d" height="%d" \n',co,co);
fprintf(fid,'\t\t\tx="0" y="0"\n');
fprintf(fid,'\t\t\tstyle="fill-opacity:1;stroke:#00d900;stroke-opacity:1"\n');
fprintf(fid,'\t\t\tid="squa"\n');
fprintf(fid,'\t\t/>	\n');
fprintf(fid,'\t</defs>\n');
fprintf(fid,'\t<g transform="translate(%d %d) rotate(180) " >\n',...
                round(co*max_x),round(co*max_y));
for i = 0:n
    fprintf(fid,'\t\t<g style="fill:#%s;" >\n',...
                generate_color_hexadecimal(ColorM(i+1,:)));            
    Offset = 2^i-1;
    for j = 1:2^i
        k = j + Offset;
        fprintf(fid,['\t\t\t<use xlink:href="#squa" ',...
                     'transform="translate(%+010.5f %+010.5f)'...
                     ' rotate(%+010.5f) scale(%8.6f)" />\n'],...
                    co*M(k,1),co*M(k,2),M(k,3)*180/pi,M(k,4));   
    end
    fprintf(fid,'\t\t</g>\n');
end
fprintf(fid,'\t</g>\n');
fprintf(fid,'</svg>\n');
fclose(fid);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function M = mat_rot(x)
c = cos(x);
s = sin(x);
M=[c -s; s c];

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function H = Pythagor_tree_plot(D,ColorM)
if numel(ColorM) == 1
    ColorM = flipud(summer(ColorM+1));
end
H = figure('color','w');
hold on
axis equal
axis off
for i=1:size(D,1)
    cx    = D(i,1);
    cy    = D(i,2);
    theta = D(i,3);
    si    = D(i,4);    
    M     = mat_rot(theta);
    x     = si*[0 1 1 0 0];
    y     = si*[0 0 1 1 0];
    pts   = M*[x;y];
    fill(cx+pts(1,:),cy+pts(2,:),ColorM(D(i,5)+1,:));
    % plot(cx+pts(1,1:2),cy+pts(2,1:2),'r');
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function Scolor = generate_color_hexadecimal(color)
Scolor = '000000';
for i=1:3
    c = dec2hex(round(255*color(i)));
    if numel(c)==1
        Scolor(2*(i-1)+1) = c;
    else
        Scolor(2*(i-1)+(1:2)) = c;
    end
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function  res = iscolormap(cmap)
% This function returns true if 'cmap' is a valid colormap
LCmap = {...
    'autumn'
    'bone'
    'colorcube'
    'cool'
    'copper'
    'flag'
    'gray'
    'hot'
    'hsv'
    'jet'
    'lines'
    'pink'
    'prism'
    'spring'
    'summer'
    'white'
    'winter'
};

res = ~isempty(strmatch(cmap,LCmap,'exact'));

Lizentzia

Nik, lan honen egileak, argitaratzen dut ondorengo lizentzia hauen pean:

Baimena duzu dokumentu hau kopiatu, banatu edo/eta aldatzeko GNU Free Documentation License baldintzapean, Free Software Foundationek argitaratutako 1.2 edo ondorengo bertsioan; sekzio aldaezinik gabe, azaleko testurik gabe, eta atzeko azaleko testurik gabe. Lizentziaren kopia dago GNU Free Documentation License izenburudun atalean.

This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported, 2.5 Generic, 2.0 Generic and 1.0 Generic license.

Askea zara:

partekatzeko – lana kopiatzeko, banatzeko eta bidaltzeko
birnahasteko – lana moldatzeko

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eskuduntza – Egiletza behar bezala aitortu behar duzu, lizentzia ikusteko esteka gehitu, eta ea aldaketak egin diren aipatu. Era egokian egin behar duzu hori guztia, baina inola ere ez egileak zure lana edo zure erabilera babesten duela irudikatuz.
berdin partekatu – Lan honetan oinarrituta edo aldatuta berria eraikitzen baduzu, emaitza lana hau bezalako lizentzia batekin argitaratu behar duzu.

Nahiago duzun lizentzia erabil dezakezu.

Fitxategiaren historia

Data/orduan klik egin fitxategiak orduan zuen itxura ikusteko.

	Data/Ordua	Iruditxoa	Neurriak	Erabiltzailea	Iruzkina
oraingoa	01:16, 1 martxoa 2010		618 × 420 (2,05 MB)	Gjacquenot	{{Information \|Description={{en\|1=Pythagoras tree}} {{fr\|1=Arbre de Pythagore}} \|Source={{own}} \|Author=Gjacquenot \|Date=2010-03-01 \|Permission= \|other_versions= }} Category:Pythagoras trees

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