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The line integral 
, where C is the boundary of the square 
oriented counterclockwise, can be evaluated in two ways:
, where C is the boundary of the square oriented counterclockwise, can be evaluated in two ways: Using the definition of the line integral:
% Initialize the integral sum
integral_sum = 0;
% Segment C1: x = -1, y goes from -1 to 1
y = linspace(-1, 1);
x = -1 * ones(size(y));
dy = diff(y);
integral_sum = integral_sum + sum(-x(1:end-1) .* dy);
% Segment C2: y = 1, x goes from -1 to 1
x = linspace(-1, 1);
y = ones(size(x));
dx = diff(x);
integral_sum = integral_sum + sum(y(1:end-1).^2 .* dx);
% Segment C3: x = 1, y goes from 1 to -1
y = linspace(1, -1);
x = ones(size(y));
dy = diff(y);
integral_sum = integral_sum + sum(-x(1:end-1) .* dy);
% Segment C4: y = -1, x goes from 1 to -1
x = linspace(1, -1);
y = -1 * ones(size(x));
dx = diff(x);
integral_sum = integral_sum + sum(y(1:end-1).^2 .* dx);
disp(['Direct Method Integral: ', num2str(integral_sum)]);
Plotting the square path
% Define the square's vertices
vertices = [-1 -1; -1 1; 1 1; 1 -1; -1 -1];
% Plot the square
figure;
plot(vertices(:,1), vertices(:,2), '-o');
title('Square Path for Line Integral');
xlabel('x');
ylabel('y');
grid on;
axis equal;
% Add arrows to indicate the path direction (counterclockwise)
hold on;
for i = 1:size(vertices,1)-1
% Calculate direction
dx = vertices(i+1,1) - vertices(i,1);
dy = vertices(i+1,2) - vertices(i,2);
% Reduce the length of the arrow for better visibility
scale = 0.2;
dx = scale * dx;
dy = scale * dy;
% Calculate the start point of the arrow
startx = vertices(i,1) + (1 - scale) * dx;
starty = vertices(i,2) + (1 - scale) * dy;
% Plot the arrow
quiver(startx, starty, dx, dy, 'MaxHeadSize', 0.5, 'Color', 'r', 'AutoScale', 'off');
end
hold off;
Apply Green's Theorem for the line integral
% Define the partial derivatives of P and Q
f = @(x, y) -1 - 2*y; % derivative of -x with respect to x is -1, and derivative of y^2 with respect to y is 2y
% Compute the double integral over the square [-1,1]x[-1,1]
integral_value = integral2(f, -1, 1, 1, -1);
disp(['Green''s Theorem Integral: ', num2str(integral_value)]);
Plotting the vector field related to Green’s theorem
% Define the grid for the vector field
[x, y] = meshgrid(linspace(-2, 2, 20), linspace(-2 ,2, 20));
% Define the vector field components
P = y.^2; % y^2 component
Q = -x; % -x component
% Plot the vector field
figure;
quiver(x, y, P, Q, 'b');
hold on; % Hold on to plot the square on the same figure
% Define the square's vertices
vertices = [-1 -1; -1 1; 1 1; 1 -1; -1 -1];
% Plot the square path
plot(vertices(:,1), vertices(:,2), '-o', 'Color', 'k'); % 'k' for black color
title('Vector Field (P = y^2, Q = -x) with Square Path');
xlabel('x');
ylabel('y');
axis equal;
% Add arrows to indicate the path direction (counterclockwise)
for i = 1:size(vertices,1)-1
% Calculate direction
dx = vertices(i+1,1) - vertices(i,1);
dy = vertices(i+1,2) - vertices(i,2);
% Reduce the length of the arrow for better visibility
scale = 0.2;
dx = scale * dx;
dy = scale * dy;
% Calculate the start point of the arrow
startx = vertices(i,1) + (1 - scale) * dx;
starty = vertices(i,2) + (1 - scale) * dy;
% Plot the arrow
quiver(startx, starty, dx, dy, 'MaxHeadSize', 0.5, 'Color', 'r', 'AutoScale', 'off');
end
hold off;
To solve a surface integral for example the
over the sphere 
easily in MATLAB, you can leverage the symbolic toolbox for a direct and clear solution. Here is a tip to simplify the process:
over the sphere easily in MATLAB, you can leverage the symbolic toolbox for a direct and clear solution. Here is a tip to simplify the process:- Use Symbolic Variables and Functions: Define your variables symbolically, including the parameters of your spherical coordinates θ and ϕ and the radius r . This allows MATLAB to handle the expressions symbolically, making it easier to manipulate and integrate them.
- Express in Spherical Coordinates Directly: Since you already know the sphere's equation and the relationship in spherical coordinates, define x, y, and z in terms of r , θ and ϕ directly.
- Perform Symbolic Integration: Use MATLAB's `int` function to integrate symbolically. Since the sphere and the function
are symmetric, you can exploit these symmetries to simplify the calculation.
Here’s how you can apply this tip in MATLAB code:
% Include the symbolic math toolbox
syms theta phi
% Define the limits for theta and phi
theta_limits = [0, pi];
phi_limits = [0, 2*pi];
% Define the integrand function symbolically
integrand = 16 * sin(theta)^3 * cos(phi)^2;
% Perform the symbolic integral for the surface integral
surface_integral = int(int(integrand, theta, theta_limits(1), theta_limits(2)), phi, phi_limits(1), phi_limits(2));
% Display the result of the surface integral symbolically
disp(['The surface integral of x^2 over the sphere is ', char(surface_integral)]);
% Number of points for plotting
num_points = 100;
% Define theta and phi for the sphere's surface
[theta_mesh, phi_mesh] = meshgrid(linspace(double(theta_limits(1)), double(theta_limits(2)), num_points), ...
linspace(double(phi_limits(1)), double(phi_limits(2)), num_points));
% Spherical to Cartesian conversion for plotting
r = 2; % radius of the sphere
x = r * sin(theta_mesh) .* cos(phi_mesh);
y = r * sin(theta_mesh) .* sin(phi_mesh);
z = r * cos(theta_mesh);
% Plot the sphere
figure;
surf(x, y, z, 'FaceColor', 'interp', 'EdgeColor', 'none');
colormap('jet'); % Color scheme
shading interp; % Smooth shading
camlight headlight; % Add headlight-type lighting
lighting gouraud; % Use Gouraud shading for smooth color transitions
title('Sphere: x^2 + y^2 + z^2 = 4');
xlabel('x-axis');
ylabel('y-axis');
zlabel('z-axis');
colorbar; % Add color bar to indicate height values
axis square; % Maintain aspect ratio to be square
view([-30, 20]); % Set a nice viewing angle
I am often talking to new MATLAB users. I have put together one script. If you know how this script works, why, and what each line means, you will be well on your way on your MATLAB learning journey.
% Clear existing variables and close figures
clear;
close all;
% Print to the Command Window
disp('Hello, welcome to MATLAB!');
% Create a simple vector and matrix
vector = [1, 2, 3, 4, 5];
matrix = [1, 2, 3; 4, 5, 6; 7, 8, 9];
% Display the created vector and matrix
disp('Created vector:');
disp(vector);
disp('Created matrix:');
disp(matrix);
% Perform element-wise multiplication
result = vector .* 2;
% Display the result of the operation
disp('Result of element-wise multiplication of the vector by 2:');
disp(result);
% Create plot
x = 0:0.1:2*pi; % Generate values from 0 to 2*pi
y = sin(x); % Calculate the sine of these values
% Plotting
figure; % Create a new figure window
plot(x, y); % Plot x vs. y
title('Simple Plot of sin(x)'); % Give the plot a title
xlabel('x'); % Label the x-axis
ylabel('sin(x)'); % Label the y-axis
grid on; % Turn on the grid
disp('This is the end of the script. Explore MATLAB further to learn more!');
Happy Pi Day!
3.14 π Day has arrived, and this post provides some very cool pi implementations and complete MATLAB code.
Firstly, in order to obtain the first n decimal places of pi, we need to write the following code (to prevent inaccuracies, we need to take a few more tails and perform another operation of taking the first n decimal places when needed):
function Pi=getPi(n)
if nargin<1,n=3;end
Pi=char(vpa(sym(pi),n+10));
Pi=abs(Pi)-48;
Pi=Pi(3:n+2);
end
With this function to obtain the decimal places of pi, our visualization journey has begun~Step by step, from simple to complex~(Please try to use newer versions of MATLAB to run, at least R17b)
1 Pie chart
Just calculate the proportion of each digit to the first 1500 decimal places:
% 获取pi前1500位小数
Pi=getPi(1500);
% 统计各个数字出现次数
numNum=find([diff(sort(Pi)),1]);
numNum=[numNum(1),diff(numNum)];
% 配色列表
CM=[20,164,199;43,187,170;53,165,81;189,190,28;248,167,22;
232,74,27;244,57,99;240,118,177;168,109,195;78,125,187]./255;
% 绘图并修饰
pieHdl=pie(numNum);
set(gcf,'Color',[1,1,1],'Position',[200,100,620,620]);
for i=1:2:20
pieHdl(i).EdgeColor=[1,1,1];
pieHdl(i).LineWidth=1;
pieHdl(i).FaceColor=CM((i+1)/2,:);
end
for i=2:2:20
pieHdl(i).Color=CM(i/2,:);
pieHdl(i).FontWeight='bold';
pieHdl(i).FontSize=14;
end
% 绘制图例并修饰
lgdHdl=legend(num2cell('0123456789'));
lgdHdl.FontWeight='bold';
lgdHdl.FontSize=11;
lgdHdl.TextColor=[.5,.5,.5];
lgdHdl.Location='southoutside';
lgdHdl.Box='off';
lgdHdl.NumColumns=10;
lgdHdl.ItemTokenSize=[20,15];
title("VISUALIZING \pi 'Pi' Chart | 1500 digits",'FontSize',18,...
'FontName','Times New Roman','Color',[.5,.5,.5])
2 line chart
Calculate the change in the proportion of each number:
% 获取pi前1500位小数
Pi=getPi(1500);
% 计算比例变化
Ratio=cumsum(Pi==(0:9)',2);
Ratio=Ratio./sum(Ratio);
D=1:length(Ratio);
% 配色列表
CM=[20,164,199;43,187,170;53,165,81;189,190,28;248,167,22;
232,74,27;244,57,99;240,118,177;168,109,195;78,125,187]./255;
hold on
% 循环绘图
for i=1:10
plot(D(20:end),Ratio(i,20:end),'Color',[CM(i,:),.6],'LineWidth',1.8)
end
% 坐标区域修饰
ax=gca;box on;grid on
ax.YLim=[0,.2];
ax.YTick=0:.05:.2;
ax.XTick=0:200:1400;
ax.YTickLabel={'0%','5%','10%','15%','20%'};
ax.XMinorTick='on';
ax.YMinorTick='on';
ax.LineWidth=.8;
ax.GridLineStyle='-.';
ax.FontName='Cambria';
ax.FontSize=11;
ax.XLabel.String='Decimals';
ax.YLabel.String='Proportion';
ax.XLabel.FontSize=13;
ax.YLabel.FontSize=13;
% 绘制图例并修饰
lgdHdl=legend(num2cell('0123456789'));
lgdHdl.NumColumns=5;
lgdHdl.FontWeight='bold';
lgdHdl.FontSize=11;
lgdHdl.TextColor=[.5,.5,.5];
3 stacked area diagram
% 获取pi前500位小数
Pi=getPi(500);
% 计算比例变化
Ratio=cumsum(Pi==(0:9)',2);
Ratio=Ratio./sum(Ratio);
% 配色列表
CM=[231,98,84;239,138,71;247,170,88;255,208,111;255,230,183;
170,220,224;114,188,213;82,143,173;55,103,149;30,70,110]./255;
% 绘制堆叠面积图
hold on
areaHdl=area(Ratio');
for i=1:10
areaHdl(i).FaceColor=CM(i,:);
areaHdl(i).FaceAlpha=.9;
end
% 图窗和坐标区域修饰
set(gcf,'Position',[200,100,720,420]);
ax=gca;
ax.YLim=[0,1];
ax.XMinorTick='on';
ax.YMinorTick='on';
ax.LineWidth=.8;
ax.FontName='Cambria';
ax.FontSize=11;
ax.TickDir='out';
ax.XLabel.String='Decimals';
ax.YLabel.String='Proportion';
ax.XLabel.FontSize=13;
ax.YLabel.FontSize=13;
ax.Title.String='Area Chart of Proportion — 500 digits';
ax.Title.FontSize=14;
% 绘制图例并修饰
lgdHdl=legend(num2cell('0123456789'));
lgdHdl.NumColumns=5;
lgdHdl.FontSize=11;
lgdHdl.Location='southeast';
4 connected stacked bar chart
% 获取pi前100位小数
Pi=getPi(100);
% 计算比例变化
Ratio=cumsum(Pi==(0:9)',2);
Ratio=Ratio./sum(Ratio);
X=Ratio(:,10:10:80)';
barHdl=bar(X,'stacked','BarWidth',.2);
CM=[231,98,84;239,138,71;247,170,88;255,208,111;255,230,183;
170,220,224;114,188,213;82,143,173;55,103,149;30,70,110]./255;
for i=1:10
barHdl(i).FaceColor=CM(i,:);
end
% 以下是生成连接的部分
hold on;axis tight
yEndPoints=reshape([barHdl.YEndPoints]',length(barHdl(1).YData),[])';
zeros(1,length(barHdl(1).YData));
yEndPoints=[zeros(1,length(barHdl(1).YData));yEndPoints];
barWidth=barHdl(1).BarWidth;
for i=1:length(barHdl)
for j=1:length(barHdl(1).YData)-1
y1=min(yEndPoints(i,j),yEndPoints(i+1,j));
y2=max(yEndPoints(i,j),yEndPoints(i+1,j));
if y1*y2<0
ty=yEndPoints(find(yEndPoints(i+1,j)*yEndPoints(1:i,j)>=0,1,'last'),j);
y1=min(ty,yEndPoints(i+1,j));
y2=max(ty,yEndPoints(i+1,j));
end
y3=min(yEndPoints(i,j+1),yEndPoints(i+1,j+1));
y4=max(yEndPoints(i,j+1),yEndPoints(i+1,j+1));
if y3*y4<0
ty=yEndPoints(find(yEndPoints(i+1,j+1)*yEndPoints(1:i,j+1)>=0,1,'last'),j+1);
y3=min(ty,yEndPoints(i+1,j+1));
y4=max(ty,yEndPoints(i+1,j+1));
end
fill([j+.5.*barWidth,j+1-.5.*barWidth,j+1-.5.*barWidth,j+.5.*barWidth],...
[y1,y3,y4,y2],barHdl(i).FaceColor,'FaceAlpha',.4,'EdgeColor','none');
end
end
% 图窗和坐标区域修饰
set(gcf,'Position',[200,100,720,420]);
ax=gca;box off
ax.YLim=[0,1];
ax.XMinorTick='on';
ax.YMinorTick='on';
ax.LineWidth=.8;
ax.FontName='Cambria';
ax.FontSize=11;
ax.TickDir='out';
ax.XTickLabel={'10','20','30','40','50','60','70','80'};
ax.XLabel.String='Decimals';
ax.YLabel.String='Proportion';
ax.XLabel.FontSize=13;
ax.YLabel.FontSize=13;
ax.Title.String='Area Chart of Proportion — 10-80 digits';
ax.Title.FontSize=14;
% 绘制图例并修饰
lgdHdl=legend(barHdl,num2cell('0123456789'));
lgdHdl.NumColumns=5;
lgdHdl.FontSize=11;
lgdHdl.Location='southeast';
5 bichord chart
Need to use this tool:
% 构建连接矩阵
dataMat=zeros(10,10);
Pi=getPi(1001);
for i=1:1000
dataMat(Pi(i)+1,Pi(i+1)+1)=dataMat(Pi(i)+1,Pi(i+1)+1)+1;
end
BCC=biChordChart(dataMat,'Arrow','on','Label',num2cell('0123456789'));
BCC=BCC.draw();
% 添加刻度
BCC.tickState('on')
% 修改字体，字号及颜色
BCC.setFont('FontName','Cambria','FontSize',17)
set(gcf,'Position',[200,100,820,820]);
6 Gravity simulation diagram
Imagine each decimal as a small ball with a mass of
For example, if
, the weight of ball 0 is 1, ball 9 is 1.2589, the initial velocity of the ball is 0, and it is attracted by other balls. Gravity follows the inverse square law, and if the balls are close enough, they will collide and their value will become 
, the weight of ball 0 is 1, ball 9 is 1.2589, the initial velocity of the ball is 0, and it is attracted by other balls. Gravity follows the inverse square law, and if the balls are close enough, they will collide and their value will become After adding, take the mod, add the velocity direction proportionally, and recalculate the weight.
Pi=[3,getPi(71)];K=.18;
% 基础配置
CM=[239,32,120;239,60,52;247,98,32;255,182,60;247,235,44;
142,199,57;55,180,70;0,170,239;40,56,146;147,37,139]./255;
T=linspace(0,2*pi,length(Pi)+1)';
T=T(1:end-1);
ct=linspace(0,2*pi,100);
cx=cos(ct).*.027;
cy=sin(ct).*.027;
% 初始数据
Pi=Pi(:);
N=Pi;
X=cos(T);Y=sin(T);
VX=T.*0;VY=T.*0;
PX=X;PY=Y;
% 未碰撞时初始质量
getM=@(x)(x+1).^K;
M=getM(N);
% 绘制初始圆圈
hold on
for i=1:length(N)
fill(cx+X(i),cy+Y(i),CM(N(i)+1,:),'EdgeColor','w','LineWidth',1)
end
for k=1:800
% 计算加速度
Rn2=1./squareform(pdist([X,Y])).^2;
Rn2(eye(length(X))==1)=0;
MRn2=Rn2.*(M');
AX=X'-X;AY=Y'-Y;
normXY=sqrt(AX.^2+AY.^2);
AX=AX./normXY;AX(eye(length(X))==1)=0;
AY=AY./normXY;AY(eye(length(X))==1)=0;
AX=sum(AX.*MRn2,2)./150000;
AY=sum(AY.*MRn2,2)./150000;
% 计算速度及新位置
VX=VX+AX;X=X+VX;PX=[PX,X];
VY=VY+AY;Y=Y+VY;PY=[PY,Y];
% 检测是否有碰撞
R=squareform(pdist([X,Y]));
R(triu(ones(length(X)))==1)=inf;
[row,col]=find(R<=0.04);
if length(X)==1
break;
end
if ~isempty(row)
% 碰撞的点合为一体
XC=(X(row)+X(col))./2;YC=(Y(row)+Y(col))./2;
VXC=(VX(row).*M(row)+VX(col).*M(col))./(M(row)+M(col));
VYC=(VY(row).*M(row)+VY(col).*M(col))./(M(row)+M(col));
PC=nan(length(row),size(PX,2));
NC=mod(N(row)+N(col),10);
% 删除碰撞点并绘图
uniNum=unique([row;col]);
X(uniNum)=[];VX(uniNum)=[];
Y(uniNum)=[];VY(uniNum)=[];
for i=1:length(uniNum)
plot(PX(uniNum(i),:),PY(uniNum(i),:),'LineWidth',2,'Color',CM(N(uniNum(i))+1,:))
end
PX(uniNum,:)=[];PY(uniNum,:)=[];N(uniNum,:)=[];
% 绘制圆形
for i=1:length(XC)
fill(cx+XC(i),cy+YC(i),CM(NC(i)+1,:),'EdgeColor','w','LineWidth',1)
end
% 补充合体点
X=[X;XC];Y=[Y;YC];VX=[VX;VXC];VY=[VY;VYC];
PX=[PX;PC];PY=[PY;PC];N=[N;NC];M=getM(N);
end
end
for i=1:size(PX,1)
plot(PX(i,:),PY(i,:),'LineWidth',2,'Color',CM(N(i)+1,:))
end
text(-1,1,{['Num=',num2str(length(Pi))];['K=',num2str(K)]},'FontSize',13,'FontName','Cambria')
% 图窗及坐标区域修饰
set(gcf,'Position',[200,100,820,820]);
ax=gca;
ax.Position=[0,0,1,1];
ax.DataAspectRatio=[1,1,1];
ax.XLim=[-1.1,1.1];
ax.YLim=[-1.1,1.1];
ax.XTick=[];
ax.YTick=[];
ax.XColor='none';
ax.YColor='none';
7 forest chart
The method comes from
The digits of π are shown as a forest. Each tree in the forest represents the digits of π up to the next 9. The first 10 trees are "grown" from the digit sets 314159, 2653589, 79, 3238462643383279, 50288419, 7169, 39, 9, 3751058209, and 749.
BRANCHES
The first digit of a tree controls how many branches grow from the trunk of the tree. For example, the first tree's first digit is 3, so you see 3 branches growing from the trunk.
The next digit's branches grow from the end of a branch of the previous digit in left-to-right order. This process continues until all the tree's digits have been used up.
Each tree grows from a set of consecutive digits sampled from the digits of π up to the next 9. The first tree, shown here, grows from 314159. Each of the digits determine how many branches grow at each fork in the tree — the branches here are colored by their corresponding digit to illustrate this. Leaves encode the digits in a left-to-right order. The digit 9 spawns a flower on one of the branches of the previous digit. The branching exception is 0, which terminates the current branch — 0 branches grow!
LEAVES AND FLOWERS
The tree's digits themselves are drawn as circular leaves, color-coded by the digit.
The leaf exception is 9, which causes one of the branches of the previous digit to sprout a flower! The petals of the flower are colored by the digit before the 9 and the center is colored by the digit after the 9, which is on the next tree. This is how the forest propagates.
The colors of a flower are determined by the first digit of the next tree and the penultimate digit of the current tree. If the current tree only has one digit, then that digit is used. Leaves are placed at the tips of branches in a left-to-right order — you can "easily" read them off. Additionally, the leaves are distributed within the tree (without disturbing their left-to-right order) to spread them out as much as possible and avoid overlap. This order is deterministic.
The leaf placement exception are the branch set that sprouted the flower. These are not used to grow leaves — the flower needs space!
function PiTree(X,pos,D)
lw=2;
theta=pi/2+(rand(1)-.5).*pi./12;
% 树叶及花朵颜色
CM=[237,32,121;237,62,54;247,99,33;255,183,59;245,236,43;
141,196,63;57,178,74;0,171,238;40,56,145;146,39,139]./255;
hold on
if all(X(1:end-2)==0)
endSet=[pos,pos,theta];
else
kplot(pos(1)+[0,cos(theta)],pos(2)+[0,sin(theta)],lw./.6)
endSet=[pos,pos+[cos(theta),sin(theta)],theta];
% 计算层级
Layer=0;
for i=1:length(X)
Layer=[Layer,ones(1,X(i)).*i];
end
% 计算树枝
if D
for i=1:length(X)-2
if X(i)==0 % 若数值为0则不长树枝
newSet=endSet(1,:);
elseif X(i)==1 % 若数值为1则一长一短两个树枝
tTheta=endSet(1,5);
tTheta=linspace(tTheta+pi/8,tTheta-pi/8,2)'+(rand([2,1])-.5).*pi./8;
newSet=repmat(endSet(1,3:4),[X(i),1]);
newSet=[newSet.*[1;1],newSet+[cos(tTheta),sin(tTheta)].*.7^Layer(i).*[1;.1],tTheta];
else % 其他情况数值为几长几个树枝
tTheta=endSet(1,5);
tTheta=linspace(tTheta+pi/5,tTheta-pi/5,X(i))'+(rand([X(i),1])-.5).*pi./8;
newSet=repmat(endSet(1,3:4),[X(i),1]);
newSet=[newSet,newSet+[cos(tTheta),sin(tTheta)].*.7^Layer(i),tTheta];
end
% 绘制树枝
for j=1:size(newSet,1)
kplot(newSet(j,[1,3]),newSet(j,[2,4]),lw.*.6^Layer(i))
end
endSet=[endSet;newSet];
endSet(1,:)=[];
end
end
end
% 计算叶子和花朵位置
FLSet=endSet(:,3:4);
[~,FLInd]=sort(FLSet(:,1));
FLSet=FLSet(FLInd,:);
[~,tempInd]=sort(rand([1,size(FLSet,1)]));
tempInd=sort(tempInd(1:length(X)-2));
flowerInd=tempInd(randi([1,length(X)-2],[1,1]));
leafInd=tempInd(tempInd~=flowerInd);
% 绘制树叶
for i=1:length(leafInd)
scatter(FLSet(leafInd(i),1),FLSet(leafInd(i),2),70,'filled','CData',CM(X(i)+1,:))
end
% 绘制花朵
for i=1:5
% if ~D
% tC=CM(X(end)+1,:);
% else
% tC=CM(X(end-2)+1,:);
% end
scatter(FLSet(flowerInd,1)+cos(pi*2*i/5).*.18,FLSet(flowerInd,2)+sin(pi*2*i/5).*.18,60,...
'filled','CData',CM(X(end-2)+1,:),'MarkerEdgeColor',[1,1,1])
end
scatter(FLSet(flowerInd,1),FLSet(flowerInd,2),60,'filled','CData',CM(X(end)+1,:),'MarkerEdgeColor',[1,1,1])
drawnow;%axis tight
% =========================================================================
function kplot(XX,YY,LW,varargin)
LW=linspace(LW,LW*.6,10);%+rand(1,20).*LW./10;
XX=linspace(XX(1),XX(2),11)';
XX=[XX(1:end-1),XX(2:end)];
YY=linspace(YY(1),YY(2),11)';
YY=[YY(1:end-1),YY(2:end)];
for ii=1:10
plot(XX(ii,:),YY(ii,:),'LineWidth',LW(ii),'Color',[.1,.1,.1])
end
end
end
main part:
Pi=[3,getPi(800)];
pos9=[0,find(Pi==9)];
set(gcf,'Position',[200,50,900,900],'Color',[1,1,1]);
ax=gca;hold on
ax.Position=[0,0,1,1];
ax.DataAspectRatio=[1,1,1];
ax.XLim=[.5,36];
ax.XTick=[];
ax.YTick=[];
ax.XColor='none';
ax.YColor='none';
for j=1:8
for i=1:11
n=i+(j-1)*11;
if n<=85
tPi=Pi((pos9(n)+1):pos9(n+1)+1);
if length(tPi)>2
PiTree(tPi,[0+i*3,0-j*4],true);
else
PiTree([Pi(pos9(n)),tPi],[0+i*3,0-j*4],false);
end
end
end
end
8 random walk
n=1200;
% 获取pi前n位小数
Pi=getPi(n);
CM=[239,65,75;230,115,48;229,158,57;232,136,85;239,199,97;
144,180,116;78,166,136;81,140,136;90,118,142;43,121,159]./255;
hold on
endPoint=[0,0];
t=linspace(0,2*pi,100);
T=linspace(0,2*pi,11)+pi/2;
fill(endPoint(1)+cos(t).*.5,endPoint(2)+sin(t).*.5,CM(Pi(1)+1,:),'EdgeColor','none')
for i=1:n
theta=T(Pi(i)+1);
plot(endPoint(1)+[0,cos(theta)],endPoint(2)+[0,sin(theta)],'Color',[CM(Pi(i)+1,:),.8],'LineWidth',1.2);
endPoint=endPoint+[cos(theta),sin(theta)];
end
fill(endPoint(1)+cos(t).*.5,endPoint(2)+sin(t).*.5,CM(Pi(n)+1,:),'EdgeColor','none')
% 图窗和坐标区域修饰
set(gcf,'Position',[200,100,820,820]);
ax=gca;
ax.XTick=[];
ax.YTick=[];
ax.Color=[0,0,0];
ax.DataAspectRatio=[1,1,1];
ax.XLim=[-30,5];
ax.YLim=[-5,40];
% 绘制图例
endPoint=[1,35];
for i=1:10
theta=T(i);
plot(endPoint(1)+[0,cos(theta).*2],endPoint(2)+[0,sin(theta).*2],'Color',[CM(i,:),.8],'LineWidth',3);
text(endPoint(1)+cos(theta).*2.7,endPoint(2)+sin(theta).*2.7,num2str(i-1),'Color',[1,1,1].*.7,...
'FontSize',12,'FontWeight','bold','FontName','Cambria','HorizontalAlignment','center')
end
text(-15,35,'Random walk of \pi digits','Color',[1,1,1],'FontName','Cambria',...
'HorizontalAlignment','center','FontSize',25,'FontAngle','italic')
9 grid chart
Pi=[3,getPi(399)];
% 配色数据
CM=[248,65,69;246,152,36;249,198,81;67,170,139;87,118,146]./255;
% 绘制圆圈
hold on
t=linspace(0,2*pi,100);
x=cos(t).*.8.*.5;
y=sin(t).*.8.*.5;
for i=1:400
[col,row]=ind2sub([20,20],i);
if mod(Pi(i),2)==0
fill(x+col,y+row,CM(round((Pi(i)+1)/2),:),'LineWidth',1,'EdgeAlpha',.8)
else
fill(x+col,y+row,[0,0,0],'EdgeColor',CM(round((Pi(i)+1)/2),:),'LineWidth',1,'EdgeAlpha',.7)
end
end
text(10.5,-.4,'\pi on a grid — 400 digits','Color',[1,1,1],'FontName','Cambria',...
'HorizontalAlignment','center','FontSize',25,'FontAngle','italic')
% 图窗和坐标区域修饰
set(gcf,'Position',[200,100,820,820]);
ax=gca;
ax.YDir='reverse';
ax.XLim=[.5,20.5];
ax.YLim=[-1,20.5];
ax.XTick=[];
ax.YTick=[];
ax.Color=[0,0,0];
ax.DataAspectRatio=[1,1,1];
10 scale grid diagram
Let's still put the numbers in the form of circles, but the difference is that six numbers are grouped together, and the pure purple circle at the end is the six 9s that we are familiar with decimal places 762-767
Pi=[3,getPi(767)];
% 762-767
% 配色数据
CM=[239,32,120;239,60,52;247,98,32;255,182,60;247,235,44;
142,199,57;55,180,70;0,170,239;40,56,146;147,37,139]./255;
% 绘制圆圈
hold on
t=linspace(0,2*pi,100);
x=cos(t).*.9.*.5;
y=sin(t).*.9.*.5;
for i=1:6:length(Pi)
n=round((i-1)/6+1);
[col,row]=ind2sub([10,13],n);
tNum=Pi(i:i+5);
numNum=find([diff(sort(tNum)),1]);
numNum=[numNum(1),diff(numNum)];
cumNum=cumsum(numNum);
uniNum=unique(tNum);
for j=length(cumNum):-1:1
fill(x./6.*cumNum(j)+col,y./6.*cumNum(j)+row,CM(uniNum(j)+1,:),'EdgeColor','none')
end
end
% 绘制图例
for i=1:10
fill(x./4+5.5+(i-5.5)*.32,y./4+14.5,CM(i,:),'EdgeColor','none')
text(5.5+(i-5.5)*.32,14.9,num2str(i-1),'Color',[1,1,1],'FontSize',...
9,'FontName','Cambria','HorizontalAlignment','center')
end
text(5.5,-.2,'FEYNMAN POINT of \pi','Color',[1,1,1],'FontName','Cambria',...
'HorizontalAlignment','center','FontSize',25,'FontAngle','italic')
% 图窗和坐标区域修饰
set(gcf,'Position',[200,100,600,820]);
ax=gca;
ax.YDir='reverse';
ax.Position=[0,0,1,1];
ax.XLim=[.3,10.7];
ax.YLim=[-1,15.5];
ax.XTick=[];
ax.YTick=[];
ax.Color=[0,0,0];
ax.DataAspectRatio=[1,1,1];
11 text chart
First, write a code to generate an image of each letter:
function getLogo
if ~exist('image','dir')
mkdir('image\')
end
logoSet=['.',char(65:90)];
for i=1:27
figure();
ax=gca;
ax.XLim=[-1,1];
ax.YLim=[-1,1];
ax.XColor='none';
ax.YColor='none';
ax.DataAspectRatio=[1,1,1];
logo=logoSet(i);
hold on
text(0,0,logo,'HorizontalAlignment','center','FontSize',320,'FontName','Segoe UI Black')
exportgraphics(ax,['image\',logo,'.png'])
close
end
dotPic=imread('image\..png');
newDotPic=uint8(ones([400,size(dotPic,2),3]).*255);
newDotPic(end-size(dotPic,1)+1:end,:,1)=dotPic(:,:,1);
newDotPic(end-size(dotPic,1)+1:end,:,2)=dotPic(:,:,2);
newDotPic(end-size(dotPic,1)+1:end,:,3)=dotPic(:,:,3);
imwrite(newDotPic,'image\..png')
S=20;
for i=1:27
logo=logoSet(i);
tPic=imread(['image\',logo,'.png']);
sz=size(tPic,[1,2]);
sz=round(sz./sz(1).*400);
tPic=imresize(tPic,sz);
tBox=uint8(255.*ones(size(tPic,[1,2])+S));
tBox(S+1:S+size(tPic,1),S+1:S+size(tPic,2))=tPic(:,:,1);
imwrite(cat(3,tBox,tBox,tBox),['image\',logo,'.png'])
end
end
Pi=[3,-1,getPi(150)];
CM=[109,110,113;224,25,33;244,126,26;253,207,2;154,203,57;111,150,124;
121,192,235;6,109,183;190,168,209;151,118,181;233,93,163]./255;
ST={'.','ZERO','ONE','TWO','THREE','FOUR','FIVE','SIX','SEVEN','EIGHT','NINE'};
n=1;
hold on
% 循环绘制字母
for i=1:20%:10
STList='';
NMList=[];
PicListR=uint8(zeros(400,0));
PicListG=uint8(zeros(400,0));
PicListB=uint8(zeros(400,0));
% PicListA=uint8(zeros(400,0));
for j=1:6
STList=[STList,ST{Pi(n)+2}];
NMList=[NMList,ones(size(ST{Pi(n)+2})).*(Pi(n)+2)];
n=n+1;
if length(STList)>15&&length(STList)+length(ST{Pi(n)+2})>20
break;
end
end
for k=1:length(STList)
tPic=imread(['image\',STList(k),'.png']);
% PicListA=[PicListA,tPic(:,:,1)];
PicListR=[PicListR,(255-tPic(:,:,1)).*CM(NMList(k),1)];
PicListG=[PicListG,(255-tPic(:,:,2)).*CM(NMList(k),2)];
PicListB=[PicListB,(255-tPic(:,:,3)).*CM(NMList(k),3)];
end
PicList=cat(3,PicListR,PicListG,PicListB);
image([-1200,1200],[0,150]-(i-1)*150,flipud(PicList))
end
% 图窗及坐标区域修饰
set(gcf,'Position',[200,100,600,820]);
ax=gca;
ax.DataAspectRatio=[1,1,1];
ax.XLim=[-1300,1300];
ax.Position=[0,0,1,1];
ax.XTick=[];
ax.YTick=[];
ax.Color=[0,0,0];
ax.YLim=[-19*150-80,230];
12 spiral chart
Pi=getPi(600);
% 配色列表
CM=[78,121,167;242,142,43;225,87,89;118,183,178;89,161,79;
237,201,72;176,122,161;255,157,167;156,117,95;186,176,172]./255;
% 绘制圆圈
hold on
t=linspace(0,2*pi,100);
x=cos(t).*.8;
y=sin(t).*.8;
for i=1:600
X=i.*cos(i./10)./10;
Y=i.*sin(i./10)./10;
fill(X+x,Y+y,CM(Pi(i)+1,:),'EdgeColor','none','FaceAlpha',.9)
end
text(0,65,'The Circle of \pi','Color',[1,1,1],'FontName','Cambria',...
'HorizontalAlignment','center','FontSize',25,'FontAngle','italic')
% 图窗和坐标区域修饰
set(gcf,'Position',[200,100,820,820]);
ax=gca;
ax.XLim=[-60,60];
ax.YLim=[-60,70];
ax.XTick=[];
ax.YTick=[];
ax.Color=[0,0,0];
ax.DataAspectRatio=[1,1,1];
13 Archimedean spiral diagram
a=1;b=.227;
Pi=getPi(500);
% 配色列表
CM=[78,121,167;242,142,43;225,87,89;118,183,178;89,161,79;
237,201,72;176,122,161;255,157,167;156,117,95;186,176,172]./255;
% 绘制圆圈
hold on
T=0;R=1;
t=linspace(0,2*pi,100);
x=cos(t).*.7;
y=sin(t).*.7;
for i=1:500
X=R.*cos(T);Y=R.*sin(T);
fill(X+x,Y+y,CM(Pi(i)+1,:),'EdgeColor','none','FaceAlpha',.9)
T=T+1./R.*1.4;
R=a+b*T;
end
text(17.25,22,{'The Archimedes spiral of \pi';'—— 500 digits'},...
'Color',[1,1,1],'FontName','Cambria',...
'HorizontalAlignment','right','FontSize',25,'FontAngle','italic')
% 图窗和坐标区域修饰
set(gcf,'Position',[200,100,820,820]);
ax=gca;
ax.XLim=[-19,18.5];
ax.YLim=[-20,25];
ax.XTick=[];
ax.YTick=[];
ax.Color=[0,0,0];
ax.DataAspectRatio=[1,1,1];
14 proportional Archimedean spiral diagram
Pi=[3,getPi(1199)];
% 配色数据
CM=[239,32,120;239,60,52;247,98,32;255,182,60;247,235,44;
142,199,57;55,180,70;0,170,239;40,56,146;147,37,139]./255;
% CM=slanCM(184,10);
% 绘制圆圈
hold on
T=0;R=1;
t=linspace(0,2*pi,100);
x=cos(t).*.7;
y=sin(t).*.7;
for i=1:4:length(Pi)
X=R.*cos(T);Y=R.*sin(T);
tNum=Pi(i:i+3);
numNum=find([diff(sort(tNum)),1]);
numNum=[numNum(1),diff(numNum)];
cumNum=cumsum(numNum);
uniNum=unique(tNum);
for j=length(cumNum):-1:1
fill(x./4.*cumNum(j)+X,y./4.*cumNum(j)+Y,CM(uniNum(j)+1,:),'EdgeColor','none')
end
T=T+1./R.*1.4;
R=a+b*T;
end
text(14,16.5,{'The ratio of four numbers from \pi';'—— 1200 digits'},...
'Color',[1,1,1],'FontName','Cambria',...
'HorizontalAlignment','right','FontSize',23,'FontAngle','italic')
% 图窗和坐标区域修饰
set(gcf,'Position',[200,100,820,820]);
ax=gca;
ax.XLim=[-15,15.5];
ax.YLim=[-15,19];
ax.XTick=[];
ax.YTick=[];
ax.Color=[0,0,0];
ax.DataAspectRatio=[1,1,1];
15 graph
% 构建连接矩阵
corrMat=zeros(10,10);
Pi=getPi(401);
for i=1:400
corrMat(Pi(i)+1,Pi(i+1)+1)=corrMat(Pi(i)+1,Pi(i+1)+1)+1;
end
% 配色列表
colorList=[0.3725 0.2745 0.5647
0.1137 0.4118 0.5882
0.2196 0.6510 0.6471
0.0588 0.5216 0.3294
0.4510 0.6863 0.2824
0.9294 0.6784 0.0314
0.8824 0.4863 0.0196
0.8000 0.3137 0.2431
0.5804 0.2039 0.4314
0.4353 0.2510 0.4392];
t=linspace(0,2*pi,11);t=t(1:10)';
posXY=[cos(t),sin(t)];
maxWidth=max(corrMat(corrMat>0));
minWidth=min(corrMat(corrMat>0));
ttList=linspace(0,1,3)';
% 循环绘图
hold on
for i=1:size(corrMat,1)
for j=i+1:size(corrMat,2)
if corrMat(i,j)>0
tW=(corrMat(i,j)-minWidth)./(maxWidth-minWidth);
colorData=(1-ttList).*colorList(i,:)+ttList.*colorList(j,:);
CData(:,:,1)=colorData(:,1);
CData(:,:,2)=colorData(:,2);
CData(:,:,3)=colorData(:,3);
% 绘制连线
fill(linspace(posXY(i,1),posXY(j,1),3),...
linspace(posXY(i,2),posXY(j,2),3),[0,0,0],'LineWidth',tW.*12+1,...
'CData',CData,'EdgeColor','interp','EdgeAlpha',.7,'FaceAlpha',.7)
end
end
% 绘制圆点
scatter(posXY(i,1),posXY(i,2),200,'filled','LineWidth',1.2,...
'MarkerFaceColor',colorList(i,:),'MarkerEdgeColor',[.7,.7,.7]);
text(posXY(i,1).*1.13,posXY(i,2).*1.13,num2str(i-1),'Color',[1,1,1].*.7,...
'FontSize',30,'FontWeight','bold','FontName','Cambria','HorizontalAlignment','center')
end
text(0,1.3,'Numerical adjacency of \pi — 400 digits','Color',[1,1,1],'FontName','Cambria',...
'HorizontalAlignment','center','FontSize',25,'FontAngle','italic')
% 图窗和坐标区域修饰
set(gcf,'Position',[200,100,820,820]);
ax=gca;
ax.XLim=[-1.2,1.2];
ax.YLim=[-1.21,1.5];
ax.XTick=[];
ax.YTick=[];
ax.Color=[0,0,0];
ax.DataAspectRatio=[1,1,1];
16 circos chart
Need to use this tool:
Class=getPi(1001)+1;
Data=diag(ones(1,1000),-1);
className={'0','1','2','3','4','5','6','7','8','9'};
colorOrder=[239,65,75;230,115,48;229,158,57;232,136,85;239,199,97;
144,180,116;78,166,136;81,140,136;90,118,142;43,121,159]./255;
CC=circosChart(Data,Class,'ClassName',className,'ColorOrder',colorOrder);
CC=CC.draw();
ax=gca;
ax.Color=[0,0,0];
CC.setClassLabel('Color',[1,1,1],'FontSize',25,'FontName','Cambria')
CC.setLine('LineWidth',.7)
YOU CAN GET ALL CODE HERE:
Chord diagrams are very common in Python and R, but there are no related functions in MATLAB before. It is not easy to draw chord diagrams of the same quality as R language, But I created a MATLAB tool that could almost do it.
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
Here is the help document:
1 Data Format
The data requirement is a numerical matrix with all values greater than or equal to 0, or a table array, or a numerical matrix and cell array for names. First, give an example of a numerical matrix:
1.1 Numerical Matrix
dataMat=randi([0,5],[5,4]);
% 绘图(draw)
CC=chordChart(dataMat);
CC=CC.draw();
Since each object is not named, it will be automatically named Rn and Cn
1.2 Numerical Matrix and Cell Array for Names
dataMat=[2 0 1 2 5 1 2;
3 5 1 4 2 0 1;
4 0 5 5 2 4 3];
colName={'G1','G2','G3','G4','G5','G6','G7'};
rowName={'S1','S2','S3'};
CC=chordChart(dataMat,'rowName',rowName,'colName',colName);
CC=CC.draw();
RowName should be the same size as the rows of the matrix
ColName should be the same size as the columns of the matrix
For this example, if the value in the second row and third column is 1, it indicates that there is an energy flow from S2 to G3, and a chord with a width of 1 is needed between these two.
1.3 Table Array
A table array in the following format is required:
2 Decorate Chord
2.1 Batch modification of chords
Batch modification of chords can be done using the setChordProp function, and all properties of the Patch object can be modified. For example, modifying the color of the string, edge color, edge line sstyle, etc.:
CC.setChordProp('EdgeColor',[.3,.3,.3],'LineStyle','--',...
'LineWidth',.1,'FaceColor',[.3,.3,.3])
2.2 Individual Modification of Chord
The individual modification of chord can be done using the setChordMN function, where the values of m and n correspond exactly to the rows and columns of the original numerical matrix. For example, changing the color of the strings flowing from S2 to G4 to red:
CC.setChordMN(2,4,'FaceColor',[1,0,0])
2.3 Color Mapping of Chords
Just use function colormap to do so:
% version 1.7.0更新
% 可使用colormap函数直接修改颜色
% Colors can be adjusted directly using the function colormap(demo4)
colormap(flipud(pink))
3 Arc Shaped Block Decoration
3.1 Batch Decoration of Arc-Shaped Blocks
use:
- setSquareT_Prop
- setSquareF_Prop
to modify the upper and lower blocks separately, and all attributes of the Patch object can be modified. For example, batch modify the upper blocks (change to black):
CC.setSquareT_Prop('FaceColor',[0,0,0])
3.2 Arc-Shaped Blocks Individually Decoration
use:
- setSquareT_N
- setSquareF_N
to modify the upper and lower blocks separately. For example, modify the second block above separately (changed to red):
CC.setSquareT_N(2,'FaceColor',[.8,0,0])
4 Font Adjustment
Use the setFont function to adjust the font, and all properties of the text object can be modified. For example, changing the font size, font, and color of the text:
CC.setFont('FontSize',25,'FontName','Cambria','Color',[0,0,.8])
5 Show and Hide Ticks
Usage:
CC.tickState('on')
% CC.tickState('off')
6 Attribute 'Sep' with Adjustable Square Spacing
If the matrix size is large, the drawing will be out of scale:
dataMat=randi([0,1],[20,10]);
CC=chordChart(dataMat);
CC=CC.draw();
% CC.tickState('on')
We can modify its Sep attribute:
dataMat=randi([0,1],[20,10]);
% use Sep to decrease space (separation)
% 使用 sep 减小空隙
CC=chordChart(dataMat,'Sep',1/120);
CC=CC.draw();
7 Modify Text Direction
dataMat=randi([0,1],[20,10]);
% use Sep to decrease space (separation)
% 使用 sep 减小空隙
CC=chordChart(dataMat,'Sep',1/120);
CC=CC.draw();
CC.tickState('on')
% version 1.7.0更新
% 函数labelRatato用来旋转标签
% The function labelRatato is used to rotate the label
CC.labelRotate('on')
8 Add Tick Labels
dataMat=[2 0 1 2 5 1 2;
3 5 1 4 2 0 1;
4 0 5 5 2 4 3];
colName={'G1','G2','G3','G4','G5','G6','G7'};
rowName={'S1','S2','S3'};
CC=chordChart(dataMat,'rowName',rowName,'colName',colName);
CC=CC.draw();
CC.setFont('FontSize',17,'FontName','Cambria')
% 显示刻度和数值
% Displays scales and numeric values
CC.tickState('on')
CC.tickLabelState('on')
% 调节标签半径
% Adjustable Label radius
CC.setLabelRadius(1.3);
% figure()
% dataMat=[2 0 1 2 5 1 2;
% 3 5 1 4 2 0 1;
% 4 0 5 5 2 4 3];
% dataMat=dataMat+rand(3,7);
% dataMat(dataMat<1)=0;
%
% CC=chordChart(dataMat,'rowName',rowName,'colName',colName);
% CC=CC.draw();
% CC.setFont('FontSize',17,'FontName','Cambria')
%
% % 显示刻度和数值
% % Displays scales and numeric values
% CC.tickState('on')
% CC.tickLabelState('on')
%
% % 调节标签半径
% % Adjustable Label radius
% CC.setLabelRadius(1.4);
9 Custom Tick Label Format
A function handle is required to input numeric output strings. The format can be set through the setTickLabelFormat function, such as Scientific notation:
dataMat=[2 0 1 2 5 1 2;
3 5 1 4 2 0 1;
4 0 5 5 2 4 3];
dataMat=dataMat+rand(3,7);
dataMat(dataMat<1)=0;
dataMat=dataMat.*1000;
CC=chordChart(dataMat);
CC=CC.draw();
CC.setFont('FontSize',17,'FontName','Cambria')
% 显示刻度和数值
% Displays scales and numeric values
CC.tickState('on')
CC.tickLabelState('on')
% 调节标签半径
% Adjustable Label radius
CC.setLabelRadius(1.4);
% 调整数值字符串格式
% Adjust numeric string format
CC.setTickLabelFormat(@(x)sprintf('%0.1e',x))
10 A Demo
rng(2)
dataMat=randi([1,7],[11,5]);
colName={'Fly','Beetle','Leaf','Soil','Waxberry'};
rowName={'Bartomella','Bradyrhizobium','Dysgomonas','Enterococcus',...
'Lactococcus','norank','others','Pseudomonas','uncultured',...
'Vibrionimonas','Wolbachia'};
CC=chordChart(dataMat,'rowName',rowName,'colName',colName,'Sep',1/80);
CC=CC.draw();
% 修改上方方块颜色(Modify the color of the blocks above)
CListT=[0.7765 0.8118 0.5216;0.4431 0.4706 0.3843;0.5804 0.2275 0.4549;
0.4471 0.4039 0.6745;0.0157 0 0 ];
for i=1:5
CC.setSquareT_N(i,'FaceColor',CListT(i,:))
end
% 修改下方方块颜色(Modify the color of the blocks below)
CListF=[0.5843 0.6863 0.7843;0.1098 0.1647 0.3255;0.0902 0.1608 0.5373;
0.6314 0.7961 0.2118;0.0392 0.2078 0.1059;0.0157 0 0 ;
0.8549 0.9294 0.8745;0.3882 0.3255 0.4078;0.5020 0.7216 0.3843;
0.0902 0.1843 0.1804;0.8196 0.2314 0.0706];
for i=1:11
CC.setSquareF_N(i,'FaceColor',CListF(i,:))
end
% 修改弦颜色(Modify chord color)
for i=1:5
for j=1:11
CC.setChordMN(j,i,'FaceColor',CListT(i,:),'FaceAlpha',.5)
end
end
CC.tickState('on')
CC.labelRotate('on')
CC.setFont('FontSize',17,'FontName','Cambria')
Hope to have your Reviews and Stars!!!
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
Most of my projects are the tension between what exists and what I must create to make the project work. The more I leverage what exists, the faster I can complete my project in a shorter amount of time. I often find out later in the process that a MATLAB function already exists and has existed for 10 releases. The faster that I discover built-in functions, the faster I can get to the new stuff that I have to invent.
I asked the AI Chat Playground why to use MATLAB's builtin functions and I got some interesting ideas. What are your thoughts?
AI Chat Playground:
- Efficiency and Performance: MATLAB's built-in functions are highly optimized for performance. They are written in optimized and compiled languages like C or C++, which means they run faster than equivalent functions written directly in MATLAB language. When you use these functions, your code can execute more efficiently, which is especially beneficial for handling large datasets or complex mathematical computations.
- Reliability and Stability: Built-in functions have been rigorously tested and debugged by the developers at MathWorks (the company behind MATLAB). They are less likely to contain errors compared to custom-written code, which means you can rely on them to produce accurate and consistent results.
- Ease of Use: Built-in functions are designed to be user-friendly, with clear documentation and support materials available. This can make it easier to perform complex operations without needing a deep understanding of the underlying algorithms. It also reduces the amount of code you need to write and debug, saving time and reducing the potential for errors.
- Compatibility: Built-in functions are updated alongside MATLAB itself, ensuring compatibility with new features and versions of the software. Using built-in functions can help ensure that your code continues to work correctly as MATLAB evolves.
- Advanced Features: MATLAB’s built-in functions often include advanced features and options that are not readily available or would be difficult to implement on your own. For example, MATLAB’s mathematical functions can handle a wide range of special cases and data types, providing more flexibility than custom implementations.
- Community and Support: Using built-in functions means you're working with standard tools recognized by the MATLAB community. This can make it easier to get help from other users or from MathWorks support, as more people will be familiar with the functionality and potential issues of these functions.
Given a vector v whose order we would like to randomly permute, many would perform the permutation by explicitly querying the length/size of v, e.g.,
I=randperm(numel(v));
v=v(I);
However, one can instead do as follows, avoiding the size query.
v=v(randperm(end))
Analogous things can be done with matrices, e.g.,
A=A(randperm(end), randperm(end));
The creativity comes from the copper sulfate crystal heart made in junior high school. Copper sulfate is a triclinic crystal, and the same structure was not used here for convenience in drawing.
Part 1. Coordinate transformation
To draw a crystal heart, one must first be able to draw crystal clusters. To draw a crystal cluster, one must first be able to draw a crystal. To draw a crystal, we need this kind of structure:
We first need a point with a certain distance from the straight line and a perpendicular point of cutPnt, which is very easy to find, for example, cutPnt=[x0, y0, z0]; The direction of the central axis is V=[x1, y1, z1]; If the distance to the straight line is L, the following points clearly meet the conditions:
v2=[z1,z1,-x1-y1];
v2=v2./norm(v2).*L;
pnt=cutPnt+v2;
But finding only one point is not enough. We need to find four points, and each point is obtained by rotating the previous point around a straight line by 
degrees. Therefore, we need to obtain our point rotation transformation matrix around a straight line
degrees. Therefore, we need to obtain our point rotation transformation matrix around a straight line
quite complex,right?
rotateMat=[u^2+(v^2+w^2)*cos(theta) , u*v*(1-cos(theta))-w*sin(theta), u*w*(1-cos(theta))+v*sin(theta), (a*(v^2+w^2)-u*(b*v+c*w))*(1-cos(theta))+(b*w-c*v)*sin(theta);
u*v*(1-cos(theta))+w*sin(theta), v^2+(u^2+w^2)*cos(theta) , v*w*(1-cos(theta))-u*sin(theta), (b*(u^2+w^2)-v*(a*u+c*w))*(1-cos(theta))+(c*u-a*w)*sin(theta);
u*w*(1-cos(theta))-v*sin(theta), v*w*(1-cos(theta))+u*sin(theta), w^2+(u^2+v^2)*cos(theta) , (c*(u^2+v^2)-w*(a*u+b*v))*(1-cos(theta))+(a*v-b*u)*sin(theta);
0 , 0 , 0 , 1];
Where [u, v, w] is the directional unit vector, and [a, b, c] is the initial coordinate of the axis:
Part 2. Crystal Cluster Drawing
function crystall
hold on
for i=1:50
len=rand(1)*8+5;
tempV=rand(1,3)-0.5;
tempV(3)=abs(tempV(3));
tempV=tempV./norm(tempV).*len;
tempEpnt=tempV;
drawCrystal([0 0 0],tempEpnt,pi/6,0.8,0.1,rand(1).*0.2+0.2)
disp(i)
end
ax=gca;
ax.XLim=[-15,15];
ax.YLim=[-15,15];
ax.ZLim=[-2,15];
grid on
ax.GridLineStyle='--';
ax.LineWidth=1.2;
ax.XColor=[1,1,1].*0.4;
ax.YColor=[1,1,1].*0.4;
ax.ZColor=[1,1,1].*0.4;
ax.DataAspectRatio=[1,1,1];
ax.DataAspectRatioMode='manual';
ax.CameraPosition=[-67.6287 -204.5276 82.7879];
function drawCrystal(Spnt,Epnt,theta,cl,w,alpha)
%plot3([Spnt(1),Epnt(1)],[Spnt(2),Epnt(2)],[Spnt(3),Epnt(3)])
mainV=Epnt-Spnt;
cutPnt=cl.*(mainV)+Spnt;
cutV=[mainV(3),mainV(3),-mainV(1)-mainV(2)];
cutV=cutV./norm(cutV).*w.*norm(mainV);
cornerPnt=cutPnt+cutV;
cornerPnt=rotateAxis(Spnt,Epnt,cornerPnt,theta);
cornerPntSet(1,:)=cornerPnt';
for ii=1:3
cornerPnt=rotateAxis(Spnt,Epnt,cornerPnt,pi/2);
cornerPntSet(ii+1,:)=cornerPnt';
end
F = [1,3,4;1,4,5;1,5,6;1,6,3;...
2,3,4;2,4,5;2,5,6;2,6,3];
V = [Spnt;Epnt;cornerPntSet];
patch('Faces',F,'Vertices',V,'FaceColor',[0 71 177]./255,...
'FaceAlpha',alpha,'EdgeColor',[0 71 177]./255.*0.8,...
'EdgeAlpha',0.6,'LineWidth',0.5,'EdgeLighting',...
'gouraud','SpecularStrength',0.3)
end
function newPnt=rotateAxis(Spnt,Epnt,cornerPnt,theta)
V=Epnt-Spnt;V=V./norm(V);
u=V(1);v=V(2);w=V(3);
a=Spnt(1);b=Spnt(2);c=Spnt(3);
cornerPnt=[cornerPnt(:);1];
rotateMat=[u^2+(v^2+w^2)*cos(theta) , u*v*(1-cos(theta))-w*sin(theta), u*w*(1-cos(theta))+v*sin(theta), (a*(v^2+w^2)-u*(b*v+c*w))*(1-cos(theta))+(b*w-c*v)*sin(theta);
u*v*(1-cos(theta))+w*sin(theta), v^2+(u^2+w^2)*cos(theta) , v*w*(1-cos(theta))-u*sin(theta), (b*(u^2+w^2)-v*(a*u+c*w))*(1-cos(theta))+(c*u-a*w)*sin(theta);
u*w*(1-cos(theta))-v*sin(theta), v*w*(1-cos(theta))+u*sin(theta), w^2+(u^2+v^2)*cos(theta) , (c*(u^2+v^2)-w*(a*u+b*v))*(1-cos(theta))+(a*v-b*u)*sin(theta);
0 , 0 , 0 , 1];
newPnt=rotateMat*cornerPnt;
newPnt(4)=[];
end
end
Part 3. Drawing of Crystal Heart
function crystalHeart
clc;clear;close all
hold on
% drawCrystal([1,1,1],[3,3,3],pi/6,0.8,0.14)
sep=pi/8;
t=[0:0.2:sep,sep:0.02:pi-sep,pi-sep:0.2:pi+sep,pi+sep:0.02:2*pi-sep,2*pi-sep:0.2:2*pi];
x=16*sin(t).^3;
y=13*cos(t)-5*cos(2*t)-2*cos(3*t)-cos(4*t);
z=zeros(size(t));
plot3(x,y,z,'Color',[186,110,64]./255,'LineWidth',1)
for i=1:length(t)
for j=1:6
len=rand(1)*2.5+1.5;
tempV=rand(1,3)-0.5;
tempV=tempV./norm(tempV).*len;
tempSpnt=[x(i),y(i),z(i)];
tempEpnt=tempV+tempSpnt;
drawCrystal(tempSpnt,tempEpnt,pi/6,0.8,0.14)
disp([i,j])
end
end
ax=gca;
ax.XLim=[-22,22];
ax.YLim=[-20,20];
ax.ZLim=[-10,10];
grid on
ax.GridLineStyle='--';
ax.LineWidth=1.2;
ax.XColor=[1,1,1].*0.4;
ax.YColor=[1,1,1].*0.4;
ax.ZColor=[1,1,1].*0.4;
ax.DataAspectRatio=[1,1,1];
ax.DataAspectRatioMode='manual';
function drawCrystal(Spnt,Epnt,theta,cl,w)
%plot3([Spnt(1),Epnt(1)],[Spnt(2),Epnt(2)],[Spnt(3),Epnt(3)])
mainV=Epnt-Spnt;
cutPnt=cl.*(mainV)+Spnt;
cutV=[mainV(3),mainV(3),-mainV(1)-mainV(2)];
cutV=cutV./norm(cutV).*w.*norm(mainV);
cornerPnt=cutPnt+cutV;
cornerPnt=rotateAxis(Spnt,Epnt,cornerPnt,theta);
cornerPntSet(1,:)=cornerPnt';
for ii=1:3
cornerPnt=rotateAxis(Spnt,Epnt,cornerPnt,pi/2);
cornerPntSet(ii+1,:)=cornerPnt';
end
F = [1,3,4;1,4,5;1,5,6;1,6,3;...
2,3,4;2,4,5;2,5,6;2,6,3];
V = [Spnt;Epnt;cornerPntSet];
patch('Faces',F,'Vertices',V,'FaceColor',[0 71 177]./255,...
'FaceAlpha',0.2,'EdgeColor',[0 71 177]./255.*0.9,...
'EdgeAlpha',0.25,'LineWidth',0.01,'EdgeLighting',...
'gouraud','SpecularStrength',0.3)
end
function newPnt=rotateAxis(Spnt,Epnt,cornerPnt,theta)
V=Epnt-Spnt;V=V./norm(V);
u=V(1);v=V(2);w=V(3);
a=Spnt(1);b=Spnt(2);c=Spnt(3);
cornerPnt=[cornerPnt(:);1];
rotateMat=[u^2+(v^2+w^2)*cos(theta) , u*v*(1-cos(theta))-w*sin(theta), u*w*(1-cos(theta))+v*sin(theta), (a*(v^2+w^2)-u*(b*v+c*w))*(1-cos(theta))+(b*w-c*v)*sin(theta);
u*v*(1-cos(theta))+w*sin(theta), v^2+(u^2+w^2)*cos(theta) , v*w*(1-cos(theta))-u*sin(theta), (b*(u^2+w^2)-v*(a*u+c*w))*(1-cos(theta))+(c*u-a*w)*sin(theta);
u*w*(1-cos(theta))-v*sin(theta), v*w*(1-cos(theta))+u*sin(theta), w^2+(u^2+v^2)*cos(theta) , (c*(u^2+v^2)-w*(a*u+b*v))*(1-cos(theta))+(a*v-b*u)*sin(theta);
0 , 0 , 0 , 1];
newPnt=rotateMat*cornerPnt;
newPnt(4)=[];
end
end
So, how to draw a roseball just like this ?
To begin with, we need to know how to draw a single rose in MATLAB:
function drawrose
set(gca,'CameraPosition',[2 2 2])
hold on
grid on
[x,t]=meshgrid((0:24)./24,(0:0.5:575)./575.*20.*pi+4*pi);
p=(pi/2)*exp(-t./(8*pi));
change=sin(15*t)/150;
u=1-(1-mod(3.6*t,2*pi)./pi).^4./2+change;
y=2*(x.^2-x).^2.*sin(p);
r=u.*(x.*sin(p)+y.*cos(p));
h=u.*(x.*cos(p)-y.*sin(p));
surface(r.*cos(t),r.*sin(t),h,'EdgeAlpha',0.1,...
'EdgeColor',[0 0 0],'FaceColor','interp')
end
Tts pretty easy, Now we are trying to dye it the desired color:
function drawrose
set(gca,'CameraPosition',[2 2 2])
hold on
grid on
[x,t]=meshgrid((0:24)./24,(0:0.5:575)./575.*20.*pi+4*pi);
p=(pi/2)*exp(-t./(8*pi));
change=sin(15*t)/150;
u=1-(1-mod(3.6*t,2*pi)./pi).^4./2+change;
y=2*(x.^2-x).^2.*sin(p);
r=u.*(x.*sin(p)+y.*cos(p));
h=u.*(x.*cos(p)-y.*sin(p));
map=[0.9176 0.9412 1.0000
0.8353 0.8706 0.9922
0.8196 0.8627 0.9804
0.7020 0.7569 0.9412
0.5176 0.5882 0.9255
0.3686 0.4824 0.9412
0.3059 0.4000 0.9333
0.2275 0.3176 0.8353
0.1216 0.2275 0.6471];
Xi=1:size(map,1);Xq=linspace(1,size(map,1),100);
map=[interp1(Xi,map(:,1),Xq,'linear')',...
interp1(Xi,map(:,2),Xq,'linear')',...
interp1(Xi,map(:,3),Xq,'linear')'];
surface(r.*cos(t),r.*sin(t),h,'EdgeAlpha',0.1,...
'EdgeColor',[0 0 0],'FaceColor','interp')
colormap(map)
end
I try to take colors from real roses and interpolate them to make them more realistic
Then, how can I put these colorful flowers on to a ball ?
We need to place the drawn flowers on each face of the polyhedron sphere through coordinate transformation. Here, we use a regular dodecahedron:
Move the flower using the following rotation formula:
We place a flower on each plane, which means that the angle between every two flowers is 
degrees. We can place each flower at the appropriate angle through multiple x-axis rotations and multiple z-axis rotations. The code is as follows:
degrees. We can place each flower at the appropriate angle through multiple x-axis rotations and multiple z-axis rotations. The code is as follows:function roseBall(colorList)
%曲面数据计算
%==========================================================================
[x,t]=meshgrid((0:24)./24,(0:0.5:575)./575.*20.*pi+4*pi);
p=(pi/2)*exp(-t./(8*pi));
change=sin(15*t)/150;
u=1-(1-mod(3.6*t,2*pi)./pi).^4./2+change;
y=2*(x.^2-x).^2.*sin(p);
r=u.*(x.*sin(p)+y.*cos(p));
h=u.*(x.*cos(p)-y.*sin(p));
%颜色映射表
%==========================================================================
hMap=(h-min(min(h)))./(max(max(h))-min(min(h)));
col=size(hMap,2);
if nargin<1
colorList=[0.0200 0.0400 0.3900
0 0.0900 0.5800
0 0.1300 0.6400
0.0200 0.0600 0.6900
0 0.0800 0.7900
0.0100 0.1800 0.8500
0 0.1300 0.9600
0.0100 0.2600 0.9900
0 0.3500 0.9900
0.0700 0.6200 1.0000
0.1700 0.6900 1.0000];
end
colorFunc=colorFuncFactory(colorList);
dataMap=colorFunc(hMap');
colorMap(:,:,1)=dataMap(:,1:col);
colorMap(:,:,2)=dataMap(:,col+1:2*col);
colorMap(:,:,3)=dataMap(:,2*col+1:3*col);
function colorFunc=colorFuncFactory(colorList)
xx=(0:size(colorList,1)-1)./(size(colorList,1)-1);
y1=colorList(:,1);y2=colorList(:,2);y3=colorList(:,3);
colorFunc=@(X)[interp1(xx,y1,X,'linear')',interp1(xx,y2,X,'linear')',interp1(xx,y3,X,'linear')'];
end
%曲面旋转及绘制
%==========================================================================
surface(r.*cos(t),r.*sin(t),h+0.35,'EdgeAlpha',0.05,...
'EdgeColor',[0 0 0],'FaceColor','interp','CData',colorMap)
hold on
surface(r.*cos(t),r.*sin(t),-h-0.35,'EdgeAlpha',0.05,...
'EdgeColor',[0 0 0],'FaceColor','interp','CData',colorMap)
Xset=r.*cos(t);
Yset=r.*sin(t);
Zset=h+0.35;
yaw_z=72*pi/180;
roll_x=pi-acos(-1/sqrt(5));
R_z_2=[cos(yaw_z),-sin(yaw_z),0;
sin(yaw_z),cos(yaw_z),0;
0,0,1];
R_z_1=[cos(yaw_z/2),-sin(yaw_z/2),0;
sin(yaw_z/2),cos(yaw_z/2),0;
0,0,1];
R_x_2=[1,0,0;
0,cos(roll_x),-sin(roll_x);
0,sin(roll_x),cos(roll_x)];
[nX,nY,nZ]=rotateXYZ(Xset,Yset,Zset,R_x_2);
surface(nX,nY,nZ,'EdgeAlpha',0.05,...
'EdgeColor',[0 0 0],'FaceColor','interp','CData',colorMap)
for k=1:4
[nX,nY,nZ]=rotateXYZ(nX,nY,nZ,R_z_2);
surface(nX,nY,nZ,'EdgeAlpha',0.05,...
'EdgeColor',[0 0 0],'FaceColor','interp','CData',colorMap)
end
[nX,nY,nZ]=rotateXYZ(nX,nY,nZ,R_z_1);
for k=1:5
[nX,nY,nZ]=rotateXYZ(nX,nY,nZ,R_z_2);
surface(nX,nY,-nZ,'EdgeAlpha',0.05,...
'EdgeColor',[0 0 0],'FaceColor','interp','CData',colorMap)
end
%--------------------------------------------------------------------------
function [nX,nY,nZ]=rotateXYZ(X,Y,Z,R)
nX=zeros(size(X));
nY=zeros(size(Y));
nZ=zeros(size(Z));
for i=1:size(X,1)
for j=1:size(X,2)
v=[X(i,j);Y(i,j);Z(i,j)];
nv=R*v;
nX(i,j)=nv(1);
nY(i,j)=nv(2);
nZ(i,j)=nv(3);
end
end
end
%axes属性调整
%==========================================================================
ax=gca;
grid on
ax.GridLineStyle='--';
ax.LineWidth=1.2;
ax.XColor=[1,1,1].*0.4;
ax.YColor=[1,1,1].*0.4;
ax.ZColor=[1,1,1].*0.4;
ax.DataAspectRatio=[1,1,1];
ax.DataAspectRatioMode='manual';
ax.CameraPosition=[-6.5914 -24.1625 -0.0384];
end
TRY DIFFERENT COLORS !!
colorList1=[0.2000 0.0800 0.4300
0.2000 0.1300 0.4600
0.2000 0.2100 0.5000
0.2000 0.2800 0.5300
0.2000 0.3700 0.5800
0.1900 0.4500 0.6200
0.2000 0.4800 0.6400
0.1900 0.5400 0.6700
0.1900 0.5700 0.6900
0.1900 0.7500 0.7800
0.1900 0.8000 0.8100
];
colorList2=[0.1300 0.1000 0.1600
0.2000 0.0900 0.2000
0.2800 0.0800 0.2300
0.4200 0.0800 0.3000
0.5100 0.0700 0.3400
0.6600 0.1200 0.3500
0.7900 0.2200 0.4000
0.8800 0.3500 0.4700
0.9000 0.4500 0.5400
0.8900 0.7800 0.7900
];
colorList3=[0.3200 0.3100 0.7600
0.3800 0.3400 0.7600
0.5300 0.4200 0.7500
0.6400 0.4900 0.7300
0.7200 0.5500 0.7200
0.7900 0.6100 0.7100
0.9100 0.7100 0.6800
0.9800 0.7600 0.6700
];
colorList4=[0.2100 0.0900 0.3800
0.2900 0.0700 0.4700
0.4000 0.1100 0.4900
0.5500 0.1600 0.5100
0.7500 0.2400 0.4700
0.8900 0.3200 0.4100
0.9700 0.4900 0.3700
1.0000 0.5600 0.4100
1.0000 0.6900 0.4900
1.0000 0.8200 0.5900
0.9900 0.9200 0.6700
0.9800 0.9500 0.7100];
Let us consider how to draw a Happy Sheep. A Happy Sheep was introduced in the MATLAB Mini Hack contest: Happy Sheep!
In this contest there was the strict limitation on the code length. So the code of the Happy Sheep is very compact and is only 280 characters long. We will analyze the process of drawing the Happy Sheep in MATLAB step by step. The explanations of the even more compact version of the code of the same sheep are given below.
So, how to draw a sheep? It is very easy. We could notice that usually a sheep is covered by crimped wool. Therefore, a sheep could be painted using several geometrical curves of similar types. Of course, then it will be an abstract model of the real sheep. Let us select two mathematical curves, which are the most appropriate for our goal. They are an ellipse for smooth parts of the sheep and an ellipse combined with a rose for woolen parts of the sheep.
Let us recall the mathematical formulas of these curves. A parametric representation of the standard ellipse is the following:
Also we will use the following parametric representation of the rose (rhodonea) curve:
This curve was named by the mathematician Guido Grandi.
Let us combine them in one curve and add possible shifts:
Now if we would like to create an ellipse, we will set 
and 
. If we would like to create a rose, we will set 
and 
. If we would like to shift our curve, we will set 
and 
to the required values. Of course, we could set all non-zero parameters to combine both chosen curves and use the shifts.
and . If we would like to create a rose, we will set and . If we would like to shift our curve, we will set and to the required values. Of course, we could set all non-zero parameters to combine both chosen curves and use the shifts.Let us describe how to create these curves using the MATLAB code. To make the code more compact, it is possible to program both formulas for the combined curve in one line using the anonymous function. We could make the code more compact using the function handles for sine and cosine functions. Then the MATLAB code for an example of the ellipse curve will be the following.
% Handles
s=@sin;
c=@cos;
% Ellipse + Polar Rose
F=@(t,a,f) a(1)*f(t)+s(a(2)*t).*f(t)+a(3);
% Angles
t=0:.1:7;
% Parameters
E = [5 7;0 0;0 0];
% Painting
figure;
plot(F(t,E(:,1),c),F(t,E(:,2),s),'LineWidth',10);
axis equal
The parameter t varies from 0 to 7, which is the nearest integer greater than 
, with the step 0.1. The result of this code is the following ellipse curve with 
and 
.
, with the step 0.1. The result of this code is the following ellipse curve with and .
This ellipse is described by the following parametric equations:
The MATLAB code for an example of the rose curve will be the following.
% Handles
s=@sin;
c=@cos;
% Ellipse + Polar Rose
F=@(t,a,f) a(1)*f(t)+s(a(2)*t).*f(t)+a(3);
% Angles
t=0:.1:7;
% Parameters
R = [0 0;4 4;0 0];
% Painting
figure;
plot(F(t,R(:,1),c),F(t,R(:,2),s),'LineWidth',10);
axis equal
The result of this code is the following rose curve with 
and 
.
and .
This rose is described by the following parametric equations:
Obviously, now we are ready to draw main parts of our sheep! As we reproduce an abstract model of the sheep, let us select the following main parts for the representation: head, eyes, hoofs, body, crown, and tail. We will use ellipses for the first three parts in this list and ellipses combined with roses for the last three ones.
First let us describe drawing of each part independently.
The following MATLAB code will be used to do this.
% Handles
s=@sin;
c=@cos;
% Ellipse + Polar Rose
F=@(t,a,f) a(1)*f(t)+s(a(2)*t).*f(t)+a(3);
% Angles
t=0:.1:7;
% Parameters
Head = 1;
Eyes = 2:3;
Hoofs = 4:7;
Body = 8;
Crown = 9;
Tail = 10;
G=-13;
P=[5 7 repmat([.1 .5],1,6) 6 4 14 9 3 3;zeros(1,14) 8 8 12 12 4 4;...
-15 2 G 3 -17 3 -3 G 0 G 9 G 12 G -15 12 4 3 20 7];
% Painting
figure;
hold;
for i=Head
plot(F(t,P(:,2*i-1),c),F(t,P(:,2*i),s),'LineWidth',10);
end
axis([-25 25 -15 20]);
figure;
hold;
for i=Eyes
plot(F(t,P(:,2*i-1),c),F(t,P(:,2*i),s),'LineWidth',10);
end
axis([-25 25 -15 20]);
figure;
hold;
for i=Hoofs
plot(F(t,P(:,2*i-1),c),F(t,P(:,2*i),s),'LineWidth',10);
end
axis([-25 25 -15 20]);
figure;
hold;
for i=Body
plot(F(t,P(:,2*i-1),c),F(t,P(:,2*i),s),'LineWidth',10);
end
axis([-25 25 -15 20]);
figure;
hold;
for i=Crown
plot(F(t,P(:,2*i-1),c),F(t,P(:,2*i),s),'LineWidth',10);
end
axis([-25 25 -15 20]);
figure;
hold;
for i=Tail
plot(F(t,P(:,2*i-1),c),F(t,P(:,2*i),s),'LineWidth',10);
end
axis([-25 25 -15 20]);
The parameters 
, 
, 
, 
, 
, and 
are written in the different submatrices of the matrix P. The code generates the following curves to illustrate the different parts of our sheep.
, , , , , and are written in the different submatrices of the matrix P. The code generates the following curves to illustrate the different parts of our sheep.The following ellipse describes the head of the sheep.
The following submatrix of the matrix P represents its parameters.
The parametric equations of the head are the following:
The following ellipses describe the eyes of the sheep.
The following submatrices of the matrix P represent their parameters.
The parametric equations of the left and right eyes correspondingly are the following:
The following ellipses describe the hoofs of the sheep.
The following submatrices of the matrix P represent their parameters.
The parametric equations of the right front, left front, right hind, and left hind hoofs correspondingly are the following:
The following ellipse combined with the rose describes the crown of the sheep.
The following submatrix of the matrix P represents its parameters.
The parametric equations of the crown are the following:
The following ellipse combined with the rose describes the body of the sheep.
The following submatrix of the matrix P represents its parameters.
The parametric equations of the body are the following:
The following ellipse combined with the rose describes the tail of the sheep.
The following submatrix of the matrix P represents its parameters.
The parametric equations of the tail are the following:
Now all the parts of our sheep should be put together! It is very easy because all the parts are described by the same equations with different parameters.
The following code helps us to accomplish this goal and ultimately draw a Happy Sheep in MATLAB!
% Happy Sheep!
% By Victoria A. Sablina
% Handles
s=@sin;
c=@cos;
% Ellipse + Rose
F=@(t,a,f) a(1)*f(t)+s(a(2)*t).*f(t)+a(3);
% Angles
t=0:.1:7;
% Parameters
% Head (1:2)
% Eyes (3:6)
% Hoofs (7:14)
% Crown (15:16)
% Body (17:18)
% Tail (19:20)
G=-13;
P=[5 7 repmat([.1 .5],1,6) 6 4 14 9 3 3;zeros(1,14) 8 8 12 12 4 4;...
-15 2 G 3 -17 3 -3 G 0 G 9 G 12 G -15 12 4 3 20 7];
% Painting
hold;
for i=1:10
plot(F(t,P(:,2*i-1),c),F(t,P(:,2*i),s),'LineWidth',10);
end
This code is even more compact than the original code from the contest. It is only 253 instead of 280 characters long and generates the same Happy Sheep!
Our sheep is happy, because of becoming famous in the MATLAB community, a star!
Congratulations! Now you know how to draw a Happy Sheep in MATLAB!
Thank you for reading!
MATLAB O/X Quiz
Answer BEFORE Googling!
- An infinite loop can be made using "for".
- "A == A" is always true.
- "round(2.5)" is 3.
- "round(-0.5)" is 0.
MATLAB Support Package for Quantum Computing lets you build, simulate, and run quantum algorithms.
Check out the Cheat Sheet here!
I think that MATLAB's Flipbook Mini Hack had quite some inspiring entries. My work largely deals with digital elevation models (DEMs). Hence I really liked the random renderings of landscapes, in particular this one written by Tim which inspired me to adopt the code and apply to the example data that comes with my software TopoToolbox. The results and code are shown here.
and immeditaely everyone wanted the code! It turns out that this is the result of my remix of @Zhaoxu Liu / slandarer's entry on the MATLAB Flipbook Mini Hack.
I pointed people to the Flipbook entry but, of course, that just gave the code to render a single frame and people wanted the full code to render the animated gif. That way, they could make personalised versions
I just published a blog post that gives the code used by the team behind the Mini Hack to produce the animated .gifs https://blogs.mathworks.com/matlab/2024/02/16/producing-animated-gifs-from-matlab-flipbook-mini-hack-entries/
Thanks again to @Zhaoxu Liu / slandarer for a great entry that seems like it will live for a long time :)
If you've dabbled in "procedural generation," (algorithmically generating natural features), you may have come across the problem of sphere texturing. How to seamlessly texture a sphere is not immediately obvious. Watch what happens, for example, if you try adding power law noise to an evenly sampled grid of spherical angle coordinates (i.e. a "UV sphere" in Blender-speak):
% Example: how [not] to texture a sphere:
rng(2, 'twister'); % Make what I have here repeatable for you
% Make our radial noise, mapped onto an equal spaced longitude and latitude
% grid.
N = 51;
b = linspace(-1, 1, N).^2;
r = abs(ifft2(exp(6i*rand(N))./(b'+b+1e-5))); % Power law noise
r = rescale(r, 0, 1) + 5;
[lon, lat] = meshgrid(linspace(0, 2*pi, N), linspace(-pi/2, pi/2, N));
[x2, y2, z2] = sph2cart(lon, lat, r);
r2d = @(x)x*180/pi;
% Radial surface texture
subplot(1, 3, 1);
imagesc(r, 'Xdata', r2d(lon(1,:)), 'Ydata', r2d(lat(:, 1)));
xlabel('Longitude (Deg)');
ylabel('Latitude (Deg)');
title('Texture (radial variation)');
% View from z axis
subplot(1, 3, 2);
surf(x2, y2, z2, r);
axis equal
view([0, 90]);
title('Top view');
% Side view
subplot(1, 3, 3);
surf(x2, y2, z2, r);
axis equal
view([-90, 0]);
title('Side view');
The created surface shows "pinching" at the poles due to different radial values mapping to the same location. Furthermore, the noise statistics change based on the density of the sampling on the surface.
How can this be avoided? One standard method is to create a textured volume and sample the volume at points on a sphere. Code for doing this is quite simple:
rng default % Make our noise realization repeatable
% Create our 3D power-law noise
N = 201;
b = linspace(-1, 1, N);
[x3, y3, z3] = meshgrid(b, b, b);
b3 = x3.^2 + y3.^2 + z3.^2;
r = abs(ifftn(ifftshift(exp(6i*randn(size(b3)))./(b3.^1.2 + 1e-6))));
% Modify it - make it more interesting
r = rescale(r);
r = r./(abs(r - 0.5) + .1);
% Sample on a sphere
[x, y, z] = sphere(500);
% Plot
ir = interp3(x3, y3, z3, r, x, y, z, 'linear', 0);
surf(x, y, z, ir);
shading flat
axis equal off
set(gcf, 'color', 'k');
colormap(gray);
The result of evaluating this code is a seamless, textured sphere with no discontinuities at the poles or variation in the spatial statistics of the noise texture:
But what if you want to smooth it or perform some other local texture modification? Smoothing the volume and resampling is not equivalent to smoothing the surficial features shown on the map above.
A more flexible alternative is to treat the samples on the sphere surface as a set of interconnected nodes that are influenced by adjacent values. Using this approach we can start by defining the set of nodes on a sphere surface. These can be sampled almost arbitrarily, though the noise statistics will vary depending on the sampling strategy.
One noise realisation I find attractive can be had by randomly sampling a sphere. Normalizing a point in N-dimensional space by its 2-norm projects it to the surface of an N-dimensional unit sphere, so randomly sampling a sphere can be done very easily using randn() and vecnorm():
N = 5e3; % Number of nodes on our sphere
g=randn(3,N); % Random 3D points around origin
p=g./vecnorm(g); % Projected to unit sphere
The next step is to find each point's "neighbors." The first step is to find the convex hull. Since each point is on the sphere, the convex hull will include each point as a vertex in the triangulation:
k=convhull(p');
In the above, k is an N x 3 set of indices where each row represents a unique triangle formed by a triplicate of points on the sphere surface. The vertices of the full set of triangles containing a point describe the list of neighbors to that point.
What we want now is a large, sparse symmetric matrix where the indices of the columns & rows represent the indices of the points on the sphere and the nth row (and/or column) contains non-zero entries at the indices corresponding to the neighbors of the nth point.
How to do this? You could set up a tiresome nested for-loop searching for all rows (triangles) in k that contain some index n, or you could directly index via:
c=@(x)sparse(k(:,x)*[1,1,1],k,1,N,N);
t=c(1)|c(2)|c(3);
The result is the desired sparse connectivity matrix: a matrix with non-zero entries defining neighboring points.
So how do we create a textured sphere with this connectivity matrix? We will use it to form a set of equations that, when combined with the concept of "regularization," will allow us to determine the properties of the randomness on the surface. Our regularizer will penalize the difference of the radial distance of a point and the average of its neighbors. To do this we replace the main diagonal with the negative of the sum of the off-diagonal components so that the rows and columns are zero-mean. This can be done via:
w=spdiags(-sum(t,2)+1,0,double(t));
Now we invoke a bit of linear algebra. Pretend x is an N-length vector representing the radial distance of each point on our sphere with the noise realization we desire. Y will be an N-length vector of "observations" we are going to generate randomly, in this case using a uniform distribution (because it has a bias and we want a non-zero average radius, but you can play around with different distributions than uniform to get different effects):
Y=rand(N,1);
and A is going to be our "transformation" matrix mapping x to our noisy observations:
Ax = Y
In this case both x and Y are N length vectors and A is just the identity matrix:
A = speye(N);
Y, however, doesn't create the noise realization we want. So in the equation above, when solving for x we are going to introduce a regularizer which is going to penalize unwanted behavior of x by some amount. That behavior is defined by the point-neighbor radial differences represented in matrix w. Our estimate of x can then be found using one of my favorite Matlab assets, the "\" operator:
smoothness = 10; % Smoothness penalty: higher is smoother
x = (A+smoothness*w'*w)\Y; % Solving for radii
The vector x now contains the radii with the specified noise realization for the sphere which can be created simply by multiplying x by p and plotting using trisurf:
p2 = p.*x';
trisurf(k,p2(1,:),p2(2,:),p2(3,:),'FaceC', 'w', 'EdgeC', 'none','AmbientS',0,'DiffuseS',0.6,'SpecularS',1);
light;
set(gca, 'color', 'k');
axis equal
The following images show what happens as you change the smoothness parameter using values [.1, 1, 10, 100] (left to right):
Now you know a couple ways to make a textured sphere: that's the starting point for having a lot of fun with basic procedural planet, moon, or astroid generation! Here's some examples of things you can create based on these general ideas:
The MATLAB command window isn't just for commands and outputs—it can also host interactive hyperlinks. These can serve as powerful shortcuts, enhancing the feedback you provide during code execution. Here are some hyperlinks I frequently use in fprintf statements, warnings, or error messages.
1. Open a website.
msg = "Could not download data from website.";
url = "https://blogs.mathworks.com/graphics-and-apps/";
hypertext = "Go to website"
fprintf(1,'%s <a href="matlab: web(''%s'') ">%s</a>\n',msg,url,hypertext);
Could not download data from website. Go to website
2. Open a folder in file explorer (Windows)
msg = "File saved to current directory.";
directory = cd();
hypertext = "[Open directory]";
fprintf(1,'%s <a href="matlab: winopen(''%s'') ">%s</a>\n',msg,directory,hypertext)
File saved to current directory. [Open directory]
3. Open a document (Windows)
msg = "Created database.csv.";
filepath = fullfile(cd,'database.csv');
hypertext = "[Open file]";
fprintf(1,'%s <a href="matlab: winopen(''%s'') ">%s</a>\n',msg,filepath,hypertext)
Created database.csv. [Open file]
4. Open an m-file and go to a specific line
msg = 'Go to';
file = 'streamline.m';
line = 51;
fprintf(1,'%s <a href="matlab: matlab.desktop.editor.openAndGoToLine(which(''%s''), %d); ">%s line %d</a>', msg, file, line, file, line);
Go to streamline.m line 51
5. Display more text
msg = 'Incomplete data detected.';
extendedInfo = '\tFilename: m32c4r28\n\tDate: 12/20/2014\n\tElectrode: (3,7)\n\tDepth: ???\n';
hypertext = '[Click for more info]';
warning('%s <a href="matlab: fprintf(''%s'') ">%s</a>', msg,extendedInfo,hypertext);
<click>
- Filename: m32c4r28
- Date: 12/20/2014
- Electrode: (3,7)
- Depth: ???
6. Run a function
Similarly, you can also add hyperlinks in figures and apps
To enlarge an array with more rows and/or columns, you can set the lower right index to zero. This will pad the matrix with zeros.
m = rand(2, 3) % Initial matrix is 2 rows by 3 columns
mCopy = m;
% Now make it 2 rows by 5 columns
m(2, 5) = 0
m = mCopy; % Go back to original matrix.
% Now make it 3 rows by 3 columns
m(3, 3) = 0
m = mCopy; % Go back to original matrix.
% Now make it 3 rows by 7 columns
m(3, 7) = 0
Greetings to all MATLAB users,
Although the MATLAB Flipbook contest has concluded, the pursuit of ‘learning while having fun’ continues. I would like to take this opportunity to highlight some recent insightful technical articles from a standout contest participant – Zhaoxu Liu / slandarer.
Zhaoxu has contributed eight informative articles to both the Tips & Tricks and Fun channels in our new Discussions area. His articles offer practical advice on topics such as customizing legends, constructing chord charts, and adding color to axes. Additionally, he has shared engaging content, like using MATLAB to create an interactive dragon that follows your mouse cursor, a nod to the upcoming Year of the Dragon in 2024!
I invite you to explore these articles for both enjoyment and education, and I hope you'll find new techniques to incorporate into your work.
Our community is full of individuals skilled in MATLAB, and we're always eager to learn from one another. Who would you like to see featured next? Or perhaps you have some tips & tricks of your own to contribute. Remember, sharing knowledge is a collaborative effort, as Confucius wisely stated, 'When I walk along with two others, they may serve me as my teachers.'
Let's maintain our commitment to a continuous learning journey. This could be the perfect warm-up for the upcoming 2024 contest.
It is easy to obtain sankey plot like that using my tool:
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