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This study explores the demographic patterns and disease outcomes during the cholera outbreak in London in 1849. Utilizing historical records and scholarly accounts, the research investigates the impact of the outbreak on the city' s population. While specific data for the 1849 cholera outbreak is limited, trends from similar 19th - century outbreaks suggest a high infection rate, potentially ranging from 30% to 50% of the population, owing to poor sanitation and overcrowded living conditions . Additionally, the birth rate in London during this period was estimated at 0.037 births per person per year . Although the exact reproduction number (R₀) for cholera in 1849 remains elusive, historical evidence implies a high R₀ due to the prevalent unsanitary conditions . This study sheds light on the challenges of estimating disease parameters from historical data, emphasizing the critical role of sanitation and public health measures in mitigating the impact of infectious diseases.
Introduction
The cholera outbreak of 1849 was a significant event in the history of cholera, a deadly waterborne disease caused by the bacterium Vibrio cholera. Cholera had several major outbreaks during the 19th century, and the one in 1849 was particularly devastating.
During this outbreak, cholera spread rapidly across Europe, including countries like England, France, and Germany . The disease also affected North America, with outbreaks reported in cities like New York and Montreal. The exact number of casualties from the 1849 cholera outbreak is difficult to determine due to limited record - keeping at that time. However, it is estimated that tens of thousands of people died as a result of the disease during this outbreak.
Cholera is highly contagious and spreads through contaminated water and food . The lack of proper sanitation and hygiene practices in the 19th century contributed to the rapid spread of the disease. It wasn't until the late 19th and early 20th centuries that advancements in public health, sanitation, and clean drinking water significantly reduced the incidence and impact of cholera outbreaks in many parts of the world.
Infection Rate
Based on general patterns observed in 19th - century cholera outbreaks and the conditions of that time, it' s reasonable to assume that the infection rate was quite high. During major cholera outbreaks in densely populated and unsanitary areas, infection rates could be as high as 30 - 50% or even more.
This means that in a densely populated city like London, with an estimated population of around 2.3 million in 1849, tens of thousands of people could have been infected during the outbreak. It' s important to emphasize that this is a rough estimation based on historical patterns and not specific to the 1849 outbreak. The actual infection rate could have varied widely based on the local conditions, public health measures in place, and the effectiveness of efforts to contain the disease.
For precise and localized estimations, detailed historical records specific to the 1849 cholera outbreak in a particular city or region would be required, and such data might not be readily available due to the limitations of historical documentation from that time period
Mortality Rate
It' s challenging to provide an exact death rate for the 1849 cholera outbreak because of the limited and often unreliable historical records from that time period. However, it is widely acknowledged that the death rate was significant, with tens of thousands of people dying as a result of the disease during this outbreak.
Cholera has historically been known for its high mortality rate, particularly in areas with poor sanitation and limited access to clean water. During cholera outbreaks in the 19th century, mortality rates could be extremely high, sometimes reaching 50% or more in affected communities. This high mortality rate was due to the rapid onset of severe dehydration and electrolyte imbalance caused by the cholera toxin, leading to death if not promptly treated.
Studies and historical accounts from various cholera outbreaks suggest that the R₀ for cholera can range from 1.5 to 2.5 or even higher in conditions where sanitation is inadequate and clean water is scarce. This means that one person with cholera could potentially infect 1.5 to 2.5 or more other people in such settings.
Unfortunately, there are no specific and reliable data available regarding the recovery rates from the 1849 cholera outbreak, as detailed and accurate record - keeping during that time period was limited. Cholera outbreaks in the 19th century were often devastating due to the lack of effective medical treatments and poor sanitation conditions. Recovery from cholera largely depended on the individual's ability to rehydrate, which was difficult given the rapid loss of fluids through severe diarrhea and vomiting .
LONDON CASE OF STUDY
In 1849, the estimated population of London was around 2.3 million people. London experienced significant population growth during the 19th century due to urbanization and industrialization. It’s important to note that historical population figures are often estimates, as comprehensive and accurate record-keeping methods were not as advanced as they are today.
% Define parameters
R0 = 2.5;
beta = 0.5;
gamma = 0.2; % Recovery rate
N = 2300000; % Total population
I0 = 1; % Initial number of infected individuals
% Define the SIR model differential equations
sir_eqns = @(t, Y) [-beta * Y(1) * Y(2) / N; % dS/dt
beta * Y(1) * Y(2) / N - gamma * Y(2); % dI/dt
gamma * Y(2)]; % dR/dt
% Initial conditions
Y0 = [N - I0; I0; 0]; % Initial conditions for S, I, R
% Time span
tmax1 = 100; % Define the maximum time (adjust as needed)
tspan = [0 tmax1];
% Solve the SIR model differential equations
[t, Y] = ode45(sir_eqns, tspan, Y0);
% Plot the results
figure;
plot(t, Y(:,1), 'b', t, Y(:,2), 'r', t, Y(:,3), 'g');
legend('Susceptible', 'Infected', 'Recovered');
xlabel('Time');
ylabel('Population');
title('SIR Model');
axis tight;
grid on;
% Assuming t and Y are obtained from the ode45 solver for the SIR model
% Extract the infected population data (second column of Y)
infected = Y(:,2);
% Plot the infected population over time
figure;
plot(t, infected, 'r');
legend('Infected');
xlabel('Time');
ylabel('Population');
title('Infected Population over Time');
grid on;
The code provides a visual representation of how the disease spreads and eventually diminishes within the population over the specified time interval . It can be used to understand the impact of different parameters (such as infection and recovery rates) on the progression of the outbreak .
The study of nonlinear dynamical systems in lattices is an area of research with continuously growing interest.The first systematic studies of these systems emerged in the late 1930 s,thanks to the work of Frenkel and Kontorova on crystal dislocations.These studies led to the formulation of the discrete Klein-Gordon equation (DKG).Specifically,in 1939,Frenkel and Kontorova proposed a model that describes the structure and dynamics of a crystal lattice in a dislocation core.The FK model has become one of the fundamental models in physics,as it has been proven to reliably describe significant phenomena observed in discrete media.The equation we will examine is a variation of the following form:
The process described involves approximating a nonlinear differential equation through the Taylor method and simplifying it into a linear model.Let's analyze step by step the process from the initial equation to its final form.For small angles,
can be approximated through the Taylor series as:
can be approximated through the Taylor series as:
We substitute 
in the original equation with the Taylor approximation:
in the original equation with the Taylor approximation:
To map this equation to a linear model,we consider the angles 
to correspond to displacements 
in a mass-spring system.Thus,the equation transforms into:
to correspond to displacements in a mass-spring system.Thus,the equation transforms into:
We recognize that the term 
expresses the nonlinearity of the system,while β is a coefficient corresponding to this nonlinearity,simplifying the expression.The final form of the equation is:
expresses the nonlinearity of the system,while β is a coefficient corresponding to this nonlinearity,simplifying the expression.The final form of the equation is:
The exact value of β depends on the mapping of coefficients in the Taylor approximation and its application to the specific physical problem.Our main goal is to derive results regarding stability and convergence in nonlinear lattices under nonlinear conditions.We will examine the basic characteristics of the discrete Klein-Gordon equation:
This model is often used to describe the opening of the DNA double helix during processes such as transcription.The model focuses on the transverse motion of the base pairs,which can be represented by a set of coupled nonlinear differential equations.
% Parameters
numBases = 50; % Number of base pairs
kappa = 0.1; % Elasticity constant
omegaD = 0.2; % Frequency term
beta = 0.05; % Nonlinearity coefficient
% Initial conditions
initialPositions = 0.01 + (0.02 - 0.01) * rand(numBases, 1);
initialVelocities = zeros(numBases, 1);
Time span
tSpan = [0 50];
>> % Differential equations
odeFunc = @(t, y) [y(numBases+1:end); ... % velocities
kappa * ([y(2); y(3:numBases); 0] - 2 * y(1:numBases) + [0; y(1:numBases-1)]) + ...
omegaD^2 * (y(1:numBases) - beta * y(1:numBases).^3)]; % accelerations
% Solve the system
[T, Y] = ode45(odeFunc, tSpan, [initialPositions; initialVelocities]);
% Visualization
plot(T, Y(:, 1:numBases))
legend(arrayfun(@(n) sprintf('Base %d', n), 1:numBases, 'UniformOutput', false))
xlabel('Time')
ylabel('Position')
title('Dynamics of DNA Base Pairs')
% Choose a specific time for the snapshot
snapshotTime = 10;
% Find the index in T that is closest to the snapshot time
[~, snapshotIndex] = min(abs(T - snapshotTime));
% Extract the solution at the snapshot time
snapshotSolution = Y(snapshotIndex, 1:numBases);
% Generate discrete plot for the DNA model at the snapshot time
figure;
stem(1:numBases, snapshotSolution, 'filled')
title(sprintf('DNA Model Displacement at t = %d', snapshotTime))
xlabel('Base Pair Index')
ylabel('Displacement')
% Time vector for detailed sampling
tDetailed = 0:0.5:50;
% Initialize an empty array to hold the data
data = [];
% Generate the data for 3D plotting
for i = 1:numBases
% Interpolate to get detailed solution data for each base pair
detailedSolution = interp1(T, Y(:, i), tDetailed);
% Concatenate the current base pair's data to the main data array
data = [data; repmat(i, length(tDetailed), 1), tDetailed', detailedSolution'];
end
% 3D Plot
figure;
scatter3(data(:,1), data(:,2), data(:,3), 10, data(:,3), 'filled')
xlabel('Base Pair')
ylabel('Time')
zlabel('Displacement')
title('3D Plot of DNA Base Pair Displacements Over Time')
colormap('rainbow')
colorbar
Lots of students like me have a break from school this week or next! If y'all are looking for something interesting to do learn a bit about using hgtransform by making the transforming snake animation in MATLAB!
Code below!
⬇️⬇️⬇️
numblock=24;
v = [ -1 -1 -1 ; 1 -1 -1 ; -1 1 -1 ; -1 1 1 ; -1 -1 1 ; 1 -1 1 ];
f = [ 1 2 3 nan; 5 6 4 nan; 1 2 6 5; 1 5 4 3; 3 4 6 2 ];
clr = hsv(numblock);
shapes = [ 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 % box
0 0 .5 -.5 .5 0 1 0 -.5 .5 -.5 0 1 0 .5 -.5 .5 0 1 0 -.5 .5 -.5 0 % fluer
0 0 1 1 0 .5 -.5 1 .5 .5 -.5 -.5 1 .5 .5 -.5 -.5 1 .5 .5 -.5 -.5 1 .5 % bowl
0 .5 -.5 -.5 .5 -.5 .5 .5 -.5 .5 -.5 -.5 .5 -.5 .5 .5 -.5 .5 -.5 -.5 .5 -.5 .5 .5]; % ball
% Build the assembly
set(gcf,'color','black');
daspect(newplot,[1 1 1]);
xform=@(R)makehgtform('axisrotate',[0 1 0],R,'zrotate',pi/2,'yrotate',pi,'translate',[2 0 0]);
P=hgtransform('Parent',gca,'Matrix',makehgtform('xrotate',pi*.5,'zrotate',pi*-.8));
for i = 1:numblock
P = hgtransform('Parent',P,'Matrix',xform(shapes(end,i)*pi));
patch('Parent',P, 'Vertices', v, 'Faces', f, 'FaceColor',clr(i,:),'EdgeColor','none');
patch('Parent',P, 'Vertices', v*.75, 'Faces', f(end,:), 'FaceColor','none',...
'EdgeColor','w','LineWidth',2);
end
view([10 60]);
axis tight vis3d off
camlight
% Setup vectors for animation
h=findobj(gca,'type','hgtransform')'; h=h(2:end);
r=shapes(end,:)*pi;
steps=100;
% Animate between different shapes
for si = 1:size(shapes,1)
sh = shapes(si,:)*pi;
diff = (sh-r)/steps;
% Animate to a new shape
for s=1:steps
arrayfun(@(tx)set(h(tx),'Matrix',xform(r(tx)+diff(tx)*s)),1:numblock);
view([s*360/steps 20]); drawnow();
end
r=sh;
for s=1:steps; view([s*360/steps 20]); drawnow(); end % finish rotate
end
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.
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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!!!
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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!
Several of the colormaps are great for a 256 color surface plot, but aren't well optimized for extracting m colors for plotting several independent lines. The issue is that many colormaps have start/end colors that are too similar or are suboptimal colors for lines. There are certainly many workarounds for this, but it would be a great quality of life to adjust that directly when calling this.
Example:
x = linspace(0,2*pi,101)';
y = [1:6].*cos(x);
figure; plot(x,y,'LineWidth',2); grid on; axis tight;
And now if I wanted to color these lines, I could use something like turbo(6) or gray(6) and then apply it using colororder.
colororder(turbo(6))
But my issue is that the ends of the colormap are too similar. For other colormaps, you may get lines that are too light to be visible against the white background. There are plenty of workarounds, with my preference being to create extra colors and truncate that before using colororder.
cmap = turbo(8); cmap = cmap(2:end-1,:); % Truncate the end colors
figure; plot(x,y,'LineWidth',2); grid on; axis tight;
colororder(cmap)
I think it would be really awesome to add some name-argument input pair to these colormaps that can specify the range you want so this could even be done inside the colororder calling if desired. An example of my proposed solution would look something like this:
cmap = turbo(6,'Range',[0.1 0.8]); % Proposed idea to add functionality
Where in this scenario, the resulting colormap would be 6 equally spaced colors that range from 10% to 80% of the total color range. This would be especially nice because you could more quickly modify the range of colors, or you could set the limits regardless of whether you need to plot 3, 6, or 20 lines.
I asked my question in the general forum and a few minutes later it was deleted. Perhaps this is a better place?
Rather than using my German regional forum (as I do not speak German), I want to ask questions in an international English-speaking forum. Presumably there should be an international English forum for everyone around the world, as English is the first or second language of everyone who has gone to school. Where is it?
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!
2 x 2 행렬의 행렬식은
- 행렬의 두 row 벡터로 정의되는 평행사변형의 면적입니다.
- 물론 두 column 벡터로 정의되는 평행사변형의 면적이기도 합니다.
- 좀 더 정확히는 signed area입니다. 면적이 음수가 될 수도 있다는 뜻이죠.
- 행렬의 두 행(또는 두 열)을 맞바꾸면 행렬식의 부호도 바뀌고 면적의 부호도 바뀌어야합니다.
일반적으로 n x n 행렬의 행렬식은
- 각 row 벡터(또는 각 column 벡터)로 정의되는 N차원 공간의 평행면체(?)의 signed area입니다.
- 제대로 이해하려면 대수학의 개념을 많이 가지고 와야 하는데 자세한 설명은 생략합니다.(=저도 모른다는 뜻)
- 더 자세히 알고 싶으시면 수학하는 만화의 '넓이 이야기' 편을 추천합니다.
- 수학적인 정의를 알고 싶으시면 위키피디아를 보시면 됩니다.
- 이렇게 생겼습니다. 좀 무섭습니다.
아래 코드는...
- 2 x 2 행렬에 대해서 이것을 수식 없이 그림만으로 증명하는 과정입니다.
- gif 생성에는 ScreenToGif를 사용했습니다. (gif 만들기엔 이게 킹왕짱인듯)
Determinant of 2 x 2 matrix is...
- An area of a parallelogram defined by two row vectors.
- Of course, same one defined by two column vectors.
- Precisely, a signed area, which means area can be negative.
- If two rows (or columns) are swapped, both the sign of determinant and area change.
More generally, determinant of n x n matrix is...
- Signed area of parallelepiped defined by rows (or columns) of the matrix in n-dim space.
- For a full understanding, a lot of concepts from abstract algebra should be brought, which I will not write here. (Cuz I don't know them.)
- For a mathematical definition of determinant, visit wikipedia.
- A little scary, isn't it?
The code below is...
- A process to prove the equality of the determinant of 2 x 2 matrix and the area of parallelogram.
- ScreenToGif is used to generate gif animation (which is, to me, the easiest way to make gif).
% 두 점 (a, b), (c, d)의 좌표
a = 4;
b = 1;
c = 1;
d = 3;
% patch 색 pre-define
lightgreen = [144, 238, 144]/255;
lightblue = [169, 190, 228]/255;
lightorange = [247, 195, 160]/255;
% animation params.
anim_Nsteps = 30;
% create window
figure('WindowStyle','docked')
ax = axes;
ax.XAxisLocation = 'origin';
ax.YAxisLocation = 'origin';
ax.XTick = [];
ax.YTick = [];
hold on
ax.XLim = [-.4, a+c+1];
ax.YLim = [-.4, b+d+1];
% create ad-bc patch
area = patch([0, a, a+c, c], [0, b, b+d, d], lightgreen);
p_ab = plot(a, b, 'ko', 'MarkerFaceColor', 'k');
p_cd = plot(c, d, 'ko', 'MarkerFaceColor', 'k');
p_ab.UserData = text(a+0.1, b, '(a, b)', 'FontSize',16);
p_cd.UserData = text(c+0.1, d-0.2, '(c, d)', 'FontSize',16);
area.UserData = text((a+c)/2-0.5, (b+d)/2, 'ad-bc', 'FontSize', 18);
pause
%% Is this really ad-bc?
area.UserData.String = 'ad-bc...?';
pause
%% fade out ad-bc
fadeinout(area, 0)
area.UserData.Visible = 'off';
pause
%% fade in ad block
rect_ad = patch([0, a, a, 0], [0, 0, d, d], lightblue, 'EdgeAlpha', 0, 'FaceAlpha', 0);
uistack(rect_ad, 'bottom');
fadeinout(rect_ad, 1, t_pause=0.003)
draw_gridline(rect_ad, ["23", "34"])
rect_ad.UserData = text(mean(rect_ad.XData), mean(rect_ad.YData), 'ad', 'FontSize', 20, 'HorizontalAlignment', 'center');
pause
%% fade-in bc block
rect_bc = patch([0, c, c, 0], [0, 0, b, b], lightorange, 'EdgeAlpha', 0, 'FaceAlpha', 0);
fadeinout(rect_bc, 1, t_pause=0.0035)
draw_gridline(rect_bc, ["23", "34"])
rect_bc.UserData = text(b/2, c/2, 'bc', 'FontSize', 20, 'HorizontalAlignment', 'center');
pause
%% slide ad block
patch_slide(rect_ad, ...
[0, 0, 0, 0], [0, b, b, 0], t_pause=0.004)
draw_gridline(rect_ad, ["12", "34"])
pause
%% slide ad block
patch_slide(rect_ad, ...
[0, 0, d/(d/c-b/a), d/(d/c-b/a)],...
[0, 0, b/a*d/(d/c-b/a), b/a*d/(d/c-b/a)], t_pause=0.004)
draw_gridline(rect_ad, ["14", "23"])
pause
%% slide bc block
uistack(p_cd, 'top')
patch_slide(rect_bc, ...
[0, 0, 0, 0], [d, d, d, d], t_pause=0.004)
draw_gridline(rect_bc, "34")
pause
%% slide bc block
patch_slide(rect_bc, ...
[0, 0, a, a], [0, 0, 0, 0], t_pause=0.004)
draw_gridline(rect_bc, "23")
pause
%% slide bc block
patch_slide(rect_bc, ...
[d/(d/c-b/a), 0, 0, d/(d/c-b/a)], ...
[b/a*d/(d/c-b/a), 0, 0, b/a*d/(d/c-b/a)], t_pause=0.004)
pause
%% finalize: fade out ad, bc, and fade in ad-bc
rect_ad.UserData.Visible = 'off';
rect_bc.UserData.Visible = 'off';
fadeinout([rect_ad, rect_bc, area], [0, 0, 1])
area.UserData.String = 'ad-bc';
area.UserData.Visible = 'on';
%% functions
function fadeinout(objs, inout, options)
arguments
objs
inout % 1이면 fade-in, 0이면 fade-out
options.anim_Nsteps = 30
options.t_pause = 0.003
end
for alpha = linspace(0, 1, options.anim_Nsteps)
for i = 1:length(objs)
switch objs(i).Type
case 'patch'
objs(i).FaceAlpha = (inout(i)==1)*alpha + (inout(i)==0)*(1-alpha);
objs(i).EdgeAlpha = (inout(i)==1)*alpha + (inout(i)==0)*(1-alpha);
case 'constantline'
objs(i).Alpha = (inout(i)==1)*alpha + (inout(i)==0)*(1-alpha);
end
pause(options.t_pause)
end
end
end
function patch_slide(obj, x_dist, y_dist, options)
arguments
obj
x_dist
y_dist
options.anim_Nsteps = 30
options.t_pause = 0.003
end
dx = x_dist/options.anim_Nsteps;
dy = y_dist/options.anim_Nsteps;
for i=1:options.anim_Nsteps
obj.XData = obj.XData + dx(:);
obj.YData = obj.YData + dy(:);
obj.UserData.Position(1) = mean(obj.XData);
obj.UserData.Position(2) = mean(obj.YData);
pause(options.t_pause)
end
end
function draw_gridline(patch, where)
ax = patch.Parent;
for i=1:length(where)
v1 = str2double(where{i}(1));
v2 = str2double(where{i}(2));
x1 = patch.XData(v1);
x2 = patch.XData(v2);
y1 = patch.YData(v1);
y2 = patch.YData(v2);
if x1==x2
xline(x1, 'k--')
else
fplot(@(x) (y2-y1)/(x2-x1)*(x-x1)+y1, [ax.XLim(1), ax.XLim(2)], 'k--')
end
end
end
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
Sie können auch eine Website aus der folgenden Liste auswählen:
Amerika
- América Latina (Español)
- Canada (English)
- United States (English)
Europa
- Belgium (English)
- Denmark (English)
- Deutschland (Deutsch)
- España (Español)
- Finland (English)
- France (Français)
- Ireland (English)
- Italia (Italiano)
- Luxembourg (English)
- Netherlands (English)
- Norway (English)
- Österreich (Deutsch)
- Portugal (English)
- Sweden (English)
- Switzerland
- United Kingdom(English)
Asien-Pazifik
- Australia (English)
- India (English)
- New Zealand (English)
- 中国
- 日本Japanese (日本語)
- 한국Korean (한국어)