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How to Draw a Happy Sheep in MATLAB!

Victoria on 29 Feb 2024 (Edited on 29 Feb 2024)

Latest activity Reply by David on 1 Mar 2024

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 a21 = 0

and

. If we would like to create a rose, we will set a11 = 0

and

. If we would like to shift our curve, we will set a31

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 2pi

, with the step 0.1. The result of this code is the following ellipse curve with a11 = 5

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 a21 = 4

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 a11

, 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!