14.2: Affine Transforms

In this section, we are going to explore different types of transformation matrices. The following code is designed to demonstrate the properties of some different transformation matrices.

Do This

Review the following code.

#Some python packages we will be using
%matplotlib inline
import numpy as np
import matplotlib.pylab as plt
from mpl_toolkits.mplot3d import Axes3D #Lets us make 3D plots
import numpy as np
import sympy as sym
sym.init_printing(use_unicode=True) # Trick to make matrixes look nice in jupyter

# Define some points
x = [0.0,  0.0,  2.0,  8.0, 10.0, 10.0, 8.0, 4.0, 3.0, 3.0, 4.0, 6.0, 7.0, 7.0, 10.0, 
     10.0,  8.0,  2.0, 0.0, 0.0, 2.0, 6.0, 7.0,  7.0,  6.0,  4.0,  3.0, 3.0, 0.0]
y = [0.0, -2.0, -4.0, -4.0, -2.0,  2.0, 4.0, 4.0, 5.0, 7.0, 8.0, 8.0, 7.0, 6.0,  6.0,
     8.0, 10.0, 10.0, 8.0, 4.0, 2.0, 2.0, 1.0, -1.0, -2.0, -2.0, -1.0, 0.0, 0.0]
con = [ 1.0 for i in range(len(x))] 

p = np.matrix([x,y,con])


mp = p.copy()

#Plot Points
plt.plot(mp[0,:].tolist()[0],mp[1,:].tolist()[0], color='green');
plt.axis('scaled');
plt.axis([-10,20,-15,15]);
plt.title('Start Location');

Example Scaling Matrix

#Example Scaling Matrix

#Define Matrix
scale = 0.5  #The amount that coordinates are scaled.
S = np.matrix([[scale,0,0], [0,scale,0], [0,0,1]])

#Apply matrix

mp = p.copy()
mp = S*mp

#Plot points after transform
plt.plot(mp[0,:].tolist()[0],mp[1,:].tolist()[0], color='green')
plt.axis('scaled')
plt.axis([-10,20,-15,15])
plt.title('After Scaling')

#Uncomment the next line if you want to see the original.
# plt.plot(p[0,:].tolist()[0],p[1,:].tolist()[0], color='blue',alpha=0.3);

sym.Matrix(S)

Example Translation Matrix

#Example Translation Matrix

#Define Matrix
dx = 1  #The amount shifted in the x-direction
dy = 1  #The amount shifted in the y-direction
T = np.matrix([[1,0,dx], [0,1,dy], [0,0,1]])

#Apply matrix

mp = p.copy()

mp = T*mp

#Plot points after transform
plt.plot(mp[0,:].tolist()[0],mp[1,:].tolist()[0], color='green')
plt.axis('scaled')
plt.axis([-10,20,-15,15])
plt.title('After Translation')

#Uncomment the next line if you want to see the original.
# plt.plot(p[0,:].tolist()[0],p[1,:].tolist()[0], color='blue',alpha=0.3);

sym.Matrix(T)

Example Reflection Matrix

#Example Reflection Matrix

#Define Matrix
Re = np.matrix([[1,0,0],[0,-1,0],[0,0,1]]) ## Makes all y-values opposite so it reflects over the x-axis.

#Apply matrix

mp = p.copy()

mp = Re*mp

#Plot points after transform
plt.plot(mp[0,:].tolist()[0],mp[1,:].tolist()[0], color='green')
plt.axis('scaled')
plt.axis([-10,20,-15,15])

#Uncomment the next line if you want to see the original.
# plt.plot(p[0,:].tolist()[0],p[1,:].tolist()[0], color='blue',alpha=0.3);

sym.Matrix(Re)

Example Rotation Matrix

#Example Rotation Matrix

#Define Matrix
degrees = 30
theta = degrees * np.pi / 180  ##Make sure to always convert from degrees to radians. 

# Rotates the points 30 degrees counterclockwise.
R = np.matrix([[np.cos(theta),-np.sin(theta),0],[np.sin(theta), np.cos(theta),0],[0,0,1]]) 

#Apply matrix
mp = p.copy()

mp = R*mp

#Plot points after transform
plt.plot(mp[0,:].tolist()[0],mp[1,:].tolist()[0], color='green')
plt.axis('scaled')
plt.axis([-10,20,-15,15])

#Uncomment the next line if you want to see the original.
# plt.plot(p[0,:].tolist()[0],p[1,:].tolist()[0], color='blue',alpha=0.3);

sym.Matrix(R)

Example Shear Matrix

Combine Transforms

We have five transforms \(R\), \(S\), \(T\), \(Re\), and \(SH\)

Do This

Construct a (\(3 \times 3\)) transformation Matrix (called \(M\)) which combines these five transforms into a single matrix. You can choose different orders for these five matrix, then compare your result with other students.

#Put your code here

#Plot combined transformed points
mp = p.copy()
mp = M*mp
plt.plot(mp[0,:].tolist()[0],mp[1,:].tolist()[0], color='green');
plt.axis('scaled');
plt.axis([-10,20,-15,15]);
plt.title('Start Location');

Questions

Did you can get the same result with others? You can compare the matrix \(M\) to see the difference. If not, can you explain why it happens?

Interactive Example

from ipywidgets import interact,interact_manual

def affine_image(angle=0,scale=1.0,dx=0,dy=0, shx=0, shy=0):
    theta = -angle/180  * np.pi
    
    plt.plot(p[0,:].tolist()[0],p[1,:].tolist()[0], color='green')
    
    S = np.matrix([[scale,0,0], [0,scale,0], [0,0,1]])
    SH = np.matrix([[1,shx,0], [shy,1,0], [0,0,1]])
    T = np.matrix([[1,0,dx], [0,1,dy], [0,0,1]])
    R = np.matrix([[np.cos(theta),-np.sin(theta),0],[np.sin(theta), np.cos(theta),0],[0,0,1]])
    
    #Full Transform
    FT = T*SH*R*S;
    #Apply Transforms
    p2 =  FT*p;
    
    #Plot Output
    plt.plot(p2[0,:].tolist()[0],p2[1,:].tolist()[0], color='black')
    plt.axis('scaled')
    plt.axis([-10,20,-15,15])
    return sym.Matrix(FT)

##TODO: Modify this line of code
interact(affine_image, angle=(-180,180), scale_manual=(0.01,2), 
         dx=(-5,15,0.5), dy=(-15,15,0.5), shx = (-1,1,0.1), shy = (-1,1,0.1));

The following command can also be used but it may be slow on some peoples computers.

##TODO: Modify this line of code
#interact(affine_image, angle=(-180,180), scale=(0.01,2),
#			dx=(-5,15,0.5), dy=(-15,15,0.5), shx = (-1,1,0.1), shy = (-1,1,0.1));

Do This

Using the above interactive enviornment to see if you can figure out the transformation matrix to make the following image:

Question

What where the input values?