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# 14.2: Transformaciones afín

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En esta sección, vamos a explorar diferentes tipos de matrices de transformación. El siguiente código está diseñado para demostrar las propiedades de algunas matrices de transformación diferentes.

##### Hacer esto

Revisa el siguiente código.

#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');

### Ejemplo de matriz de escalado

#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)

### Ejemplo de Matriz de Traducción

#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)

### Ejemplo de matriz de reflexión

#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)

### Ejemplo de matriz de rotación

#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)

### Combinar Transformas

Tenemos cinco transformaciones$$R$$,$$S$$$$T$$,$$Re$$, y$$SH$$

##### Hacer esto

Construir una ($$3 \times 3$$) Matriz de transformación (llamada$$M$$) que combina estas cinco transformaciones en una sola matriz. Puedes elegir diferentes órdenes para estas cinco matrices, luego comparar tu resultado con otros alumnos.

#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');
##### Preguntas

¿Se pudo obtener el mismo resultado con otros? Se puede comparar la matriz$$M$$ para ver la diferencia. Si no, ¿puedes explicar por qué sucede?

### Ejemplo Interactivo

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));

También se puede usar el siguiente comando pero puede ser lento en las computadoras de algunas personas.

##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));
##### Hacer esto

Usando el ambiente interactivo anterior para ver si puedes averiguar la matriz de transformación para hacer la siguiente imagen:

##### Pregunta

¿Cuáles son los valores de entrada?

This page titled 14.2: Transformaciones afín is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Dirk Colbry via source content that was edited to the style and standards of the LibreTexts platform.