Question

Instructions: Solve using Python programming language. Submit a single script file

6) Using composition of functions, show how rotation and translation are not commutative transformations.

Answer 1

I have pasted the code and screenshot of sample run of code below. We have mainly 2 functions rotate2d and translate2d. We compute new point n1 by first applying translation and then rotatation and n2 by first applying rotation and then translation. We can see the picture below that the points n1 and n2 are not same so we can say that, rotation and translation are not commutative transformations.

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import numpy as np

sin = np.sin
cos = np.cos

def rotate2d(point,theta):
   # convert angle form degree to radians
   theta = theta/180*np.pi
   x,y = point
   new_point = [x*cos(theta) - y*sin(theta), x*sin(theta) + y*cos(theta)]
   return new_point

def translate2d(point,t):
   new_point = [point[0]+t[0], point[1]+t[1]]
   return new_point

thetas = [30,45,90]
translations = [[1,2],[2,2],[1,0]]
point = [1,1]

for theta,t in zip(thetas,translations):
   n1 = rotate2d(translate2d(point,t),theta)
   n2 = translate2d(rotate2d(point,theta),t)
   print("%s %s"%(str(n1),str(n2)))