## Monday, 13 January 2014

### Problem 3 formulation:

Given a line described by ax+by+c=0, create a function script with the values a,b,c as input that prints:

1-The affine transformation matrix obtained by creating a symmetry respect to the line

2-A plot of a circle with 1 unit radius centered on (0,0) and its reflection

### Solution:

##### Libraries #####

from numpy import *

import math as mt # Mistake corrected

import pyqtgraph as pg

pg.setConfigOption('background','w') # Changes plot window background to white

####################

def tafin(a,b,c):       # function declaration
window=pg.plot(title="Reflection") # Creates plot window called window
Ax=linspace(-5,5)    # Generates values from -5 to 5
Ay=-a*Ax/b-c/b       # Line equation
window.plot(Ax,Ay)       # Plots line
alpha=linspace(0,2*pi) # Generates alpha values
Bx=cos(alpha)          # X coordinates of the (0,0) centered circumference
By=sin(alpha)          # Y coordinates of the (0,0) centered circumference
window.plot(Bx,By,pen='r') # Plots (0,0) centered circumference in red colour
B1s=ones((1,size(Bx)))        # Line vector filled with 1's
F=vstack((Bx,By,B1s)) # Compounds an array from Bx,By and B1s
XA=0     # random point
YA=(-a/b)*XA-(c/b)     # Obtains one point of the symetry line
XB=1     # another random point
YB=(-a/b)*XB-(c/b)     # Obtains another point of the symetry line
# Symetry matrix
Ta1=array(([XA,XB,-a],[YA,YB,-b],[1,1,0])) # Mirrored points coincident with line
Ta2=linalg.inv(array(([XA,XB,a],[YA,YB,b],[1,1,0])))  # Original line coincident points
Ta=dot(Ta1,Ta2) # Symetry matrix
S=dot(Ta,F)     # Mirrors the circumference coordinates stored in F
window.plot(S[0,:],S[1,:],pen='b') # Plots mirrored circumference in blue
print "The affine transformation matix is"  #
print Ta   # prints affine transformation matrix

Open a terminal, paste the function and then type something like tafin(2.0,1.0,4.0)  the output should be very similar to the screenshot.

Solving this problem with python has allowed me to understand more things about this language, for example, I used to import math as * alongside with numpy as *, this should not be done. The first thing where you can see how that affects to coding is the absence of "for" bucles to operate within arrays and their elements.
This clearly makes the code easier, cleaner and more efficient and shorter than the matlab equivalent.

The new thing with the pyqtgraph module are the color plots, and plot background, easy to perform.

With this problem I finish this little section about algebra and python, hope that you got something good from it.

Bye!