### 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

###
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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!