I'm trying to compute the optical phenomenon called Gravitational Lensing. In simple words its when a massive object (or with massiva mass) its between me as an observer and a star or some clase of light source. Because its massive mass the light will bend and for us it will apparently come from another location than it real position. There is a particular case (and simpler) where we suppose the mass is spheric, so from our perspective its circular in a 2D plane (or photo).
My idea for code that was changing the coordinates of a 2D plane in function of where my source light its. In other words, if I have a spheric light source, if it is far from my massive object it will image no change, but if its close to te spheric mass it will change (in fact, if its exactly behind the massive object I as an observer will see the called Einstein Ring).
For compute that I first write a mapping of this function. I take the approximation of a = x + sin(t)/exp(x) , b = y + cos(t)/exp(y). So when the source light its far from the mass, the exponential will be approximately zero, and if it is just behind the mass the source light coordinates will be (0,0), so the imagen will return (sin(t),cos(t)) the Einstein circle I was expected to get.
I code that in this way, first I define my approximation:
def coso1(x,y):
t = arange(0,2*pi, .01);
a = x + sin(t)/exp(x)
b = y + cos(t)/exp(y)
plt.plot(a,b)
plt.show()
Then I try to plot that to see how the coordinate map is changing:
from numpy import *
from matplotlib.pyplot import *
x=linspace(-10,10,10)
y=linspace(-10,10,10)
y = y.reshape(y.size, 1)
x = x.reshape(x.size, 1)
plot(coso1(x,y))
And I get this plot.
Notice that it looks that way because the intervale I choose to take values for the x and y coordinates. If I take place in the "frontier" case where x={-1,0,1} and y={-1,0,1} it will show how the space its been deformed (or I'm guessing thats what Im seeing).
I then have a few questions. An easy question but that I hadnt find an easy answer its if I can manipulate this transformation (rotate with the mouse to aprecciate the deformation, a controller of how x or y change). And the two hard questions: Can I plot the countour lines to see how exactly are changing the topography of my map in every level of x (suppose I let y be constant), and the other question: If this is my "new" way of how the map is acting, can I use this new coordinate map as a tool where If a project any image it will be distorted in function of this "new" map. Something analogous of how cameras works with fish lens effect.