Bilinear Form: Weil pairing I am doing an example on Weil pairings, and for that purpose I follow the thesis of Alex Edward Aftuck, The Weil Pairing on Elliptic Curves and Its Cryptographic Applications.

By following his thesis, on page #39 he selects 4 points on the elliptic curve  $Y^2=X^3+2X^2-3X$ and calculates six divisors from it 
as shown in the figure:

Using divisors, he calculated four rational functions as  but in the last step, he calculates the Weil pairing between $P_1 $ and $P_3$. The sums of points are $P_3+S=(-2.496,-2.047) $ and $P_1-S=(20.798,-98.990) $.

My question is: how can I can put the points into the functions $f_{p_1} $ and $f_{p_3} $?  All of the functions have three variables $X,Y,Z$, but we have  only two points. What is the value of $Z$, and how can I calculate functions using these points? Please help me as I'm really confused.
See page 17 of the thesis you linked to. There he mentions affine coordinates and homogenization. For your points, $Z=1$. So just set $Z=1$ in your functions above.


  To find the poles, we must remember that as a rational
  function, $Y = Y/1$. Homogenizing, we have $f(X, Y, Z) = Y/Z$. Then to find the poles
  of $f$, we must analyze $Z$ as a polynomial. For all affine points on $E$, $Z = 1$, so $Z = 0$
  only at $\mathcal O$.


Here's more about projective coordinates.

Keep in mind that in general, the sum of two points in projective coordinates having $Z=1$ is not a point with $Z=1$. In this case, the author already did the sums $P_3 + S$ and $P_1 - S$ in affine coordinates.