So my questions are:
1. Is it possible to know when the point addition process has gone outside of the field?
And
2. Is there a way (like a Python script or something) that I could use to add 2 points and it would tell me if the result went outside of the field?
I know that the result of the addition is still a valid point on the curve, but I would love to know if it passed out of bounds first.
This would be fantastic for my project to show others how this works.
I hope that made sense and thanks in advance for your help.
The point at infinity is not "outside the field", since by definition it is included.
For secp256k1 adding two points results in the infinity point (zero), if and only if the x coordinates are equal, and y coordinates differ: x1 = x2, and y1 ≠ y2. In jacobian coordinates you'd need to check some intermediate calculations for equality: x1*z2^2 = x2*z1^2, and y1*z2^3 ≠ y2*z1^3.
Usually the point at infinity is denoted as (0,0), or (0,0,1) in jacobian coordinates. Check if y is zero.