Deriving the addition equation for a Weierstrass curve is simple and straightforward (I started with this video that covers the simple case. If you know the basic derivative rules you can find the second case -- adding a point to itself -- easily).

How does one go about deriving the addition equation for an Untwisted Edwards Curve with a form of $x^2+y^2 = 1 +dx^2y^2$?

I know the formula itself is $(x_1,y_1)+(x_2,y_2)=(\frac{x_1y_2+x_2y_1}{1+dx_1x_2y_1y_2},\frac{y_1y_2-x_1x_2}{1-dx_1x_2y_1y_2})$

I have seen multiple proofs of correctness for this formula. I have also seen and understand the analogue of this with the unit circle.

But how do I derive this equation? If someone just handed me an Edwards curve and said "work out how to do point addition similar to point/angle addition on the unit circle" -- where would I begin?

I've seen a few sources reference Euler as the original one to discover it, but the papers referenced were in a language I don't speak and didn't have much context. -- Oh and I only have a basic, high school level understanding of calculus.


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