Geometric interpretation of an Edwards curve

Question

Addition on an elliptic curve in Weierstrass form (over the rationals) is typically depicted with the following figure:

(Image CC SA 3.0 https://en.wikipedia.org/wiki/File:ECClines.svg)

To add two points, one draws the line that connects these points. The third intersection point is mirrored to get the result of the addition.

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A curve in Edwards form might look like this:

(Image CC SA 3.0 https://commons.wikimedia.org/wiki/File:Edward-curves.svg)

However, the classical geometric interpretation for addition on Weierstrass curves does not seem to work on these Edwards curves. Take for example the point $(0,-1)$. When doubled, this becomes $(0,1)$, the neutral point, according to the addition law $$(x_1, y_1) + (x_2, y_2) = \left(\frac{x_1y_2 + x_2y_1}{1-dx_1x_2y_1y_2}, \frac{y_1y_2 + x_1x_2}{1-dx_1x_2y_1y_2}\right).$$

When using the "classical" Weierstrass geometric interpretation (case 4 in the first image), I would become the point at infinity (which of course does not exist for an Edwards curve).

Clearly, Edwards curves follow a different way of life. Does there exist a similar geometric interpretation of the addition law for Edwards curves?

Answer 1

The normal form (later Edwards form) of an elliptic curve was first introduced by Harlod Edwards in his AMS bulletin by its addition law but gave no geometric interpretation. To give an interpretation of the addition law of two points $P$ and $Q$ you need a function $g_{P,Q}=\frac{f_1}{f_2}$ with $div(g_{P,Q})=(P)+(Q)-(\mathcal{O})-(P+Q)$ where $\mathcal{O}=(0,1)$ is the neutral element. The curve has degree 4, so it has $4\times deg(f)$ intersection points with the function $f$. We can choose $f_i$ to be quadratic functions to offer enough freedom of cancellation (8 intersections). Quadratic functions (conic sections) are determined by 5 points. Observing that points at infinity $\Omega_1 = (1:0:0)$ and $\Omega_2 = (0:1:0)$ are singular and have multiplicity 2, let us determine the conic by passing through $P$, $Q$, $(0,-1)$, $\Omega_1$ and $\Omega_2$. This let only one more intersection point $P+Q$.

(addition and doubling over $\mathbb{R}$ for $d<0$)

This was the first suggestion by Arène, Lange, Naehrig and Ritzenthaler to give a geometric interpretation of the addition law.

Answer 2

The geometric interpretation of the "addition law" on Edward Curves is not the same as for Weierstrass Curves.

The correct interpretation for this kind of curves is "adding their angles". It works as on a clock. Of course, as for the Weierstrass curves, the geometric interpretation stands for the curve over the real numbers and not in a finite field (useful for cryptography).

You can give a look to the ECCHacks: a gentle introduction to elliptic-curve cryptography (starting at page 6) by Daniel J. Bernstein and Tanja Lange