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 intersections 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.

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.

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 intersections point $P+Q$.

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

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 intersections 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.