What are the homogeneous coordinates of the point at infinity on elliptic curve?

Question

Elliptic curve in a projective plane has a structure of a group where the point at infinity serves as the identity element. However, I have seen different definitions of this point, so I would appreciate an advice.

1. On Wikipedia, the point at infinity is said to have coordinates $[0:1:0]$.
2. In all other articles I have found, the point at infinity is said to have coordinates $[x, y, 0]$.

I have basic knowledge about homogeneous coordinates, so maybe I am missing something here. Also, I am interested if the notations $[a:b:c]$ and $[a,b,c]$ are the same?

Thank you.

Answer 1

The projective plane $\Bbb P^2(K)$ over a field $K$ can be represented as union of

$$\{(x:y:1) \mid x,y \in K\}$$

with the hyperplane $H$ generated by $z=0$:

$$H = \{(x:y:0) \mid x,y \in K \text{ and not both zero}\}$$ Sometimes $H$ is dubbed "the points at infinity" or such.

In order to represent projective cubics in the affine plane, one identifies the affine point $(x/z, y/z)$ with the projective point $(x:y:z) = (x/z:y/z:1)$ provided $z\neq 0$.

In the case of an elliptic curve, the definig equation has exactly one solution with $z=0$, namely the point ${\cal O} = (0:1:0)$, dubbed the "point at infinity".

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I don't understand your last sentence — how does the $y^2=x^3+ax+b$ have "exactly one solution with $z=0$"? How does one conclude that it must be $x=0$ and $y=1$ if $z$ is $0$?

I projective form, the defining equation is an homogenious polynomial of degree 3 with some additional constraints. In the case of a short Weierstrass form:

$$E:\ y^2z=x^3 + axz^2 + bz^3$$

If $z=0$, this implies $x=0$ and thus the points have the form $(0:y:0)$. The is no restriction on $y$ except that $y\neq0$ as we are in projective space. Hence we may divide by $y$ which yields the one point $(0:1:0)$ without loss of generality.

The case of a short Weierstrass equation is a bit too restrictive, e.g. if $K$ has characteristic 2 or 3. But with more elaborate forms of homogenious polynomials of degree 3 with more terms like $yz^2$, $x^2z$ or $xyz$, it's the same outcome of one point at infinity. Depending on the exact normal form in use, the point at infinity might attain a different representation, though, see Point at Infinity of E.C. in Jacobian Coordinates for example.

I also found an answer to "What is the projective space?" in the context of Elliptic Curve Cryptography. It's just over finite fields but has some nice images and more explanation.