A -rational point is a point on an algebraic curve , where and are in a field . For example, rational point in the field of ordinary rational numbers is a point
satisfying the given equation such
that both
and
are rational numbers.

and homogenize it by introducing a third variable so that each term has degree 3 as follows:

(2)

Now, find the points at infinity by setting , obtaining

(3)

Solving gives ,
equal to any value, and (by definition)
. Despite freedom in the choice of
, there is only a single point
at infinity because the two triples (, , ), (, , ) are considered to be equivalent (or identified) only if
one is a scalar multiple of the other. Here, (0, 0, 0) is not considered to be a
valid point. The triples (, ,
1) correspond to the ordinary points (, ),
and the triples (,
, 0) correspond to the points
at infinity, usually called the line at infinity.