Elliptic Curve - Elliptic Curves Over The Real Numbers

Elliptic Curves Over The Real Numbers

Although the formal definition of an elliptic curve is fairly technical and requires some background in algebraic geometry, it is possible to describe some features of elliptic curves over the real numbers using only high school algebra and geometry.

In this context, an elliptic curve is a plane curve defined by an equation of the form

where a and b are real numbers. This type of equation is called a Weierstrass equation.

The definition of elliptic curve also requires that the curve be non-singular. Geometrically, this means that the graph has no cusps, self-intersections, or isolated points. Algebraically, this involves calculating the discriminant

The curve is non-singular if and only if the discriminant is not equal to zero. (Although the factor −16 seems irrelevant here, it turns out to be convenient in a more advanced study of elliptic curves.)

The (real) graph of a non-singular curve has two components if its discriminant is positive, and one component if it is negative. For example, in the graphs shown in figure to the right, the discriminant in the first case is 64, and in the second case is −368.