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.
Read more about this topic: Elliptic Curve
Famous quotes containing the words curves, real and/or numbers:
“One way to do it might be by making the scenery penetrate the automobile. A polished black sedan was a good subject, especially if parked at the intersection of a tree-bordered street and one of those heavyish spring skies whose bloated gray clouds and amoeba-shaped blotches of blue seem more physical than the reticent elms and effusive pavement. Now break the body of the car into separate curves and panels; then put it together in terms of reflections.”
—Vladimir Nabokov (1899–1977)
“The real price of everything, what everything really costs to the man who wants to acquire it, is the toil and trouble of acquiring it.”
—Adam Smith (1723–1790)
“What culture lacks is the taste for anonymous, innumerable germination. Culture is smitten with counting and measuring; it feels out of place and uncomfortable with the innumerable; its efforts tend, on the contrary, to limit the numbers in all domains; it tries to count on its fingers.”
—Jean Dubuffet (1901–1985)