Elliptic Curve Explorer

The 3D form is the real surface of the homogenization of the curve.

We begin with an algebraic curve; in this case, say y^2 = x^3 - x. The equation defining this curve is degree 3 but inhomogeneous, meaning that not all monomials have the same total degree.

In order to reveal hidden "structure at infinity" (essential for defining and studying the addition law on elliptic curves), we perform so-called homogenization. In our example, we take the inhomogeneous equation y^2 = x^3 - x and introduce a third variable z to balance out the degrees on all monomials leading to the equation y^2 z = x^3 - x z^2; now each of the three monomials has total degree 3. This equation can be thought of as defining a homogeneous surface in real 3D space, or from a more advanced perspective, a homogeneous curve in projective space.

The original curve lies at the slice z=1 of this homogeneous surface, and the entire surface can be built by emitting all possible lines from the origin connecting to points on that sliced curve.

Geometrically, if you wiggle the model around you can see this structure: the plane z=1 slicing through the surface, cutting out the curve, and lines through the points of the curve tracing out the entire surface.

Algebraically, if you have a point (a,b,c) on the surface xz=y^2, that is, satisfying ac=b^2, the entire parametrized line (ra,rb,rc) satisfies rarb=(rc)^2, and so lies on the surface.

There is not much to illustrate regarding the operation itself: pick two points on the curve P and Q, draw the line through them, find the third point of intersection with the curve (which may be the point at infinity, in which case P+Q=0) and reflect it through the x-axis; that is your sum. I think Wikipedia has this visual description in the Group Law subsection of elliptic curves. Now visualizing properties of the group law convincingly, such as commutativity and associativity, that is a challenge!

I am not sure what you are looking for in a subtraction. Like in all groups P-Q is perfectly well defined as P+(-Q), so as before: first reflect the point Q through the x axis to get -Q, and then connect that point to P to get -(P+(-Q)), and do one final reflection to get P+(-Q) which is the only consistent interpretation of P-Q.