Adding points
on a curve
Before elliptic curves become the engine of modern cryptography, they are just a graceful loop with one surprising trick: you can add two points on the curve and land on a third. This exhibit introduces that trick slowly, by hand, over the ordinary real numbers.
Not an ellipse
An elliptic curve is the graph of a cubic in Weierstrass form. The name is a historical hand-me-down from the integrals that measure an ellipse's arc — the curve itself is this looping shape, symmetric across the x-axis.
The chord-and-tangent rule
To add points P and Q: draw the line through them, find the third point where it meets the curve, and reflect that across the x-axis. The reflection is P + Q. Tap two points and watch the construction draw itself.
Why an addition at all?
This rule isn't arbitrary — it turns the points of the curve into a group, with all the structure that word carries: an identity (the "point at infinity"), inverses (the reflection), and associativity. That group is exactly what cryptography moves into a finite field to build keys on.
Two doors from here
Follow this into Finite Fields to watch the smooth curve shatter into the scatter that makes the discrete log hard — the security payoff. Or follow it up into The Ground Beneath the Curve to meet the deep number theory of why these curves are so astonishingly well-behaved.
Applied in: ECDSA — this same chord-and-tangent group law, moved to a finite field, signing TLS certificates and blockchain transactions. The point addition you just drew by hand is what dG means.