Explore cubic curves and the remarkable elliptic curves. Discover the group law that makes them central to modern cryptography.
Moving beyond conics, we enter the fascinating world of cubic curves. The most important cubic curves are elliptic curves — smooth curves of the form y² = x³ + ax + b.
What makes elliptic curves special? They carry a natural group structure: you can "add" points on the curve to get new points. This remarkable property makes them essential for cryptography, number theory, and even proving Fermat's Last Theorem.
Every elliptic curve can be written in Weierstrass form: y² = x³ + ax + b. Adjust the parameters and watch the curve change. The discriminant tells you when the curve becomes singular.
Smooth elliptic curve
Watch what happens when an elliptic curve becomes singular. The curve can develop a node (self-crossing) or a cusp(sharp point). These singular curves are no longer elliptic curves!
No singular points — the curve is smooth everywhere.
As the parameter t decreases, the curve develops a singularity at the origin. At t = 0, we get a cusp. For t < 0, we get a nodewhere the curve crosses itself.
The chord-tangent method defines a way to "add" two points on an elliptic curve. Drag points P and Q to see how P + Q is computed. This operation makes the curve into a mathematical group!
This operation makes the curve into a group! It's associative, has an identity (point at infinity), and every point has an inverse.
Elliptic curves aren't just beautiful math — they power modern cryptography, helped prove Fermat's Last Theorem, and enable deep space communication.
Bitcoin, Ethereum, and most modern cryptocurrencies use elliptic curve cryptography for digital signatures.
Easy to compute P from n, but virtually impossible to find n from P!
Elliptic curves aren't just beautiful mathematics — they're essential tools in modern technology, from securing your Bitcoin wallet to proving 350-year-old theorems.
You've discovered the remarkable world of elliptic curves:
You've completed the Algebraic Geometry module! From simple curves to elliptic curves, from intersection theory to cryptographic applications — you've seen how polynomial equations create a bridge between algebra and geometry.