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Elliptic Curve Explorer

Interactive 3D simulation of elliptic curves y² = x³ + ax + b: reshape the curve with sliders, pick points on it, and watch the chord-and-tangent group law construct P+Q geometrically, then stack repeated additions in 3D to reveal a point's order.

Mathematics3DAdvanced60 FPS

An elliptic curve in Weierstrass form, y² = x³ + ax + b, is more than a picture — its points carry a group structure: any two points on the curve can be "added" by drawing the line through them, finding where it crosses the curve a third time, and reflecting that crossing over the x-axis. This simulation lets you reshape the curve with a and b, place two points on it, and watch that chord-and-tangent construction happen step by step in 3D. A separate "trace nP" mode repeatedly adds a point to itself and stacks each result at increasing height, turning the search for a point's order — the torsion structure at the heart of the curve's group — into a shape you can rotate and inspect.

⚙ Under the hood

Interactive 3D exploration of the elliptic curve y² = x³ + ax + b: reshape the curve live with sliders on a and b, place points on it, and watch the chord-and-tangent group law construct P+Q geometrically, then stack repeated additions in 3D to reveal a point's order.

Three.jsmathematicselliptic curvesalgebranumber theory

3D · Three.js / WebGL renderer · 60 FPS target · runs fully client-side, no install