1"""Figure: geometry of the elliptic-curve group law over the reals (illustrative; curve y² = x³ − 3x + 3)."""2import numpy as np, matplotlib3matplotlib.use("Agg")4import matplotlib.pyplot as plt, pathlib5ROOT = pathlib.Path(__file__).resolve().parents[1]6plt.rcParams.update({"font.family": "serif", "font.size": 9, "mathtext.fontset": "stix"})7a, b = -3.0, 3.08f = lambda x: x**3 + a * x + b9xs = np.linspace(-2.2, 2.6, 2000); ys = np.sqrt(np.clip(f(xs), 0, None)); m = f(xs) >= 010fig, axes = plt.subplots(1, 2, figsize=(7.0, 3.2))11for ax in axes:12 ax.plot(xs[m], ys[m], "k-", lw=1.2); ax.plot(xs[m], -ys[m], "k-", lw=1.2)13 ax.axhline(0, color="gray", lw=0.4); ax.axvline(0, color="gray", lw=0.4); ax.set_xlim(-2.3, 2.7); ax.set_ylim(-4.5, 4.5)14 ax.set_xlabel("$x$"); ax.set_ylabel("$y$")15# addition P + Q16xP, xQ = -1.6, 0.317P = (xP, np.sqrt(f(xP))); Q = (xQ, np.sqrt(f(xQ)))18lam = (Q[1] - P[1]) / (Q[0] - P[0]); x3 = lam**2 - P[0] - Q[0]; y3 = lam * (P[0] - x3) - P[1]19ax = axes[0]20ax.plot(xs, P[1] + lam * (xs - P[0]), "C0--", lw=0.9)21for pt, lab in ((P, "$P$"), (Q, "$Q$"), ((x3, -y3), "$-(P+Q)$"), ((x3, y3), "$P+Q$")):22 ax.plot(*pt, "o", color="C3", ms=4); ax.annotate(lab, pt, xytext=(4, 4), textcoords="offset points")23ax.plot([x3, x3], [y3, -y3], "C2:", lw=0.9); ax.set_title("Chord: $P+Q$")24# doubling25xP = -0.9; P = (xP, np.sqrt(f(xP))); lam = (3 * xP**2 + a) / (2 * P[1]); x3 = lam**2 - 2 * xP; y3 = lam * (xP - x3) - P[1]26ax = axes[1]27ax.plot(xs, P[1] + lam * (xs - P[0]), "C0--", lw=0.9)28for pt, lab in ((P, "$P$"), ((x3, -y3), "$-2P$"), ((x3, y3), "$2P$")):29 ax.plot(*pt, "o", color="C3", ms=4); ax.annotate(lab, pt, xytext=(4, 4), textcoords="offset points")30ax.plot([x3, x3], [y3, -y3], "C2:", lw=0.9); ax.set_title("Tangent: $2P$")31fig.suptitle("Group law on $y^2 = x^3 - 3x + 3$ over $\\mathbb{R}$ (illustrative)", fontsize=9)32fig.tight_layout()33out = ROOT / "figures/ec_geometry.pdf"; out.parent.mkdir(exist_ok=True); fig.savefig(out); fig.savefig(ROOT / "paper/figures/ec_geometry.pdf")34print(out)35