"""Gera as figuras do Capitulo 3 (Tensores no espaco-tempo plano). Uso: python3 cap03_figuras.py Salva os PDFs em ../figuras/. """ import numpy as np import matplotlib.pyplot as plt from pathlib import Path OUTDIR = Path(__file__).resolve().parent.parent / "figuras" OUTDIR.mkdir(exist_ok=True) plt.rcParams.update({ "font.family": "serif", "mathtext.fontset": "cm", "font.size": 12, "axes.linewidth": 0.9, }) # --------------------------------------------------------------------- # Figura 1: uma-forma como pilha de superficies (mesmo exemplo do texto: # omega=(2,1), V=(3,4), omega(V)=10) # --------------------------------------------------------------------- def fig_umaforma_pilha(): omega = np.array([2.0, 1.0]) V = np.array([3.0, 4.0]) fig, ax = plt.subplots(figsize=(5.2, 5.2)) # linhas de nivel de phi(x,y) = omega . (x,y) = k, para k inteiro xs = np.linspace(-1, 6, 50) k_vals = range(-2, 13) for k in k_vals: # 2x + y = k => y = k - 2x ys = k - omega[0] * xs dentro = (ys > -1.5) & (ys < 6.5) cor = "0.75" if k != 0 and k != 10 else "#B0413E" lw = 0.8 if k != 0 and k != 10 else 1.4 ax.plot(xs[dentro], ys[dentro], color=cor, lw=lw, zorder=1) ax.annotate("", xy=tuple(V), xytext=(0, 0), arrowprops=dict(arrowstyle="-|>", color="#4C72B0", lw=2.2), zorder=5) ax.plot([0], [0], "o", color="black", zorder=6, ms=4) ax.text(-0.55, -0.35, r"$\varphi=0$", color="#B0413E", fontsize=10) ax.text(V[0] + 0.15, V[1] + 0.35, r"$\varphi=10$", color="#B0413E", fontsize=10) ax.text(V[0] * 0.5 + 0.3, V[1] * 0.5 - 0.3, r"$V$", color="#4C72B0", fontsize=13) ax.set_xlim(-1, 5.5) ax.set_ylim(-1.5, 6.5) ax.set_aspect("equal") ax.set_xlabel("$x$") ax.set_ylabel("$y$") ax.set_title(r"Uma-forma $\omega=(2,1)$ como pilha de superfícies", fontsize=12) ax.text(0.03, 0.03, r"$\omega(V)=$ nº de superfícies perfuradas $=10$", transform=ax.transAxes, fontsize=9.5, bbox=dict(boxstyle="round", fc="white", ec="0.6")) fig.tight_layout() fig.savefig(OUTDIR / "cap03_umaforma_pilha.pdf", bbox_inches="tight") plt.close(fig) # --------------------------------------------------------------------- # Figura 2: base obliqua -- componentes contravariantes (paralelogramo) # vs. covariantes (projecao perpendicular) # --------------------------------------------------------------------- def fig_base_obliqua(): e1 = np.array([1.0, 0.0]) e2 = np.array([0.4, 1.0]) V = np.array([1.3, 0.9]) g = np.array([[e1 @ e1, e1 @ e2], [e2 @ e1, e2 @ e2]]) Vcontra = np.linalg.solve(np.array([e1, e2]).T, V) # V = V^1 e1 + V^2 e2 Vco = np.array([V @ e1, V @ e2]) # V_i = V . e_i fig, axs = plt.subplots(1, 2, figsize=(9.6, 4.8)) for ax, titulo in zip(axs, ["Componentes contravariantes $V^i$ (paralelogramo)", "Componentes covariantes $V_i$ (projeção perpendicular)"]): ax.annotate("", xy=tuple(e1), xytext=(0, 0), arrowprops=dict(arrowstyle="-|>", color="0.2", lw=1.8)) ax.annotate("", xy=tuple(e2), xytext=(0, 0), arrowprops=dict(arrowstyle="-|>", color="0.2", lw=1.8)) ax.text(*(e1 + [0.05, -0.13]), r"$\mathbf{e}_1$", fontsize=12) ax.text(*(e2 + [0.05, 0.03]), r"$\mathbf{e}_2$", fontsize=12) ax.annotate("", xy=tuple(V), xytext=(0, 0), arrowprops=dict(arrowstyle="-|>", color="#4C72B0", lw=2.2), zorder=5) ax.text(*(V + [0.03, 0.05]), r"$V$", color="#4C72B0", fontsize=13) # eixos (retas que contem e1 e e2), estendidos for e, cor in [(e1, "0.6"), (e2, "0.6")]: t = np.linspace(-0.5, 2.0, 10) pts = np.outer(t, e) ax.plot(pts[:, 0], pts[:, 1], color=cor, lw=0.7, ls=":", zorder=0) ax.set_xlim(-0.5, 1.8) ax.set_ylim(-0.5, 1.6) ax.set_aspect("equal") ax.set_title(titulo, fontsize=10.5) # --- painel 1: paralelogramo (contravariante) --- ax = axs[0] p1 = Vcontra[0] * e1 p2 = Vcontra[1] * e2 ax.plot([p1[0], V[0]], [p1[1], V[1]], color="#B0413E", lw=1.2, ls="--") ax.plot([p2[0], V[0]], [p2[1], V[1]], color="#B0413E", lw=1.2, ls="--") ax.plot(*p1, "o", color="#B0413E", ms=4) ax.plot(*p2, "o", color="#B0413E", ms=4) ax.text(p1[0], p1[1] - 0.15, rf"$V^1 e_1$", color="#B0413E", fontsize=9) ax.text(p2[0] - 0.35, p2[1] + 0.05, rf"$V^2 e_2$", color="#B0413E", fontsize=9) # --- painel 2: projecao perpendicular (covariante) --- ax = axs[1] for e, comp, cor in [(e1, Vco[0], "#B0413E"), (e2, Vco[1], "#55A868")]: ehat = e / np.linalg.norm(e) pe = (V @ ehat) * ehat ax.plot([pe[0], V[0]], [pe[1], V[1]], color=cor, lw=1.0, ls="--") ax.plot(*pe, "o", color=cor, ms=4) ax.text(V[0] - 0.55, -0.25, r"$V_1=V\!\cdot\!e_1$", color="#B0413E", fontsize=9) ax.text(0.15, 1.35, r"$V_2=V\!\cdot\!e_2$", color="#55A868", fontsize=9) fig.suptitle("Base oblíqua: duas maneiras de decompor o mesmo vetor $V$", fontsize=12, y=1.03) fig.tight_layout() fig.savefig(OUTDIR / "cap03_base_obliqua.pdf", bbox_inches="tight") plt.close(fig) return g, Vcontra, Vco # --------------------------------------------------------------------- # Figura 3: a base coordenada das polares, em tres pontos. O ponto e' que # ela muda de ponto a ponto -- em direcao E em comprimento -- embora o # plano seja o mesmo plano euclidiano de sempre. # --------------------------------------------------------------------- def fig_base_polar(): # Dois pontos no mesmo raio angular (r = 1 e 3) mostram o COMPRIMENTO # crescendo com r; o terceiro, noutro angulo, mostra a DIRECAO mudando. pontos = [(1.0, 0.25), (3.0, 0.25), (2.0, 0.75)] fig, ax = plt.subplots(figsize=(6.2, 5.6)) # linhas coordenadas: as curvas cujas tangentes sao os vetores de base for rc in (1.0, 2.0, 3.0, 3.6): t = np.linspace(0, np.pi / 2, 200) ax.plot(rc * np.cos(t), rc * np.sin(t), color="0.87", lw=0.8, zorder=0) for tc in np.linspace(0, np.pi / 2, 7): ax.plot([0, 3.9 * np.cos(tc)], [0, 3.9 * np.sin(tc)], color="0.87", lw=0.8, zorder=0) for k, (r, th) in enumerate(pontos): P = np.array([r * np.cos(th), r * np.sin(th)]) er = np.array([np.cos(th), np.sin(th)]) # |e_r| = 1 sempre et = np.array([-r * np.sin(th), r * np.cos(th)]) # |e_theta| = r ax.annotate("", xy=tuple(P + er), xytext=tuple(P), arrowprops=dict(arrowstyle="-|>", color="#B0413E", lw=2.0), zorder=5) ax.annotate("", xy=tuple(P + et), xytext=tuple(P), arrowprops=dict(arrowstyle="-|>", color="#4C72B0", lw=2.0), zorder=5) ax.plot(*P, "o", color="0.15", ms=4.5, zorder=6) ax.text(*(P + er + [0.05, -0.22]), r"$\vec e_r$", color="#B0413E", fontsize=11) ax.text(*(P + et + [-0.16, 0.10]), r"$\vec e_\theta$", color="#4C72B0", fontsize=11) ax.text(*(P + [-0.52, -0.30]), f"$r={r:g}$", color="0.35", fontsize=9.5) ax.set_xlim(-0.25, 4.0) ax.set_ylim(-0.25, 4.0) ax.set_aspect("equal") ax.set_xlabel("$x$") ax.set_ylabel("$y$") ax.set_title("A base coordenada das polares muda de ponto a ponto:\n" "direção e comprimento", fontsize=11) fig.tight_layout() fig.savefig(OUTDIR / "cap03_base_polar.pdf", bbox_inches="tight") plt.close(fig) if __name__ == "__main__": fig_umaforma_pilha() g, Vcontra, Vco = fig_base_obliqua() fig_base_polar() print("g =", g.tolist()) print("V^i =", Vcontra.tolist()) print("V_i =", Vco.tolist()) print("Figuras salvas em", OUTDIR)