"""Animated GIFs for the post "그래프 그리기 전에 적는 다섯 줄". usage: python make_gifs.py -> images/blog/sketch-graph-five-steps/*.gif five_steps.gif f(x) = (x+1)(x-2)(x-3) / ((x-1)(x+3)(x-3)), built up step by step asymptote_to_hole.gif f_eps(x) = (x-2+eps)(x+1) / ((x-2)(x-3)), eps -> 0 sign_chart.gif f(x) = (x+2)(x-1)^2(x-3), sign chart vs. curve """ from pathlib import Path import numpy as np import matplotlib matplotlib.use("Agg") import matplotlib.pyplot as plt from matplotlib.animation import PillowWriter OUT = Path(__file__).resolve().parents[3] / "images" / "blog" / "sketch-graph-five-steps" FPS = 12 HOLD_END = 24 # 2 s on the last frame INK, GREY, LIGHT = "#222222", "#8a8a8a", "#e6e6e6" BLUE, RED = "#1f5fa8", "#c8553d" plt.rcParams.update({ "font.family": "IBM Plex Sans KR", "mathtext.fontset": "dejavusans", "font.size": 11, "axes.edgecolor": "#bbbbbb", "axes.linewidth": 0.8, "xtick.color": GREY, "ytick.color": GREY, "xtick.labelsize": 9, "ytick.labelsize": 9, "figure.facecolor": "white", "axes.facecolor": "white", }) def branch_x(a, b, n=1500, dense=()): """Sample the open interval (a, b), refined near both ends and around `dense` points.""" t = np.logspace(-7, np.log10((b - a) / 2), 300) parts = [np.linspace(a, b, n), a + t, b - t] for c, w in dense: parts.append(np.linspace(c - w, c + w, 600)) x = np.unique(np.concatenate(parts)) return x[(x > a) & (x < b)] def cut(y, ylim): """Drop points far outside the window so no vertical line joins the two sides of a pole.""" y = np.array(y, dtype=float) y[np.abs(y) > 3 * ylim] = np.nan return y def base_axes(ax, xlim, ylim, xticks, yticks): ax.set_xlim(*xlim) ax.set_ylim(*ylim) ax.set_xticks(xticks) ax.set_yticks(yticks) ax.grid(color="#f0f0f0", lw=0.8) ax.set_axisbelow(True) ax.axhline(0, color=INK, lw=0.9, zorder=1) ax.axvline(0, color=INK, lw=0.9, zorder=1) ax.tick_params(length=2) def hole(ax, x, y, color=INK, size=7): ax.plot([x], [y], "o", ms=size, mfc="white", mec=color, mew=1.6, zorder=6) def dot(ax, x, y, color=INK, size=5.5): ax.plot([x], [y], "o", ms=size, color=color, zorder=5) def save(frames_fn, name, figsize): fig = plt.figure(figsize=figsize, dpi=100) writer = PillowWriter(fps=FPS) path = OUT / name n = 0 with writer.saving(fig, str(path), dpi=100): for state in frames_fn(): fig.clf() state(fig) writer.grab_frame() n += 1 plt.close(fig) print(f"{name}: {n} frames, {path.stat().st_size / 1024:.0f} KB") # ---------------------------------------------------------------- 1. five steps def g1(x): """Reduced form of (x+1)(x-2)(x-3) / ((x-1)(x+3)(x-3)).""" return (x + 1) * (x - 2) / ((x - 1) * (x + 3)) X1, Y1 = (-7, 7), (-6, 6) BR1 = [branch_x(X1[0], -3), branch_x(-3, 1), branch_x(1, X1[1])] STEPS = [ ("1 끝", [r"$x\to\pm\infty$ 에서 $y\to 1$"]), ("2 영점", [r"$x=-1,\ 2$ (홑근, 관통)"]), ("3 끊김", [r"구멍 $(3,\ 1/3)$", r"수직 점근선 $x=-3,\ 1$", r"수평 점근선 $y=1$"]), ("4 절편", [r"$y$절편 $(0,\ 2/3)$"]), ("5 부호", [r"$+\quad-\quad+\quad-\quad+$", r"경계: $-3,\ -1,\ 1,\ 2$"]), ] SIGNS1 = [(-5.0, "+"), (-2.0, "−"), (-0.5, "+"), (1.5, "−"), (4.6, "+")] CUTS1 = [X1[0], -3, -1, 1, 2, X1[1]] def draw_five(fig, step, front=None): """step = number of checklist lines shown (0..5); front = x up to which the curve is drawn.""" ax = fig.add_axes([0.06, 0.08, 0.53, 0.86]) base_axes(ax, X1, Y1, range(-6, 7, 2), range(-6, 7, 2)) hi = lambda k: RED if (k == step and front is None) else INK if step >= 1: # end stubs with arrows for a, b, d in [(-7.0, -5.6, -1), (5.6, 7.0, 1)]: xs = np.linspace(a, b, 40) ax.plot(xs, g1(xs), color=hi(1), lw=2.2, zorder=4) x0 = a if d < 0 else b ax.annotate("", xy=(x0, g1(x0)), xytext=(x0 - 0.25 * d, g1(x0 - 0.25 * d)), arrowprops=dict(arrowstyle="-|>", color=hi(1), lw=1.5), zorder=4) if step >= 2: # zeros with a short crossing piece for r in (-1, 2): xs = np.linspace(r - 0.3, r + 0.3, 30) ax.plot(xs, g1(xs), color=hi(2), lw=2.2, zorder=4) dot(ax, r, 0, hi(2)) ax.text(-1, -0.55, "−1", ha="center", va="top", fontsize=9, color=hi(2)) ax.text(2, -0.55, "2", ha="center", va="top", fontsize=9, color=hi(2)) if step >= 3: for a in (-3, 1): ax.axvline(a, color=hi(3), lw=1.1, ls=(0, (4, 3)), zorder=2) ax.axhline(1, color=hi(3), lw=1.1, ls=(0, (4, 3)), zorder=2) ax.text(-3.15, -5.6, "x = −3", ha="right", fontsize=9, color=hi(3)) ax.text(1.15, 5.5, "x = 1", ha="left", fontsize=9, color=hi(3)) ax.text(6.9, 1.25, "y = 1", ha="right", fontsize=9, color=hi(3)) if step >= 4: dot(ax, 0, 2 / 3, hi(4)) ax.text(-0.25, 1.25, "2/3", ha="right", fontsize=9, color=hi(4)) if step >= 5: for (a, b), (xm, s) in zip(zip(CUTS1[:-1], CUTS1[1:]), SIGNS1): up = s == "+" ax.fill_between([a, b], 0, 6 if up else -6, color=BLUE if up else RED, alpha=0.06, lw=0, zorder=0) ax.text(xm, -5.3 if not up else 5.2, s, ha="center", va="center", fontsize=15, color=BLUE if up else RED, zorder=3) if front is not None: for xs in BR1: m = xs <= front if m.any(): ax.plot(xs[m], cut(g1(xs[m]), Y1[1]), color=BLUE, lw=2.2, zorder=4) if step >= 3: hole(ax, 3, 1 / 3, hi(3)) fig.text(0.625, 0.90, r"$f(x)=\frac{(x+1)(x-2)(x-3)}{(x-1)(x+3)(x-3)}$", fontsize=13, color=INK, va="center") y = 0.78 for k, (head, lines) in enumerate(STEPS, start=1): c = (RED if k == step and front is None else INK) if k <= step else "#cccccc" fig.text(0.625, y, head, fontsize=12, color=c, weight="semibold", va="top") for ln in lines: fig.text(0.735, y, ln if k <= step else "", fontsize=10.5, color=c, va="top") y -= 0.055 y -= 0.03 if front is not None and front >= X1[1]: fig.text(0.625, 0.08, "곡선은 다섯 줄을 다 적은 뒤에 잇는다", fontsize=10, color=GREY) def frames_five(): for _ in range(10): yield lambda f: draw_five(f, 0) for s in range(1, 6): for _ in range(18): yield lambda f, s=s: draw_five(f, s) for fr in np.linspace(X1[0], X1[1], 36): yield lambda f, fr=fr: draw_five(f, 5, fr) for _ in range(HOLD_END): yield lambda f: draw_five(f, 5, X1[1]) # ------------------------------------------------------- 2. asymptote -> hole def fe(x, e): return (x - 2 + e) * (x + 1) / ((x - 2) * (x - 3)) X2, Y2 = (-3, 6.5), (-8, 8) def draw_eps(fig, e): ax = fig.add_axes([0.07, 0.08, 0.90, 0.88]) base_axes(ax, X2, Y2, range(-2, 7), range(-8, 9, 2)) ax.axvline(3, color=GREY, lw=1.1, ls=(0, (4, 3)), zorder=2) ax.axhline(1, color=GREY, lw=1.1, ls=(0, (4, 3)), zorder=2) ax.text(3.1, -7.6, "x = 3", fontsize=9, color=GREY) ax.text(6.4, 1.3, "y = 1", ha="right", fontsize=9, color=GREY) if e > 0: ax.axvline(2, color=RED, lw=1.2, ls=(0, (4, 3)), zorder=2) ax.text(2.1, 7.2, "x = 2", fontsize=9, color=RED) dense = [(2 - e, 4 * e)] branches = [branch_x(X2[0], 2, dense=dense), branch_x(2, 3), branch_x(3, X2[1])] for xs in branches: ax.plot(xs, cut(fe(xs, e), Y2[1]), color=BLUE, lw=2.0, zorder=4) dot(ax, 2 - e, 0, RED) status = [rf"$\varepsilon = {e:.3f}$", rf"영점 $x = 2-\varepsilon = {2 - e:.3f}$", r"수직 점근선 $x = 2$"] else: for xs in [branch_x(X2[0], 3), branch_x(3, X2[1])]: ax.plot(xs, cut((xs + 1) / (xs - 3), Y2[1]), color=BLUE, lw=2.0, zorder=4) hole(ax, 2, -3, RED, size=8) ax.annotate("구멍 (2, −3)", xy=(2, -3), xytext=(0.3, -5.6), fontsize=10, color=RED, arrowprops=dict(arrowstyle="-", color=RED, lw=0.8)) status = [r"$\varepsilon = 0$", r"$(x-2)$ 가 약분되어 $f=\frac{x+1}{x-3}$", r"$x=2$ 는 구멍"] dot(ax, -1, 0, INK) ax.text(-2.8, 7.1, r"$f_\varepsilon(x)=\frac{(x-2+\varepsilon)(x+1)}{(x-2)(x-3)}$", fontsize=13, color=INK, va="center") for i, s in enumerate(status): ax.text(-2.8, 5.4 - 1.05 * i, s, fontsize=11 if i else 13, color=RED if i == 0 else INK, va="center") def frames_eps(): for _ in range(14): yield lambda f: draw_eps(f, 1.0) for e in 10 ** np.linspace(0, -2, 50): yield lambda f, e=e: draw_eps(f, e) for _ in range(10): yield lambda f: draw_eps(f, 0.01) for _ in range(HOLD_END + 6): yield lambda f: draw_eps(f, 0.0) # --------------------------------------------------------------- 3. sign chart def p3(x): return (x + 2) * (x - 1) ** 2 * (x - 3) X3, Y3 = (-2.9, 3.9), (-22, 22) CUTS3 = [X3[0], -2, 1, 3, X3[1]] ROWS = [("x + 2", "−+++"), ("(x − 1)²", "++++"), ("x − 3", "−−−+")] FROW = "+−−+" def draw_signs(fig, k, front=None): """k = number of intervals already filled (0..4); front = x reached inside interval k.""" ax = fig.add_axes([0.14, 0.40, 0.83, 0.56]) base_axes(ax, X3, Y3, range(-2, 4), range(-20, 21, 10)) tb = fig.add_axes([0.14, 0.05, 0.83, 0.28], sharex=ax) tb.set_ylim(-0.6, 3.6) tb.set_yticks([3, 2, 1, 0]) tb.set_yticklabels([r[0] for r in ROWS] + ["f(x)"], fontsize=10.5, color=INK) tb.tick_params(axis="x", labelbottom=False, length=0) tb.tick_params(axis="y", length=0) for s in tb.spines.values(): s.set_visible(False) tb.axhline(0.5, color="#bbbbbb", lw=0.8) for c in (-2, 1, 3): ax.axvline(c, color=LIGHT, lw=1.0, zorder=0) tb.axvline(c, color="#cccccc", lw=0.8) mids = [0.5 * (a + b) for a, b in zip(CUTS3[:-1], CUTS3[1:])] for row, (_, signs) in enumerate(ROWS): for xm, s in zip(mids, signs): tb.text(xm, 3 - row, s, ha="center", va="center", fontsize=13, color=GREY) xs_all = branch_x(X3[0], X3[1], n=3000) ax.plot(xs_all, cut(p3(xs_all), Y3[1]), color=LIGHT, lw=2.0, zorder=2) for i in range(4): a, b = CUTS3[i], CUTS3[i + 1] col = BLUE if FROW[i] == "+" else RED if i == k and front is not None: ax.axvspan(a, b, color=col, alpha=0.07, lw=0, zorder=0) tb.axvspan(a, b, color=col, alpha=0.07, lw=0, zorder=0) if i < k or (i == k and front is not None): end = b if i < k else front xs = xs_all[(xs_all >= a) & (xs_all <= end)] ax.plot(xs, cut(p3(xs), Y3[1]), color=col, lw=2.4, zorder=4) if i < k or front >= b: tb.text(mids[i], 0, FROW[i], ha="center", va="center", fontsize=15, color=col, weight="semibold") for r, lab, dx, ha in [(-2, "관통", 0.12, "left"), (1, "접함", 0, "center"), (3, "관통", -0.12, "right")]: dot(ax, r, 0, INK) ax.text(r + dx, 2.5, lab, ha=ha, va="bottom", fontsize=9.5, color=INK) dot(ax, 0, -6, INK) ax.text(0.12, -7.5, "(0, −6)", fontsize=9, color=INK, va="top") ax.text(-2.8, 19, r"$f(x)=(x+2)(x-1)^2(x-3)$", fontsize=12, color=INK, va="top") if k == 4: ax.text(3.8, -19.5, "중복도 2인 x = 1 에서는 부호가 바뀌지 않는다", ha="right", fontsize=9.5, color=GREY) def frames_signs(): for _ in range(12): yield lambda f: draw_signs(f, 0) for k in range(4): a, b = CUTS3[k], CUTS3[k + 1] for fr in np.linspace(a, b, 10): yield lambda f, k=k, fr=fr: draw_signs(f, k, fr) for _ in range(8): yield lambda f, k=k, b=b: draw_signs(f, k, b) for _ in range(HOLD_END): yield lambda f: draw_signs(f, 4) if __name__ == "__main__": OUT.mkdir(parents=True, exist_ok=True) save(frames_five, "five_steps.gif", (7.2, 4.8)) save(frames_eps, "asymptote_to_hole.gif", (7.2, 4.6)) save(frames_signs, "sign_chart.gif", (7.2, 5.0))