"""GIFs for the post on function transformations and inverse functions. usage: python make_gifs.py -> images/blog/function-transformations-and-inverses/*.gif Needs matplotlib + Pillow and the 'IBM Plex Sans KR' font. """ from pathlib import Path import numpy as np import matplotlib matplotlib.use("Agg") import matplotlib.pyplot as plt from matplotlib.animation import FuncAnimation, PillowWriter ROOT = Path(__file__).resolve().parents[3] OUT = ROOT / "images" / "blog" / "function-transformations-and-inverses" FPS = 14 HOLD_END = 28 # frames (2.0 s at 14 fps) INK, GREY, LIGHT = "#222222", "#8a8a8a", "#c8c8c8" ACCENT, ACCENT2 = "#c8553d", "#2f6690" plt.rcParams.update({ "font.family": "IBM Plex Sans KR", "mathtext.fontset": "dejavusans", "axes.unicode_minus": False, "font.size": 12, "figure.facecolor": "white", "axes.facecolor": "white", "savefig.facecolor": "white", }) def ease(t): t = np.clip(t, 0.0, 1.0) return 0.5 - 0.5 * np.cos(np.pi * t) def style_axes(ax, xlim, ylim, xticks, yticks): ax.set_xlim(*xlim) ax.set_ylim(*ylim) ax.set_aspect("equal") ax.set_xticks(xticks) ax.set_yticks(yticks) ax.grid(True, color="#eeeeee", lw=0.8, zorder=0) ax.axhline(0, color=GREY, lw=0.9, zorder=1) ax.axvline(0, color=GREY, lw=0.9, zorder=1) for s in ax.spines.values(): s.set_visible(False) ax.tick_params(colors="#555555", length=0, labelsize=10) def timeline(*segments): """segments: (name, nframes). Returns list of (name, local_t in [0,1]).""" frames = [] for name, n in segments: for i in range(n): frames.append((name, i / (n - 1) if n > 1 else 1.0)) return frames def save(anim, fig, name, nframes): OUT.mkdir(parents=True, exist_ok=True) path = OUT / name anim.save(path, writer=PillowWriter(fps=FPS)) plt.close(fig) print(f"{name}: {nframes} frames, {path.stat().st_size / 1024:.0f} KB") # (a) sqrt(x) -> sqrt(2x) -> sqrt(2x+3) = sqrt(2(x+1.5)), vs the wrong sqrt(2x+6) def gif_shift(): u = np.linspace(0, 16, 400) # parent-graph inputs, points (u, sqrt u) v = np.sqrt(u) feats = [(0.0, "끝점"), (4.0, None)] # tracked parent inputs fig, ax = plt.subplots(figsize=(7.2, 4.4), dpi=100) fig.subplots_adjust(left=0.06, right=0.98, bottom=0.08, top=0.80) style_axes(ax, (-4.2, 5.2), (-0.9, 3.3), range(-4, 6), range(0, 4)) ghost, = ax.plot(u, v, color=LIGHT, lw=1.6, zorder=2) wrong, = ax.plot([], [], color=GREY, lw=1.6, ls=(0, (4, 3)), zorder=2, alpha=0) wrong_pts = ax.scatter([], [], s=28, facecolor="white", edgecolor=GREY, lw=1.3, zorder=4, alpha=0) curve, = ax.plot(u, v, color=INK, lw=2.4, zorder=3) pts = ax.scatter([0, 4], [0, 2], s=40, color=ACCENT, zorder=5) lab = [ax.text(0, 0, "", fontsize=11, color=ACCENT, zorder=6, ha="center", va="bottom") for _ in feats] arrow = ax.annotate("", xy=(0, -0.45), xytext=(0, -0.45), arrowprops=dict(arrowstyle="->", color=ACCENT, lw=1.6)) arrow_txt = ax.text(0, -0.78, "", color=ACCENT, fontsize=11, ha="center") wrong_txt = ax.text(1.15, 0.78, "", color=GREY, fontsize=11, alpha=0, va="top", linespacing=1.5) wrong_lab = ax.text(-3.15, 0.15, "", color=GREY, fontsize=11, alpha=0, ha="right") title = fig.text(0.06, 0.92, "", fontsize=15, color=INK, va="center") sub = fig.text(0.06, 0.845, "", fontsize=11.5, color="#555555", va="center") frames = timeline(("show", 18), ("scale", 24), ("hold1", 16), ("shift", 24), ("hold2", 16), ("wrong", 18), ("end", HOLD_END)) def fmt(x): return f"{x:g}".replace("-", "−") def update(fr): name, t = fr k, s = 1.0, 0.0 if name == "scale": k = 1 - 0.5 * ease(t) elif name in ("hold1",): k = 0.5 elif name == "shift": k, s = 0.5, -1.5 * ease(t) elif name in ("hold2", "wrong", "end"): k, s = 0.5, -1.5 X = k * u + s curve.set_data(X, v) fx = [k * f + s for f, _ in feats] fy = [np.sqrt(f) for f, _ in feats] pts.set_offsets(np.c_[fx, fy]) for L, x, y, (_, tag) in zip(lab, fx, fy, feats): txt = f"({fmt(round(x, 2))}, {fmt(round(y, 2))})" L.set_text(txt) if y == 0: L.set_position((x - 0.15, y + 0.15)) L.set_ha("right") else: L.set_position((x, y + 0.18)) L.set_ha("center") if name in ("show",): title.set_text(r"$y=\sqrt{x}$") sub.set_text("기준 그래프. 끝점 (0, 0)과 점 (4, 2)를 따라간다") elif name in ("scale", "hold1"): title.set_text(r"$y=\sqrt{2x}$") sub.set_text("x 자리에 2x: 가로로 1/2배. 점 (4, 2)는 (2, 2)로") else: title.set_text(r"$y=\sqrt{2x+3}=\sqrt{2(x+1.5)}$") sub.set_text("x 자리에 x+1.5: 왼쪽으로 1.5. 3이 아니다") if name in ("shift", "hold2", "wrong", "end"): arrow.set_visible(True) arrow.xy = (s, -0.45) arrow.set_position((0.0, -0.45)) arrow_txt.set_text(f"{fmt(round(s, 2))}") arrow_txt.set_position((s / 2, -0.82)) else: arrow.set_visible(False) arrow_txt.set_text("") a = 0.0 if name == "wrong": a = ease(t) elif name == "end": a = 1.0 if a > 0: wrong.set_data(0.5 * u - 3, v) wrong_pts.set_offsets(np.c_[[-3, -1], [0, 2]]) wrong_txt.set_text("회색 점선: 가로 1/2배 후 왼쪽으로 3\n" + r"$=\sqrt{2(x+3)}=\sqrt{2x+6}$" + " (다른 함수)") wrong_lab.set_text("(−3, 0)") for artist in (wrong, wrong_pts, wrong_txt, wrong_lab): artist.set_alpha(a) return [] anim = FuncAnimation(fig, update, frames=frames, blit=False) save(anim, fig, "shift-order-sqrt.gif", len(frames)) # (b) horizontal line test: x^2 on R vs x^2 on x >= 0 def gif_hline(): fig, axes = plt.subplots(1, 2, figsize=(7.2, 4.5), dpi=100) fig.subplots_adjust(left=0.05, right=0.98, bottom=0.07, top=0.80, wspace=0.12) xs = np.linspace(-2.3, 2.3, 300) titles = [r"$y=x^2$, 모든 실수 $x$", r"$y=x^2$, $x\geq 0$"] lines, dots, counts = [], [], [] for i, ax in enumerate(axes): style_axes(ax, (-2.6, 2.6), (-1.0, 4.8), range(-2, 3), range(0, 5)) if i == 0: ax.plot(xs, xs**2, color=INK, lw=2.4, zorder=3) else: ax.plot(xs[xs <= 0], xs[xs <= 0]**2, color=LIGHT, lw=1.6, ls=(0, (4, 3)), zorder=2) ax.plot(xs[xs >= 0], xs[xs >= 0]**2, color=INK, lw=2.4, zorder=3) ax.scatter([0], [0], s=30, color=INK, zorder=4) ax.set_title(titles[i], fontsize=13, color=INK, pad=8) ln = ax.axhline(0, color=ACCENT2, lw=1.6, zorder=2) d = ax.scatter([], [], s=46, color=ACCENT, zorder=5) c = ax.text(0.97, 0.04, "", transform=ax.transAxes, fontsize=11.5, color=ACCENT, ha="right", va="bottom") lines.append(ln); dots.append(d); counts.append(c) head = fig.text(0.05, 0.93, "", fontsize=13, color=INK, va="center") frames = timeline(("up", 44), ("pause", 8), ("down", 26), ("end", HOLD_END)) c_lo, c_hi, c_end = -0.6, 4.4, 2.25 def update(fr): name, t = fr if name == "up": c = c_lo + (c_hi - c_lo) * ease(t) elif name == "pause": c = c_hi elif name == "down": c = c_hi + (c_end - c_hi) * ease(t) else: c = c_end head.set_text(f"수평선 y = {c:.2f}".replace("-", "−")) for i in range(2): lines[i].set_ydata([c, c]) if c < 0: xi = [] elif i == 0: r = np.sqrt(c) xi = [-r, r] if r > 1e-9 else [0.0] else: xi = [np.sqrt(c)] dots[i].set_offsets(np.c_[xi, [c] * len(xi)] if xi else np.empty((0, 2))) n = len(xi) counts[i].set_text(f"교점 {n}개") counts[i].set_color(ACCENT if n >= 2 else ACCENT2) if name == "end": head.set_text("y = 2.25: 왼쪽은 x = ±1.5 두 점, 오른쪽은 x = 1.5 한 점") return [] anim = FuncAnimation(fig, update, frames=frames, blit=False) save(anim, fig, "horizontal-line-test.gif", len(frames)) # (c) points of y = e^x reflected in y = x become y = ln x def gif_reflect(): fig, ax = plt.subplots(figsize=(7.2, 6.6), dpi=100) fig.subplots_adjust(left=0.08, right=0.97, bottom=0.06, top=0.88) lim = (-3.2, 4.4) style_axes(ax, lim, lim, range(-3, 5), range(-3, 5)) ax.plot(lim, lim, color=GREY, lw=1.2, ls=(0, (5, 4)), zorder=2) ax.text(3.55, 3.95, "y = x", color=GREY, fontsize=11, ha="right") a = np.linspace(-3.2, np.log(4.4), 300) b = np.exp(a) ax.plot(a, b, color=INK, lw=2.4, zorder=3) ax.text(1.12, 3.9, r"$y=e^{x}$", color=INK, fontsize=13, ha="right") moving, = ax.plot([], [], color=ACCENT, lw=1.4, alpha=0.5, zorder=3) final, = ax.plot([], [], color=ACCENT, lw=2.4, zorder=3) final_lab = ax.text(0.75, -1.5, "", color=ACCENT, fontsize=13) pa = np.array([-2.0, -1.0, 0.0, 0.5, 1.0, 1.3]) pb = np.exp(pa) ax.scatter(pa, pb, s=26, color=INK, zorder=4) segs = [ax.plot([], [], color=LIGHT, lw=1.0, ls=":", zorder=2)[0] for _ in pa] pts = ax.scatter(pa, pb, s=40, color=ACCENT, zorder=5) lab_src = [ax.text(0, 0, "", fontsize=10.5, color=INK, zorder=6) for _ in range(2)] lab_dst = [ax.text(0, 0, "", fontsize=10.5, color=ACCENT, zorder=6) for _ in range(2)] title = fig.text(0.08, 0.94, "", fontsize=13, color=INK, va="center") frames = timeline(("show", 16), ("move", 40), ("trace", 16), ("end", HOLD_END)) def update(fr): name, t = fr s = 0.0 if name == "show" else (ease(t) if name == "move" else 1.0) X = (1 - s) * pa + s * pb Y = (1 - s) * pb + s * pa pts.set_offsets(np.c_[X, Y]) for g, x0, y0, x1, y1 in zip(segs, pa, pb, X, Y): g.set_data([x0, x1], [y0, y1]) moving.set_data((1 - s) * a + s * b, (1 - s) * b + s * a) moving.set_visible(name == "move") if name == "show": title.set_text("점 (a, b)를 y = x에 대해 뒤집으면 (b, a)") elif name == "move": title.set_text("각 점은 y = x에 수직으로 움직이고, 중점은 y = x 위에 있다") else: title.set_text("뒤집힌 점들이 이루는 곡선: " + r"$y=\ln x$") on = name in ("trace", "end") final.set_data(b, a) if on else final.set_data([], []) final_lab.set_text(r"$y=\ln x$" if on else "") # coordinate labels for (0,1)<->(1,0) and (1,e)<->(e,1) e = np.e lab_src[0].set_text("(0, 1)"); lab_src[0].set_position((-0.12, 1.12)) lab_src[1].set_text("(1, 2.72)"); lab_src[1].set_position((0.85, e - 0.05)) for L in lab_src: L.set_ha("right") show_dst = name == "end" or (name == "trace") lab_dst[0].set_text("(1, 0)" if show_dst else ""); lab_dst[0].set_position((1.1, -0.42)) lab_dst[1].set_text("(2.72, 1)" if show_dst else ""); lab_dst[1].set_position((e + 0.1, 0.62)) return [] anim = FuncAnimation(fig, update, frames=frames, blit=False) save(anim, fig, "inverse-reflection-exp-ln.gif", len(frames)) if __name__ == "__main__": gif_shift() gif_hline() gif_reflect()