from manim import * import numpy as np import math def safe_text(text, max_width=10, font_size=28, color="#E8E6E1"): t = Text(text, font_size=font_size, color=color) if t.width > max_width: t.scale(max_width / t.width) return t class GeneratedScene(Scene): def construct(self): self.camera.background_color = "#0E1116" CORAL = "#FF6B5B" CYAN = "#4ECDC4" TEXT_C = "#E8E6E1" A0 = 1.6 left_center = np.array([-3.2, 0, 0]) x_start = 0.6 x_end = 6.4 period_width = 2.2 theta_tracker = ValueTracker(0) nh_tracker = ValueTracker(1) periods_tracker = ValueTracker(1) trace_op_tracker = ValueTracker(1.0) # ---- math helpers ---- def get_endpoint(): theta = theta_tracker.get_value() nh = int(nh_tracker.get_value()) pos = left_center.copy() for k in range(nh): n = 2 * k + 1 r = A0 / n ang = n * theta pos = pos + r * np.array([np.cos(ang), np.sin(ang), 0]) return pos def get_wave_y(t, nh): return sum((A0 / (2 * k + 1)) * np.sin((2 * k + 1) * t) for k in range(nh)) def get_draw_x(): theta = theta_tracker.get_value() periods = periods_tracker.get_value() max_theta = periods * 2 * np.pi draw_theta = min(theta, max_theta) return x_start + (draw_theta / (2 * np.pi)) * period_width def make_static_trace(nh, periods, color, opacity): max_theta = periods * 2 * np.pi n_samples = 300 pts = [] for i in range(n_samples + 1): t = max_theta * i / n_samples x = x_start + (t / (2 * np.pi)) * period_width y = get_wave_y(t, nh) pts.append(np.array([x, y, 0])) line = VMobject() line.set_points_smoothly(np.array(pts)) line.set_stroke(color, 2, opacity=opacity) return line # ---- always-redraw epicycles ---- def make_epicycles(): theta = theta_tracker.get_value() nh = int(nh_tracker.get_value()) group = VGroup() pos = left_center.copy() for k in range(nh): n = 2 * k + 1 r = A0 / n ang = n * theta op = max(0.35, 1.0 - 0.22 * k) circle = Circle(radius=r, color=CORAL, stroke_width=1.5) circle.move_to(pos) circle.set_stroke(opacity=op * 0.5) group.add(circle) new_pos = pos + r * np.array([np.cos(ang), np.sin(ang), 0]) vec = Line(pos, new_pos, color=CORAL, stroke_width=2.5) vec.set_stroke(opacity=op) group.add(vec) pos = new_pos tip = Dot(pos, radius=0.05, color=CYAN) group.add(tip) return group epicycles = always_redraw(make_epicycles) # ---- guide line ---- def make_guide(): ep = get_endpoint() dx = get_draw_x() line = DashedLine(ep, np.array([dx, ep[1], 0]), color=TEXT_C, stroke_width=1) line.set_stroke(opacity=0.35) return line guide = always_redraw(make_guide) # ---- waveform trace ---- def make_trace(): theta = theta_tracker.get_value() nh = int(nh_tracker.get_value()) periods = periods_tracker.get_value() max_theta = periods * 2 * np.pi draw_theta = min(theta, max_theta) if draw_theta < 0.01: return VMobject() n_samples = max(10, int(draw_theta / (2 * np.pi) * 250)) pts = [] for i in range(n_samples + 1): t = draw_theta * i / n_samples x = x_start + (t / (2 * np.pi)) * period_width y = get_wave_y(t, nh) pts.append(np.array([x, y, 0])) if len(pts) < 2: return VMobject() line = VMobject() line.set_points_smoothly(np.array(pts)) line.set_stroke(CYAN, 2.5) line.set_stroke(opacity=trace_op_tracker.get_value()) return line trace = always_redraw(make_trace) # ---- static frame elements ---- zero_line = DashedLine( np.array([x_start, 0, 0]), np.array([x_end, 0, 0]), color=TEXT_C, stroke_width=1) zero_line.set_stroke(opacity=0.15) divider = DashedLine( np.array([-0.8, -2.5, 0]), np.array([-0.8, 2.5, 0]), color=TEXT_C, stroke_width=1) divider.set_stroke(opacity=0.1) tick_marks = VGroup() for i in range(4): tx = x_start + i * period_width if tx <= x_end + 0.1: tk = Line( np.array([tx, -0.06, 0]), np.array([tx, 0.06, 0]), color=TEXT_C, stroke_width=1) tk.set_stroke(opacity=0.3) tick_marks.add(tk) self.add(zero_line, divider, tick_marks) # ================================================================ # SCENE 1 — Fundamental vector (6 s) # ================================================================ self.add(epicycles, guide, trace) label_1w = safe_text("1×ω", max_width=2, font_size=28, color=CORAL) label_1w.move_to(np.array([-3.2, 2.3, 0])) self.play(FadeIn(label_1w), run_time=0.8) self.wait(0.2) self.play(theta_tracker.animate.set_value(2 * np.pi), run_time=4, rate_func=linear) subtitle = safe_text("square wave = sum of odd sines", max_width=8, font_size=24, color=TEXT_C) subtitle.to_edge(DOWN, buff=0.5) self.play(FadeIn(subtitle), run_time=0.8) self.wait(0.5) # ================================================================ # SCENE 2 — 3rd harmonic (7 s) # ================================================================ ghost1 = make_static_trace(1, 1, CYAN, 0.2) self.add(ghost1) self.play(FadeOut(subtitle), FadeOut(label_1w), run_time=0.5) theta_tracker.set_value(0) nh_tracker.set_value(2) f2 = MathTex(r"+\,\frac{1}{3}\sin(3\omega)", font_size=30, color=TEXT_C) f2.move_to(np.array([1.8, 2.8, 0])) self.play(FadeIn(f2), run_time=0.6) self.wait(0.2) self.play(theta_tracker.animate.set_value(2 * np.pi), run_time=4, rate_func=linear) self.wait(0.8) # ================================================================ # SCENE 3 — 5th harmonic (7 s) # ================================================================ ghost2 = make_static_trace(2, 1, CYAN, 0.2) self.add(ghost2) self.play(FadeOut(ghost1), run_time=0.3) theta_tracker.set_value(0) nh_tracker.set_value(3) f3 = MathTex(r"+\,\frac{1}{5}\sin(5\omega)", font_size=30, color=TEXT_C) f3.next_to(f2, DOWN, buff=0.2) self.play(FadeIn(f3), run_time=0.6) self.wait(0.2) self.play(theta_tracker.animate.set_value(2 * np.pi), run_time=4, rate_func=linear) self.wait(0.8) # ================================================================ # SCENE 4 — 7th harmonic (7 s) # ================================================================ ghost3 = make_static_trace(3, 1, CYAN, 0.2) self.add(ghost3) self.play(FadeOut(ghost2), run_time=0.3) theta_tracker.set_value(0) nh_tracker.set_value(4) f4 = MathTex(r"+\,\frac{1}{7}\sin(7\omega)", font_size=30, color=TEXT_C) f4.next_to(f3, DOWN, buff=0.2) self.play(FadeIn(f4), run_time=0.6) self.wait(0.2) self.play(theta_tracker.animate.set_value(2 * np.pi), run_time=4, rate_func=linear) self.wait(0.8) # ================================================================ # SCENE 5 — Gibbs overshoot (6 s) # ================================================================ self.play(FadeOut(ghost3), run_time=0.3) periods_tracker.set_value(2) # compute overshoot numerically t_arr = np.linspace(0.01, np.pi, 5000) y_arr = np.array([ sum(np.sin((2 * k + 1) * t) / (2 * k + 1) for k in range(4)) for t in t_arr ]) peak_idx = int(np.argmax(y_arr)) peak_y_val = y_arr[peak_idx] * A0 limit_y_val = (np.pi / 4) * A0 highlight_box = Rectangle( width=1.4, height=1.6, color=CYAN, stroke_width=1.5, fill_opacity=0.06) highlight_box.move_to(np.array([x_start + 0.5, limit_y_val + 0.15, 0])) ref_line = DashedLine( np.array([x_start - 0.1, limit_y_val, 0]), np.array([x_start + 2.8, limit_y_val, 0]), color=TEXT_C, stroke_width=1) ref_line.set_stroke(opacity=0.4) overshoot_line = DashedLine( np.array([x_start - 0.1, peak_y_val, 0]), np.array([x_start + 2.8, peak_y_val, 0]), color=CORAL, stroke_width=1) overshoot_line.set_stroke(opacity=0.5) gibbs_label = safe_text("Gibbs overshoot", max_width=3, font_size=22, color=CYAN) gibbs_label.next_to(highlight_box, UP, buff=0.15) pct_label = safe_text("≈ 9% overshoot", max_width=3, font_size=20, color=TEXT_C) pct_label.next_to(gibbs_label, UP, buff=0.1) self.play(FadeIn(highlight_box), run_time=0.8) self.play(FadeIn(ref_line), FadeIn(overshoot_line), run_time=0.6) self.play(FadeIn(gibbs_label), FadeIn(pct_label), run_time=0.6) self.play(theta_tracker.animate.set_value(4 * np.pi), run_time=3.0, rate_func=linear) self.wait(0.5) self.play( FadeOut(highlight_box), FadeOut(ref_line), FadeOut(overshoot_line), FadeOut(gibbs_label), FadeOut(pct_label), run_time=0.5) # ================================================================ # SCENE 6 — Full composition (8 s) # ================================================================ # fade trace out, reset, fade grid + trace in, draw 2.5 periods self.play(trace_op_tracker.animate.set_value(0), run_time=0.4) theta_tracker.set_value(0) periods_tracker.set_value(2.5) grid = VGroup() for i in range(-3, 4): h = Line(np.array([-6.6, i, 0]), np.array([6.6, i, 0]), color=TEXT_C, stroke_width=0.5) h.set_stroke(opacity=0.06) grid.add(h) for i in range(-6, 7): v = Line(np.array([i, -3.5, 0]), np.array([i, 3.5, 0]), color=TEXT_C, stroke_width=0.5) v.set_stroke(opacity=0.06) grid.add(v) self.play(FadeIn(grid), trace_op_tracker.animate.set_value(1.0), run_time=1.0) self.play(theta_tracker.animate.set_value(5 * np.pi), run_time=5, rate_func=linear) self.wait(1.5) # ================================================================ # SCENE 7 — Final formula & hold (7 s) # ================================================================ # ease rotation to a halt at 6π (vectors aligned rightward) self.play(theta_tracker.animate.set_value(6 * np.pi), run_time=1.5, rate_func=smooth) self.play(FadeOut(f2), FadeOut(f3), FadeOut(f4), run_time=0.8) formula = MathTex( r"f(t) = ", r"\frac{4}{\pi}", r"\sum_{k=1}^{\infty}", r"\frac{\sin((2k-1)\,t)}{2k-1}", font_size=34 ) formula[0].set_color(TEXT_C) formula[1].set_color(CORAL) formula[2].set_color(TEXT_C) formula[3].set_color(TEXT_C) if formula.width > 11: formula.scale(11 / formula.width) formula.to_edge(DOWN, buff=0.6) k_label = MathTex(r"k = 1, 2, 3, \ldots", font_size=22, color=TEXT_C) k_label.next_to(formula, DOWN, buff=0.15) self.play(Write(formula), run_time=2.5) self.play(FadeIn(k_label), run_time=0.8) self.wait(2.5)