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# -*- coding: utf-8 -*-
"""The brachistochrone: the straight line loses to the cycloid.
All motion comes from real physics (frictionless free fall)."""
import sys, os, math
sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))
from manim import *
from tiktok import *
TH1 = 4.0553 # final angle: gives D/H = 3, the curve dips below
DX = 7.0
RR = DX / (TH1 - math.sin(TH1))
HH = RR * (1 - math.cos(TH1))
X0, Y0 = -3.5, 3.15
T_C = 1.55 # how long the cycloid takes on screen
G = RR * (TH1 / T_C) ** 2 # gravity tuned to that time
LL = math.hypot(DX, HH)
AA = G * HH / LL
T_L = math.sqrt(2 * LL / AA) # 2.18 s: the straight line arrives 40% later
P0 = np.array([X0, Y0, 0.0])
P1 = np.array([X0 + DX, Y0 - HH, 0.0])
UU = (P1 - P0) / LL
VMAX = math.sqrt(4 * G * RR)
RW = 1.12 # bike wheel
XW = -PI * RW
YW = 0.55
R2, BX, BY = 1.22, 0.0, 0.30 # tautochrone bowl
T_TAU = 2.0
WW = PI / (2 * T_TAU)
def cyc(th):
return np.array([X0 + RR * (th - math.sin(th)), Y0 - RR * (1 - math.cos(th)), 0.0])
def tick_m(th):
return np.array([XW + RW * (th - math.sin(th)), YW + RW * (1 - math.cos(th)), 0.0])
def bowl(th):
return np.array([BX + R2 * (th - math.sin(th)) - R2 * PI,
BY + 2 * R2 - R2 * (1 - math.cos(th)), 0.0])
def poli(f, a, b, col, w=8, n=200):
m = VMobject(stroke_color=col, stroke_width=w)
m.set_points_as_corners([f(t) for t in np.linspace(a, b, n)])
return m
class Cycloid(TikTok):
NAME = "cicloide"
def construct(self):
self.prepare()
chip = self.chip("¿Cuál gana?", AMBER)
money = DashedLine([X0 - 0.8, Y0 + 0.66, 0], [X0 + 1.7, Y0 + 0.66, 0],
stroke_width=3, color=C_GRAY, dash_length=0.1)
pc = Dot(radius=0.19, color=AMBER).move_to([X0 - 0.4, Y0 + 0.66, 0])
pr = Dot(radius=0.19, color=C_SKY).move_to([X0 + 0.08, Y0 + 0.66, 0])
meta = DashedLine(P1 + UP * 1.35, P1 + DOWN * 0.95, stroke_width=6,
color=C_GREEN, dash_length=0.14)
rotm = Text("META", font=FONT, font_size=32, weight=BOLD, color=C_GREEN)
rotm.next_to(meta, UP, buff=0.14)
legend = VGroup(
VGroup(Dot(radius=0.15, color=AMBER),
Text("curva", font=FONT, font_size=38, weight=BOLD,
color=AMBER)).arrange(RIGHT, buff=0.22),
VGroup(Dot(radius=0.15, color=C_SKY),
Text("recta", font=FONT, font_size=38, weight=BOLD,
color=C_SKY)).arrange(RIGHT, buff=0.22),
).arrange(RIGHT, buff=1.3).move_to([0, -1.3, 0])
with self.beat(0) as b:
b.play(FadeIn(chip, shift=DOWN * 0.3), run_time=0.4)
b.play(Create(money), run_time=0.4)
b.play(FadeIn(pc, scale=0.4), FadeIn(pr, scale=0.4), run_time=0.5)
b.play(Create(meta), FadeIn(rotm), FadeIn(legend), run_time=0.7)
line_m = Line(P0, P1, stroke_width=8, color=C_SKY)
curve_obj = poli(cyc, 0, TH1, AMBER)
with self.beat(1) as b:
b.play(Create(line_m), run_time=0.9)
b.play(Create(curve_obj), run_time=max(0.9, b.floor - 1.4))
inter = Text("?", font=FONT, font_size=115, weight=BOLD,
color=WHITE).move_to([0.0, 1.62, 0])
with self.beat(2) as b:
b.play(FadeIn(inter, scale=0.5), run_time=0.5)
b.play(Indicate(inter, scale_factor=1.25, color=AMBER), run_time=0.9)
rot = Text("el más corto", font=FONT, font_size=40, weight=BOLD,
color=C_SKY).move_to([-0.15, 3.02, 0])
with self.beat(3) as b:
b.play(FadeOut(inter), run_time=0.3)
b.play(line_m.animate.set_stroke(width=12), FadeIn(rot, shift=UP * 0.2),
run_time=0.55)
b.play(ShowPassingFlash(line_m.copy().set_stroke(WHITE, 13), time_width=0.5),
run_time=1.0)
b.play(line_m.animate.set_stroke(width=8), run_time=0.3)
# --- the race ----------------------------------------------------------
T = ValueTracker(0.0)
def up_c(m):
t = min(T.get_value(), T_C)
m.move_to(cyc(min(t * math.sqrt(G / RR), TH1)))
def up_r(m):
t = min(T.get_value(), T_L)
m.move_to(P0 + UU * (0.5 * AA * t * t))
med = VGroup(Circle(radius=0.34, stroke_width=0, fill_color=AMBER, fill_opacity=1),
Text("1º", font=FONT, font_size=34, weight=BOLD, color="#0B0F1A"))
med.move_to(P1 + UP * 0.72).set_opacity(0)
med.add_updater(lambda m: m.set_opacity(1.0 if T.get_value() >= T_C else 0.0))
with self.beat(4) as b:
b.play(FadeOut(rot), FadeOut(money), pc.animate.move_to(P0),
pr.animate.move_to(P0), run_time=0.5)
ya = Text("¡LARGA!", font=FONT, font_size=66, weight=BOLD,
color=C_GREEN).move_to([0, 1.7, 0])
b.play(FadeIn(ya, scale=1.7), run_time=0.3)
b.play(FadeOut(ya), run_time=0.22)
pc.add_updater(up_c)
pr.add_updater(up_r)
b.add(med)
b.play(T.animate.set_value(T_L), run_time=T_L, rate_func=linear)
pc.clear_updaters(); pr.clear_updaters(); med.clear_updaters()
level = DashedLine([X0 - 0.2, Y0 - HH, 0], [X0 + DX + 0.2, Y0 - HH, 0],
stroke_width=4, color=C_GRAY, dash_length=0.12)
background = cyc(PI)
note_txt = Text("pasa por debajo de la meta", font=FONT, font_size=36, weight=BOLD,
color=C_PINK).move_to([0.3, -0.95, 0])
arrow_m = Arrow(note_txt.get_top() + RIGHT * 0.9, background + DOWN * 0.22, buff=0.06,
color=C_PINK, stroke_width=6, max_tip_length_to_length_ratio=0.2)
gap = VGroup(
Text("la recta tardó", font=FONT, font_size=38, weight=BOLD, color=WHITE),
Text("40% más", font=FONT, font_size=64, weight=BOLD, color=C_SKY),
).arrange(RIGHT, buff=0.3).move_to([0.0, -2.15, 0])
with self.beat(5) as b:
b.play(FadeOut(legend), Create(level), run_time=0.45)
b.play(FadeIn(note_txt, shift=UP * 0.2), GrowArrow(arrow_m), run_time=0.6)
b.play(FadeIn(gap, shift=UP * 0.2), run_time=0.55)
# --- why it wins: early speed ------------------------------------------
riel_c = Rectangle(width=6.2, height=0.32, stroke_width=0, fill_color="#1B2233",
fill_opacity=1).move_to([0.0, -1.30, 0])
riel_r = riel_c.copy().move_to([0.0, -2.20, 0])
bar_c = Rectangle(width=0.02, height=0.32, stroke_width=0, fill_color=AMBER,
fill_opacity=1).move_to(riel_c.get_left(), LEFT)
bar_r = Rectangle(width=0.02, height=0.32, stroke_width=0, fill_color=C_SKY,
fill_opacity=1).move_to(riel_r.get_left(), LEFT)
lab_c = Text("curva", font=FONT, font_size=30, weight=BOLD, color=AMBER)
lab_r = Text("recta", font=FONT, font_size=30, weight=BOLD, color=C_SKY)
lab_c.next_to(riel_c, LEFT, buff=0.18)
lab_r.next_to(riel_r, LEFT, buff=0.18)
left_c, left_r = riel_c.get_left()[0], riel_r.get_left()[0]
def bar_m(m, v, x0, y):
w = max(0.02, 6.2 * v / VMAX)
m.stretch_to_fit_width(w)
m.move_to([x0 + w / 2, y, 0])
def ub_c(m):
t = min(T.get_value(), T_C)
th = t * math.sqrt(G / RR)
bar_m(m, math.sqrt(max(0.0, 2 * G * RR * (1 - math.cos(th)))), left_c, -1.30)
def ub_r(m):
bar_m(m, AA * min(T.get_value(), T_L), left_r, -2.20)
with self.beat(6) as b:
b.play(FadeOut(VGroup(level, arrow_m, note_txt, gap, med)), run_time=0.35)
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