# Untitled

unknown
python
a year ago
7.4 kB
7
Indexable
Never
from manim import *

class Euler(Scene):
def construct(self):
# Créer les labels pour Re et Im
real_label = MathTex(r"\text{Re}").next_to(RIGHT*3.5, UP, buff=SMALL_BUFF)
imag_label = MathTex(r"\text{Im}").next_to(UP*3.5, LEFT, buff=SMALL_BUFF)

# Créer les flèches pour Re et Im
real_arrow = Arrow(LEFT*0.25, RIGHT*3.5, tip_length=0.15, stroke_width=1)
imag_arrow = Arrow(DOWN*0.25, UP*3.5, tip_length=0.15, stroke_width=1)

# Ajouter les labels à la fin des flèches
real_label.next_to(real_arrow.get_end(), RIGHT, buff=SMALL_BUFF)
imag_label.next_to(imag_arrow.get_end(), UP, buff=SMALL_BUFF)

# Créer le cercle de rayon

# Créer les points d'intersection
un_point = MathTex(r"\text{1}").next_to(RIGHT*2.5, UP*0.4)
i_point = MathTex(r"\text{i}").next_to( UP*2.5, RIGHT*0.4)

# Cos et Sin au point 0
cos_point = MathTex(r"{\cos\theta}").next_to(RIGHT*0.5, DOWN*0.2)
sin_point = MathTex(r"{\sin\theta}").next_to(UP*0.4, LEFT*0.7)
theta = MathTex(r"{\theta}", font_size=40).next_to(RIGHT*0.65, UP*0.4)

# Fleche de la démonstration
line45d = Line(start=[-0, -0, 0], end=[1.59, 1.59, 0], stroke_width=1.5)
linesin = Line(start=[1.59, 0, 0], end=[1.59, 1.59, 0], stroke_width=1.5)

# Arc theta
arc = Arc(radius=0.5, start_angle=0, angle=45*DEGREES, stroke_width=1.5)

#Objet de la formule
formule = MathTex(r"e^{i\theta}", "=", r"\cos(\theta) + i\sin(\theta)")
serie1 = MathTex(r"e^{i\theta} = 1+i\theta+\frac{i\theta^{2}}{2!}+...+\frac{i\theta^{n}}{n!}+...").next_to(formule, UP*2, LEFT*5)
serie2 = MathTex(r"e^{i\theta} = \sum^{\infty}_{n=0}(\frac{(i\theta)^n}{n!})= ").next_to(serie1,LEFT*0.9, buff=SMALL_BUFF)
serie3 = MathTex(r"\sum^{\infty}_{n=0}(\frac{i^n \theta^n}{n!})").next_to(serie2,RIGHT*0.9, buff=SMALL_BUFF)
convi = MathTex(r"[With : ", "i^{2k} = (i^2)^k = (-1)^k]").next_to(serie3,LEFT*0.9, buff=SMALL_BUFF)
equival = MathTex(r"\Leftrightarrow").next_to(convi,DOWN, buff=SMALL_BUFF)
equival2 = MathTex(r"\Leftrightarrow").next_to(convi,DOWN, buff=SMALL_BUFF)
serie4 = MathTex(r"=\sum^{\infty}_{k=0}(\frac{i^{2k} \theta^{2k}}{2k!}) + \sum^{\infty}_{k=0}(\frac{i^{2k+1} \theta^{2k+1}}{2k+1!})").next_to(serie2,RIGHT*0.9, buff=SMALL_BUFF)
serie5 = MathTex(r"\sum^{\infty}_{k=0}\frac{(-1)^{k} \theta^{2k}}{2k!} + i\sum^{\infty}_{k=0}\frac{(-1)^{k+1} \theta^{2k+1}}{(2k+1)!}").next_to(equival2, RIGHT, buff=SMALL_BUFF)
cost = MathTex(r"{\cos\theta}").next_to(serie5, DOWN*14, buff=SMALL_BUFF)
sint = MathTex(r"{i\sin\theta}").next_to(serie5, DOWN*14, buff=SMALL_BUFF)
plus=MathTex(r"+").next_to(serie5, DOWN*14, buff=SMALL_BUFF)
equal=MathTex(r"=").next_to(sint,RIGHT, buff=SMALL_BUFF)
eitheta = MathTex(r"e^{i\theta}").next_to(sint,RIGHT, buff=SMALL_BUFF)
equal2 = MathTex(r"=").next_to(serie4,RIGHT*2, buff=SMALL_BUFF)
eitheta2 = MathTex(r"e^{i\theta}").next_to(serie4,RIGHT, buff=SMALL_BUFF)
equal3 = MathTex(r"=").next_to(serie4,RIGHT*2, buff=SMALL_BUFF)
eitheta3 = MathTex(r"e^{i\theta}").next_to(serie4,RIGHT, buff=SMALL_BUFF)

# Regrouper la dernière equation dans un seul groupe
group = VGroup(cost, plus, sint, equal, eitheta2)

formule.shift(RIGHT*3.8 + UP*2)
serie1.shift(LEFT*2.5, UP*0.8)
serie2.shift(DOWN*0.8, RIGHT*1.8)
serie3.shift(DOWN*0.8, RIGHT*1.8)
convi.shift(DOWN*2.3, RIGHT*3.5)
equival.shift(UP*1, RIGHT*1.2)
equival2.shift(DOWN, RIGHT*1.2)
serie4.shift(RIGHT, UP*0.5)
serie5.shift(RIGHT*1.4, DOWN*1)
cost.shift(LEFT*0.4)
plus.shift(RIGHT*1.7)
sint.shift(RIGHT*3.5)
equal.shift(UP*2.45, RIGHT*2.1)
eitheta.shift(UP*2.6, RIGHT*2.6)
equal2.shift(UP*0.565, RIGHT)
equal3.shift(DOWN*1.5, LEFT)
eitheta2.shift(UP*0.7, RIGHT*1.6)
eitheta3.shift(DOWN*1.35, LEFT*0.42)

# Regrouper la dernière equation dans un seul groupe
group = VGroup(cost, plus, sint, equal, eitheta2)

# Animer l'apparition des labels, des flèches et du cercle
self.play(Create(real_label), Create(imag_label))
self.play(Create(real_arrow), Create(imag_arrow))
self.play(Create(circle))
self.play(Create(i_point))
self.play(Create(un_point))
self.play(Create(cos_point))
self.play(Create(sin_point))
self.play(Create(line45d))
self.play(Create(linesin))
self.play(Create(arc))
self.play(Create(theta))
self.play(Create(formule))

# Attendre quelques secondes
self.wait(1)

#Transition puis attente

# Faire bouger la formule
self.play(formule.animate.to_corner(UL, buff=1))

#Démonstraton par séries partie une
self.play(Create(serie1))
self.play(Create(serie2))
self.play(Create(serie3))
self.play(Create(convi))

self.wait(1.5)

#Transition
self.play(convi.animate.to_corner(UL, buff=1))
self.play(serie3.animate.to_corner(UL, buff=2))

# Démonstraton par séries partie deux
self.play(Create(equival))
self.play(Create(serie4))
self.play(Create(equal2))
self.play(Create(eitheta2))
self.play(Create(equival2))
self.play(Create(serie5))
self.play(Create(equal3))
self.play(Create(eitheta3))
self.wait(1)
self.play(Create(cost))
self.play(Create(sint))
self.play(Create(plus))

#Transition Finale

#Démonstration partie finale
self.play(cost.animate.shift(UP*2.5, RIGHT*0.6))
self.play(plus.animate.shift(UP*2.47, LEFT*0.62))
self.play(sint.animate.shift(UP*2.5, LEFT*1.5))
self.play(eitheta2.animate.move_to(RIGHT*0.9 + UP*0.625), eitheta3.animate.move_to(RIGHT*0.9 + UP*0.625), run_time=2)
self.play(Create(equal))

#Création du rèctangle final
framebox1 = SurroundingRectangle(group, buff = 0.3, color=WHITE)

self.play(Create(framebox1), runtime=2)

self.wait(2)