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elliptic_curves.py
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elliptic_curves.py
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from manim import *
from manimlib.mobject.mobject import *
from manim_slides import Slide
# Most of the code were taken from here: https://github.com/thud/eccdemo MIT License (credits for thud)
from manimlib.utils.space_ops import angle_of_vector
L,U,R,D = LEFT, UP, RIGHT, DOWN
NUMBER_PLANE = None
class EllipticCurveIntroduction(Slide):
def construct(self):
short_weierstrass = Tex("Short Weierstrass equation")
eq = Tex("$y^2 = x^3 + ax + b$").scale(1.5).next_to(short_weierstrass, DOWN)
self.play(Write(short_weierstrass), Write(eq))
class MoveAlongPathPiece(Animation):
def __init__(self, mobject, mobject2, path, t_min, t_max, t_min2, t_max2, **kwargs):
self.mobject = mobject
self.mobject2 = mobject2
self.path = path
self.t_min = t_min
self.t_max = t_max
self.t_min2 = t_min2
self.t_max2 = t_max2
self.label = kwargs["label"]
self.labelp = kwargs["labelp"]
self.label2 = kwargs["label2"]
self.labelp2 = kwargs["labelp2"]
super().__init__(mobject, **kwargs)
def interpolate_mobject(self, alpha):
self.point = self.path.get_point_from_function(self.t_min+alpha*(self.t_max-self.t_min))
self.point2 = self.path.get_point_from_function(self.t_min2+alpha*(self.t_max2-self.t_min2))
self.mobject.move_to(self.point)
self.label.next_to(self.point,self.labelp)
self.mobject2.move_to(self.point2)
self.label2.next_to(self.point2,self.labelp2)
class MovePointsWithLine(MoveAlongPathPiece):
def __init__(self, line, *args, **kwargs):
self.line = line
super().__init__(*args, **kwargs)
def interpolate_mobject(self, alpha):
super().interpolate_mobject(alpha)
self.line.set_angle(angle_of_vector(self.point2-self.point))
self.line.move_to(self.point)
class MovePointsWithLineAndThirdPoint(MovePointsWithLine):
def __init__(self, c2p, p2c, mobject3, *args, **kwargs):
self.mobject3 = mobject3
self.c2p = c2p
self.p2c = p2c
self.label3 = kwargs["label3"]
self.labelp3 = kwargs["labelp3"]
super().__init__(*args, **kwargs)
def interpolate_mobject(self, alpha):
super().interpolate_mobject(alpha)
if self.point[0] == self.point2[0]:
raise Exception(":skull:")
c1,c2 = self.p2c(self.point), self.p2c(self.point2)
m = (c2[1]-c1[1])/(c2[0]-c1[0])
self.point3 = [m**2-c1[0]-c2[0]]
self.point3.append(c1[1]+m*(self.point3[0]-c1[0]))
self.mobject3.move_to(self.c2p(self.point3[0], self.point3[1]))
self.label3.next_to(self.c2p(self.point3[0], self.point3[1]),self.labelp3)
class EllipticCurveScene(Slide):
def construct(self):
global NUMBER_PLANE
NUMBER_PLANE = NumberPlane(
x_range=[-5, 5, 1],
y_range=[-4, 4, 1],
background_line_style={
"stroke_opacity": 0
},
tips= False,
axis_config = {
"stroke_width": 4,
"include_ticks": True,
"line_to_number_buff": SMALL_BUFF,
"label_direction": DR,
"font_size": 24,
"stroke_color": BLUE_E,
"stroke_opacity": 1,
},
)
self.add(NUMBER_PLANE)
parex = ParametricFunction(function=ec1, t_range=[-5, 5])
parex2 = ParametricFunction(function=ec2, t_range=[-5, 5])
parex3 = ParametricFunction(function=ec3, t_range=[-5, 5])
parex4 = ParametricFunction(function=ec4, t_range=[-5, 5])
NUMBER_PLANE.add(parex)
ais = Tex("$a = -1$").scale(.75).move_to(3*UL+2*L)
bis = Tex("$b = 5$").scale(.75).next_to(ais, DOWN)
fnis = Tex("$y^2 = x^3 - x + 5$").scale(.75).next_to(bis, DOWN)
ais2 = Tex("$a = 0$").scale(.75).move_to(3*UL+2*L)
bis2 = Tex("$b = 0$").scale(.75).next_to(ais, DOWN)
fnis2 = Tex("$y^2 = x^3$").scale(.75).next_to(bis, DOWN)
ais3 = Tex("$a = -3$").scale(.75).move_to(3*UL+2*L)
bis3 = Tex("$b = 2$").scale(.75).next_to(ais, DOWN)
fnis3 = Tex("$y^2 = x^3 -3x + 2$").scale(.75).next_to(bis, DOWN)
ais4 = Tex("$a = -2$").scale(.75).move_to(3*UL+2*L)
bis4 = Tex("$b = 2$").scale(.75).next_to(ais, DOWN)
fnis4 = Tex("$y^2 = x^3 - 2x + 2$").scale(.75).next_to(bis, DOWN)
empty = Circle(radius=0.2)
self.play(
Create(parex),
Write(ais),
Write(bis),
Write(fnis),
)
self.next_slide()
self.play(Transform(parex, parex2), *[Transform(*x) for x in [[ais,ais2],[bis,bis2],[fnis,fnis2]]])
self.play(
Create(empty)
)
self.next_slide()
self.remove(empty)
self.play(
Transform(
parex,
parex3
),
*[Transform(*x) for x in [[ais,ais3],[bis,bis3],[fnis,fnis3]]]
)
self.wait(0.5)
empty.move_to(c2p(1,0))
self.play(
Create(empty)
)
determinant = Tex("$4a^3+27b^2 \\neq 0$").scale(.75).next_to(fnis, DOWN)
self.play(Write(determinant))
self.next_slide()
self.remove(empty)
self.play(
Transform(
parex,
parex4
),
*[Transform(*x) for x in [[ais,ais4],[bis,bis4],[fnis,fnis4]]]
)
self.wait(1)
gp = parex4.get_point_from_function
pp = Dot(gp(0.1),color=YELLOW)
pq = Dot(gp(2),color=YELLOW)
pplab = Tex("$P$",color=YELLOW).scale(0.75).next_to(gp(0.1), LEFT)
pqlab = Tex("$Q$",color=YELLOW).scale(0.75).next_to(gp(2), 0.707*UR)
self.play(Create(pp),Create(pq),Write(pplab),Write(pqlab))
self.play(
MoveAlongPathPiece(
pp,
pq,
parex4,
0.1,1.5,
2,2.75,
label=pplab,
labelp=LEFT,
label2=pqlab,
labelp2=0.707*UR
)
)
self.play(
MoveAlongPathPiece(
pp,
pq,
parex4,
1.5,0.1,
2.75,2.3,
label=pplab,
labelp=LEFT,
label2=pqlab,
labelp2=0.707*UR
)
)
jline = Line()
jline.set_angle(angle_of_vector(gp(2.3)-gp(0.1)))
jline.move_to(gp(0.1))
jline.set_length(20)
jline.set_opacity(0.8)
self.play(
Create(jline)
)
self.play(
MovePointsWithLine(
jline,
pp,
pq,
parex4,
0.1,0.2,
2.3,2.1,
label=pplab,
labelp=LEFT,
label2=pqlab,
labelp2=0.707*UR
)
)
point, point2 = NUMBER_PLANE.point_to_coords(gp(0.2)), NUMBER_PLANE.point_to_coords(gp(2.1))
pr = Dot(color=BLUE)
m = (point2[1]-point[1])*(point2[0]-point[0])**-1
point3 = [m**2-point[0]-point2[0]]
point3.append(point[1]+m*(point3[0]-point[0]))
pr.move_to(NUMBER_PLANE.coords_to_point(point3[0], point3[1]))
prlab = Tex("$R$",color=BLUE).scale(0.75).next_to(NUMBER_PLANE.coords_to_point(point3[0], point3[1]), 0.707*DR)
self.play(
Create(pr),
Write(prlab)
)
self.play(
MovePointsWithLineAndThirdPoint(
NUMBER_PLANE.coords_to_point,
NUMBER_PLANE.point_to_coords,
pr,
jline,
pp,
pq,
parex4,
0.2,-0.3,
2.1,2.4,
label=pplab,
labelp=LEFT,
label2=pqlab,
labelp2=0.707*UR,
label3=prlab,
labelp3=0.707*DR
)
)
self.next_slide()
self.play(
MovePointsWithLineAndThirdPoint(
NUMBER_PLANE.coords_to_point,
NUMBER_PLANE.point_to_coords,
pr,
jline,
pp,
pq,
parex4,
-0.3,-0.3,
2.4,0.301,
label=pplab,
labelp=LEFT,
label2=pqlab,
labelp2=0.707*UR,
label3=prlab,
labelp3=0.707*DR
)
)
self.next_slide()
self.play(
MovePointsWithLineAndThirdPoint(
NUMBER_PLANE.coords_to_point,
NUMBER_PLANE.point_to_coords,
pr,
jline,
pp,
pq,
parex4,
-0.3,-0.3,
0.301,2.4,
label=pplab,
labelp=LEFT,
label2=pqlab,
labelp2=0.707*UR,
label3=prlab,
labelp3=0.707*DR
)
)
self.wait(1)
m_t = Tex("$m = \\frac{\Delta y}{\Delta x} = \\frac{y_p-y_q}{x_p-x_q}$").scale(.75).move_to(D+5*R)
lineeq_t = Tex(
"""\\begin{align*}
y-y_P &= m(x-x_P)\\\\
\implies y &= y_P + m(x-x_P)
\\end{align*}
""").scale(.75).next_to(m_t, 1.25*DOWN)
self.play(Write(m_t), Write(lineeq_t))
self.next_slide()
intersectedline_t = Tex("\\begin{align*}\pm \sqrt{x^3+ax+b} \\\\ = y_P + m(x-x_P) \\end{align*}").scale(.75).next_to(m_t, 1.25*DOWN)
self.play(
Transform(
lineeq_t,
intersectedline_t
)
)
self.next_slide()
rxry_t = Tex(
"""\\begin{align*}
x_R=m^2-x_P-x_Q\\\\
y_R=y_P+m(x_R-x_P)
\\end{align*}
""").scale(.75).next_to(m_t, 1.25*DOWN)
self.play(
Transform(
lineeq_t,
rxry_t
)
)
self.play(
MovePointsWithLineAndThirdPoint(
NUMBER_PLANE.coords_to_point,
NUMBER_PLANE.point_to_coords,
pr,
jline,
pp,
pq,
parex4,
-0.3,0.1,
2.4,0.6001,
label=pplab,
labelp=LEFT,
label2=pqlab,
labelp2=0.707*UR,
label3=prlab,
labelp3=0.707*DR
)
)
self.next_slide()
self.play(
MovePointsWithLineAndThirdPoint(
NUMBER_PLANE.coords_to_point,
NUMBER_PLANE.point_to_coords,
pr,
jline,
pp,
pq,
parex4,
0.1,0.6,
0.6001,0.6001,
label=pplab,
labelp=LEFT,
label2=pqlab,
labelp2=0.707*UR,
label3=prlab,
labelp3=0.707*DR
)
)
self.wait(1)
mtaneq_t = Tex("$m_{\\textrm{tan}} = $").scale(.75).move_to(U+3*R)
mtanrhs = Tex("$\\frac{\mathrm{d}}{\mathrm{d}x} \left( \pm \sqrt{x^3+ax+b} \\right)$").scale(.75).next_to(mtaneq_t, RIGHT)
mtanrhs2 = Tex("$\pm \\frac{3x_P^2+a}{2\sqrt{x_P^3+ax+b}}$").scale(.75).next_to(mtaneq_t, RIGHT)
mtanrhs3 = Tex("$\\frac{3x_P^2+a}{2y_P}$").scale(.75).next_to(mtaneq_t, RIGHT)
self.play(Write(mtaneq_t),Write(mtanrhs))
self.next_slide()
self.play(
Transform(
mtanrhs,
mtanrhs2
)
)
self.next_slide()
self.play(
Transform(
mtanrhs,
mtanrhs3
)
)
self.wait(.25)
self.play(
mtaneq_t.animate.shift(1.25*R),
mtanrhs.animate.shift(1.25*R)
)
self.wait(1)
self.play(
MovePointsWithLineAndThirdPoint(
NUMBER_PLANE.coords_to_point,
NUMBER_PLANE.point_to_coords,
pr,
jline,
pp,
pq,
parex4,
0.6,-0.6,
0.6001,0.6001,
label=pplab,
labelp=LEFT,
label2=pqlab,
labelp2=0.707*UR,
label3=prlab,
labelp3=0.707*DR
)
)
infarrow = Arrow(
start=U*4+L*1.5+DL,
end=UP*4+L*1.5,
color=YELLOW
)
pointatinfinity_t = Tex("``The Point at Infinity''", color=YELLOW).scale(.75).next_to(infarrow, DL)
self.play(
Create(infarrow),
Write(pointatinfinity_t),
ais.animate.shift(1.5*D),
bis.animate.shift(1.5*D),
fnis.animate.shift(1.5*D),
determinant.animate.shift(1.5*D),
)
class EllipticCurveAdditionScene(Slide):
def construct(self):
global NUMBER_PLANE
NUMBER_PLANE = NumberPlane(
x_range=[-5, 5, 1],
y_range=[-4, 4, 1],
background_line_style={
"stroke_opacity": 0
},
tips= False,
axis_config = {
"stroke_width": 4,
"include_ticks": True,
"line_to_number_buff": SMALL_BUFF,
"label_direction": DR,
"font_size": 24,
"stroke_color": BLUE_E,
"stroke_opacity": 1,
},
)
self.add(NUMBER_PLANE)
parex = ParametricFunction(function=ec, t_range=[-5, 5])
gp = parex.get_point_from_function
fnis = Tex("$y^2 = x^3 - 2x + 2$").scale(.75).move_to(3*U+4*L)
NUMBER_PLANE.add(parex)
self.play(
Create(parex),
Write(fnis),
)
point, point2 = NUMBER_PLANE.point_to_coords(gp(0.1)), NUMBER_PLANE.point_to_coords(gp(2))
pp = Dot(gp(0.1),color=YELLOW)
pq = Dot(gp(2),color=YELLOW)
pr = Dot(color=BLUE)
pplab = Tex("$P$",color=YELLOW).scale(0.75).next_to(gp(0.1), LEFT)
pqlab = Tex("$Q$",color=YELLOW).scale(0.75).next_to(gp(2), 0.707*UR)
m = (point2[1]-point[1])*(point2[0]-point[0])**-1
point3 = [m**2-point[0]-point2[0]]
point3.append(point[1]+m*(point3[0]-point[0]))
pr.move_to(NUMBER_PLANE.coords_to_point(point3[0], point3[1]))
prlab = Tex("$R$",color=BLUE).scale(0.75).next_to(NUMBER_PLANE.coords_to_point(point3[0], point3[1]), 0.707*DR)
prneg = Dot(color=RED).move_to(NUMBER_PLANE.coords_to_point(point3[0],-point3[1]))
prneglab = Tex("$-R$",color=RED).scale(.75).next_to(NUMBER_PLANE.coords_to_point(point3[0],-point3[1]), 0.707*UR)
prnegline = Line(start=pr,end=prneg,color=RED)
jline = Line()
jline.set_angle(angle_of_vector(gp(2)-gp(0.1)))
jline.move_to(gp(0.1))
jline.set_length(20)
jline.set_opacity(0.8)
self.play(Create(pp),Create(pq),Create(pr),Create(jline),Write(pplab),Write(pqlab), Write(prlab))
ppqpreqz = Tex("$P + Q + R = 0$").scale(1.25).move_to(LEFT*4+D*2)
ppqpreqz2 = Tex("$P + Q = -R$").scale(1.25).move_to(LEFT*4+D*2)
self.play(Write(ppqpreqz))
self.next_slide()
self.play(
Transform(
ppqpreqz,
ppqpreqz2
),
Create(prnegline)
)
self.play(Create(prneg), Write(prneglab))
pass
class EllipticCurveOverGF(Slide):
def construct(self):
global NUMBER_PLANE
NUMBER_PLANE = NumberPlane(
x_range=[0, 36, 5],
y_range=[0, 36, 5],
x_length=9,
y_length=5,
background_line_style={
"stroke_opacity": 0
},
tips= False,
axis_config = {
"stroke_width": 4,
"include_ticks": True,
"include_numbers": True,
"label_direction": D,
"font_size": 24,
"stroke_color": BLUE_E,
"stroke_opacity": 1,
}
).move_to(2*R)
self.add(NUMBER_PLANE)
p = 37
ppp = [6,24]
pp = Dot(NUMBER_PLANE.coords_to_point(*ppp),color=YELLOW)
pqp = [13,8]
pq = Dot(NUMBER_PLANE.coords_to_point(*pqp),color=YELLOW)
pplab = Tex("$P$",color=YELLOW).scale(.75).next_to(NUMBER_PLANE.coords_to_point(*ppp), .707*UL)
pqlab = Tex("$Q$",color=YELLOW).scale(.75).next_to(NUMBER_PLANE.coords_to_point(*pqp), .707*UL)
points = []
for x in range(p):
for y in range(p):
if (x == ppp[0] and y == ppp[1]) or (x == pqp[0] and y == pqp[1]):
continue
if y**2 % p == (x**3 -2*x +2)%p:
points.append(NUMBER_PLANE.coords_to_point(x,y))
points.append(NUMBER_PLANE.coords_to_point(x,p-y))
dots = [Dot(x) for x in points]
self.play(
*[Create(x) for x in dots],
Create(pp),
Create(pq),
Write(pplab),
Write(pqlab),
)
self.wait(1)
mlfn, disc = self.modline(p,ppp,pqp)
ml = ParametricFunction(function=mlfn, t_range=[0, 36], discontinuities=disc)
self.play(Create(ml))
self.wait(1)
prp = [27,13]
pr = Dot(NUMBER_PLANE.coords_to_point(*prp),color=BLUE)
prlab = Tex("$R$",color=BLUE).scale(.75).next_to(NUMBER_PLANE.coords_to_point(*prp), .707*UL)
prpn = [27,p-13]
prn = Dot(NUMBER_PLANE.coords_to_point(*prpn),color=RED)
prnlab = Tex("$-R$",color=RED).scale(.75).next_to(NUMBER_PLANE.coords_to_point(*prpn), .707*UL)
jl = Line(start=pr, end=prn, color=RED)
self.play(
Create(pr),
Write(prlab)
)
self.play(
Create(jl)
)
self.play(
Create(prn),
Write(prnlab)
)
m_t = Tex("$m = \\frac{\Delta y}{\Delta x} = \\frac{y_P-y_Q}{x_P-x_Q}$").scale(.75).move_to(3*U+5*L)
m_t2 = Tex("\\begin{align*}m &= \Delta y (\Delta x)^{-1} \\\\ &= (y_P-y_Q)(x_P-x_Q)^{-1}\\end{align*}").scale(.7).move_to(3*U+5*L)
self.play(
Write(m_t)
)
self.next_slide()
self.play(
Transform(
m_t,
m_t2
)
)
self.next_slide()
closure_t = Tex("1. Closure",color=YELLOW).move_to(2*U+5*L)
assoc_t = Tex("2. Associativity",color=YELLOW).next_to(closure_t, D)
ident_t = Tex("3. Identity Element",color=YELLOW).scale(.75).next_to(assoc_t, D)
inverse_t = Tex("4. Inverse Elements",color=YELLOW).scale(.75).next_to(ident_t, D)
self.play(Write(closure_t))
self.wait(.5)
ppp2 = [6,13]
pp2 = Dot(NUMBER_PLANE.coords_to_point(*ppp2),color=YELLOW)
pqp2 = [11,4]
pq2 = Dot(NUMBER_PLANE.coords_to_point(*pqp2),color=YELLOW)
self.play(
*[FadeOut(x) for x in [pp,pq,pr,prn,ml,jl,prlab,prnlab]],
*[Create(x) for x in [pp2,pq2]],
pplab.animate.next_to(NUMBER_PLANE.coords_to_point(*ppp2),.707*UL),
pqlab.animate.next_to(NUMBER_PLANE.coords_to_point(*pqp2),.707*UL)
)
mlfn, disc = self.modline(p,ppp2,pqp2)
ml = ParametricFunction(function=mlfn, t_range=[0, 36], discontinuities=disc)
prp2 = [4,p-13]
pr2 = Dot(NUMBER_PLANE.coords_to_point(*prp2),color=BLUE)
prpn2 = [4,13]
prn2 = Dot(NUMBER_PLANE.coords_to_point(*prpn2),color=RED)
jl2 = Line(start=pr2, end=prn2, color=RED)
prlab.next_to(NUMBER_PLANE.coords_to_point(*prp2),.707*UL)
prnlab.next_to(NUMBER_PLANE.coords_to_point(*prpn2),.707*UL)
self.play(
Create(ml),
Create(pr2),
Write(prlab),
)
self.play(
Create(jl2),
Create(prn2),
Write(prnlab)
)
self.wait(1)
ppp3 = [4,13]
pp3 = Dot(NUMBER_PLANE.coords_to_point(*ppp3),color=YELLOW)
pqp3 = [4,13]
pq3 = Dot(NUMBER_PLANE.coords_to_point(*pqp3),color=YELLOW)
self.play(
*[FadeOut(x) for x in [pp2,pq2,pr2,prn2,ml,jl2,prlab,prnlab]],
*[Create(x) for x in [pp3,pq3]],
pplab.animate.next_to(NUMBER_PLANE.coords_to_point(*ppp3),.707*UL),
pqlab.animate.next_to(NUMBER_PLANE.coords_to_point(*pqp3),.707*UR)
)
mlfn, disc = self.modline(p,ppp3,pqp3)
ml = ParametricFunction(function=mlfn, t_range=[0, 36], discontinuities=disc)
prp3 = [26,p-5]
pr3 = Dot(NUMBER_PLANE.coords_to_point(*prp3),color=BLUE)
prpn3 = [26,5]
prn3 = Dot(NUMBER_PLANE.coords_to_point(*prpn3),color=RED)
jl3 = Line(start=pr3, end=prn3, color=RED)
prlab.next_to(NUMBER_PLANE.coords_to_point(*prp3),.707*UL)
prnlab.next_to(NUMBER_PLANE.coords_to_point(*prpn3),.707*UL)
self.play(
Create(ml),
Create(pr3),
Write(prlab),
)
self.play(
Create(jl3),
Create(prn3),
Write(prnlab)
)
self.wait(2)
ppp4 = [6,24]
pp4 = Dot(NUMBER_PLANE.coords_to_point(*ppp4),color=YELLOW)
pqp4 = [13,8]
pq4 = Dot(NUMBER_PLANE.coords_to_point(*pqp4),color=YELLOW)
self.play(
*[FadeOut(x) for x in [pp3,pq3,pr3,prn3,ml,jl3,prlab,prnlab]],
*[Create(x) for x in [pp4,pq4]],
pplab.animate.next_to(NUMBER_PLANE.coords_to_point(*ppp4),.707*UL),
pqlab.animate.next_to(NUMBER_PLANE.coords_to_point(*pqp4),.707*UL)
)
mlfn, disc = self.modline(p,ppp4,pqp4)
ml = ParametricFunction(function=mlfn, t_range=[0, 36], discontinuities=disc)
prp4 = [27,13]
pr4 = Dot(NUMBER_PLANE.coords_to_point(*prp4),color=BLUE)
prpn4 = [27,p-13]
prn4 = Dot(NUMBER_PLANE.coords_to_point(*prpn4),color=RED)
jl4 = Line(start=pr4, end=prn4, color=RED)
prlab.next_to(NUMBER_PLANE.coords_to_point(*prp4),.707*UL)
prnlab.next_to(NUMBER_PLANE.coords_to_point(*prpn4),.707*UL)
self.play(
Create(ml),
Create(pr4),
Write(prlab),
)
self.play(
Create(jl4),
Create(prn4),
Write(prnlab)
)
self.wait(1)
self.play(
FadeToColor(
closure_t,
GREEN
)
)
self.next_slide()
self.play(
Write(assoc_t)
)
self.play(
FadeOut(prn4),
FadeOut(prnlab),
FadeOut(jl4)
)
self.play(
pr4.animate.move_to(NUMBER_PLANE.coords_to_point(*ppp4)),
prlab.animate.next_to(NUMBER_PLANE.coords_to_point(*ppp4),.707*UL),
pp4.animate.move_to(NUMBER_PLANE.coords_to_point(*pqp4)),
pplab.animate.next_to(NUMBER_PLANE.coords_to_point(*pqp4),.707*UL),
pq4.animate.move_to(NUMBER_PLANE.coords_to_point(*prp4)),
pqlab.animate.next_to(NUMBER_PLANE.coords_to_point(*prp4),.707*UL),
)
self.next_slide()
self.play(
FadeToColor(
assoc_t,
GREEN
)
)
self.wait(0.3)
self.play(
Write(ident_t)
)
self.wait(1)
ppp5 = [6,24]
pp5 = Dot(NUMBER_PLANE.coords_to_point(*ppp5),color=YELLOW)
pqp5 = [6,p-24]
pq5 = Dot(NUMBER_PLANE.coords_to_point(*pqp5),color=YELLOW)
self.play(
*[FadeOut(x) for x in [pp4,pq4,pr4,prn4,ml,jl4,prlab,prnlab]],
*[Create(x) for x in [pp5,pq5]],
pplab.animate.next_to(NUMBER_PLANE.coords_to_point(*ppp5),.707*UL),
pqlab.animate.next_to(NUMBER_PLANE.coords_to_point(*pqp5),.707*UL)
)
ml = Line(start=NUMBER_PLANE.coords_to_point(ppp5[0],0), end=NUMBER_PLANE.coords_to_point(ppp5[0],p-1))
self.play(
Create(ml),
)
infarrow = Arrow(
start=U*4+.75*L+DR,
end=UP*4+.75*L,
color=RED
)
pointatinfinity_t = Tex("``The Point at Infinity''", color=RED).scale(.75).next_to(infarrow, DR)
self.play(
Create(infarrow),
Write(pointatinfinity_t)
)
self.wait(2)
self.play(
FadeToColor(
ident_t,
GREEN
)
)
self.next_slide()
self.play(
Write(inverse_t)
)
self.play(
Transform(
pqlab,
Tex("$-P$",color=GREEN).scale(.75).next_to(NUMBER_PLANE.coords_to_point(*pqp5), 0.707*UL)
),
FadeToColor(
pq5,
GREEN
)
)
self.next_slide()
self.play(
FadeToColor(
inverse_t,
GREEN
)
)
self.play(
*[FadeOut(x, shift=DOWN) for x in [closure_t, assoc_t, ident_t, inverse_t]]
)
self.wait(.5)
mmulp = Tex("$kP=\\underbrace{P+P+P+\\ldots+P}_{k\\text{-times}}$").scale(.5).next_to(U+7*L)
self.play(
Write(mmulp)
)
self.wait(1)
ppp6 = [4,13]
pp6 = Dot(NUMBER_PLANE.coords_to_point(*ppp6),color=YELLOW)
pqp6 = [4,13]
pq6 = Dot(NUMBER_PLANE.coords_to_point(*pqp6),color=YELLOW)
self.play(
*[FadeOut(x) for x in [pp5,pq5,ml,pqlab,infarrow,pointatinfinity_t]],
*[Create(x) for x in [pp6,pq6]],
pplab.animate.next_to(NUMBER_PLANE.coords_to_point(*ppp6),.707*UL)
)
mlfn, disc = self.modline(p,ppp6,pqp6)
ml = ParametricFunction(function=mlfn, t_range=[0, 36], discontinuities=disc)
prp6 = [26,p-5]
pr6 = Dot(NUMBER_PLANE.coords_to_point(*prp6),color=BLUE)
prpn6 = [26,5]
prn6 = Dot(NUMBER_PLANE.coords_to_point(*prpn6),color=RED)
jl6 = Line(start=pr6, end=prn6, color=RED)
prlab.next_to(NUMBER_PLANE.coords_to_point(*prp6),.707*UL)
prnlab = Tex("$2P$",color=RED).scale(.75).next_to(NUMBER_PLANE.coords_to_point(*prpn6),.707*UL)
self.play(
Create(ml),
Create(pr6),
)
self.play(
Create(jl6),
Create(prn6),
Write(prnlab)
)
# next slide
ppp7 = [4,13]
pp7 = Dot(NUMBER_PLANE.coords_to_point(*ppp7),color=YELLOW)
pqp7 = [26,5]
pq7 = Dot(NUMBER_PLANE.coords_to_point(*pqp7),color=YELLOW)
pqlab = Tex("$2P$",color=RED).scale(.75).next_to(NUMBER_PLANE.coords_to_point(*prpn6),.707*UL)
self.play(
*[FadeOut(x) for x in [pp6,pq6,pr6,jl6,ml,pqlab]],
*[Create(x) for x in [pp7,pq7]],
FadeIn(pqlab)
)
mlfn, disc = self.modline(p,ppp7,pqp7)
ml = ParametricFunction(function=mlfn, t_range=[0, 36], discontinuities=disc)
prp7 = [16,p-25]
pr7 = Dot(NUMBER_PLANE.coords_to_point(*prp7),color=BLUE)
prpn7 = [16,25]
prn7 = Dot(NUMBER_PLANE.coords_to_point(*prpn7),color=RED)
jl7 = Line(start=pr7, end=prn7, color=RED)
prlab.next_to(NUMBER_PLANE.coords_to_point(*prp7),.707*UL)
prnlab7 = Tex("$3P$",color=RED).scale(.75).next_to(NUMBER_PLANE.coords_to_point(*prpn7),.707*UL)
self.play(
Create(ml),
Create(pr7),
)
self.play(
Create(jl7),
Create(prn7),
Write(prnlab7)
)
# next slide
ppp8 = [4,13]
pp8 = Dot(NUMBER_PLANE.coords_to_point(*ppp8),color=YELLOW)
pqp8 = [16,25]
pq8 = Dot(NUMBER_PLANE.coords_to_point(*pqp8),color=YELLOW)
pqlab = Tex("$3P$",color=RED).scale(.75).next_to(NUMBER_PLANE.coords_to_point(*prpn6),.707*UL)
self.play(
*[FadeOut(x) for x in [pp7,pq7,pr7,jl7,ml,pqlab]],
*[Create(x) for x in [pp8,pq8]],
FadeIn(pqlab)
)
mlfn, disc = self.modline(p,ppp8,pqp8)
ml = ParametricFunction(function=mlfn, t_range=[0, 36], discontinuities=disc)
prp8 = [18,p-10]
pr8 = Dot(NUMBER_PLANE.coords_to_point(*prp8),color=BLUE)
prpn8 = [18,10]
prn8 = Dot(NUMBER_PLANE.coords_to_point(*prpn8),color=RED)
jl8 = Line(start=pr8, end=prn8, color=RED)
prlab.next_to(NUMBER_PLANE.coords_to_point(*prp8),.707*UL)
prnlab8 = Tex("$4P$",color=RED).scale(.75).next_to(NUMBER_PLANE.coords_to_point(*prpn8),L)
self.play(
Create(ml),
Create(pr8)
)
self.play(
Create(jl8),
Create(prn8),
Write(prnlab8)
)
self.next_slide()
self.play(
Transform(
prnlab8,
Tex("$dP=H$",color=RED).scale(.75).next_to(NUMBER_PLANE.coords_to_point(*prpn8),L)
),
Transform(
prnlab7,
Tex("$(d-1)P$",color=YELLOW).scale(.75).next_to(NUMBER_PLANE.coords_to_point(*prpn7),.707*UL)
),
*[FadeOut(x) for x in [pqlab,prn6,prnlab]]
)
dpeqh = Tex("$dP=H$",color=RED).scale(.75).move_to(5*L+D)
self.play(
Write(dpeqh)
)
def modline(self, p, ppp, pqp):
m = 0
if (ppp[0] == pqp[0]) and (ppp[1] == pqp[1]):
denom = pow(2*ppp[1], p-2, p)
m = ((3*ppp[0]**2-2)*denom)%p