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rs_codes.py
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from manimlib import *
random.seed(12345)
A_AQUA = "#8dd3c7"
A_YELLOW = "#ffffb3"
A_LAVENDER = "#bebada"
A_RED = "#fb8072"
A_BLUE = "#80b1d3"
A_ORANGE = "#fdb462"
A_GREEN = "#b3de69"
A_PINK = "#fccde5"
A_GREY = "#d9d9d9"
A_VIOLET = "#bc80bd"
A_UNKA = "#ccebc5"
A_UNKB = "#ffed6f"
class NumberSquare(VGroup):
def __init__(self, number, color=RED, opacity=0.5, num_scale=3, side_length=2, square_kwargs={}, tex_kwargs={}, *args, **kwargs):
VGroup.__init__(self, *args, **kwargs)
self.sq = Square(side_length=side_length, fill_color=color,
fill_opacity=opacity, **square_kwargs)
self.num = Tex(str(number), **tex_kwargs)
self.num.scale(num_scale)
self.add(self.sq, self.num)
class LabelledNumberSquare(NumberSquare):
def __init__(self, *args, **kwargs):
NumberSquare.__init__(self, *args, **kwargs)
class PartScene(Scene):
CONFIG = {
"n": 1,
"title": "",
"title_color": RED
}
def construct(self):
if not self.title:
raise NotImplementedError("Please set the title of the scene")
part = TexText(f"Part {self.n}")
part.scale(1.5)
part.shift(2 * UP)
title = TexText(self.title, color=self.title_color)
title.scale(1.5)
self.play(Write(part))
self.play(Write(title))
self.wait()
class PartOne(PartScene):
CONFIG = {
"n": 1,
"title": "Modular Arithmetic",
"title_color": A_RED
}
class PartTwo(PartScene):
CONFIG = {
"n": 2,
"title": "Lagrange Interpolation",
"title_color": A_YELLOW
}
class PartThree(PartScene):
CONFIG = {
"n": 3,
"title": "Putting it all together",
"title_color": A_ORANGE
}
class TitleScene(Scene):
CONFIG = {
"color": None,
"text": None
}
def construct(self):
if self.text is None:
raise NotImplementedError
brect = Rectangle(height=FRAME_HEIGHT, width=FRAME_WIDTH,
fill_opacity=1, color=self.color)
title = TexText(self.text)
title.scale(1.5)
title.to_edge(UP)
rect = ScreenRectangle(height=6)
rect.next_to(title, DOWN)
self.add(brect)
self.play(FadeIn(rect, DOWN), Write(title), run_time=2)
self.wait()
class TitleU(TitleScene):
CONFIG = {
"color": "#5b6190",
"text": "Upcoming"
}
class TitleR(TitleScene):
CONFIG = {
"color": GREY,
"text": "Recap"
}
class RS(Scene):
def construct(self):
shades = ["#fc998e", "#fb8d80", "#fb8072", "#e27367", "#c9665b"]
numbers = [2, 4, 3, 1, "?", "?"]
numbers2 = [2, 4, 3, 1, 0, "?"]
c = [2, 4, 0, 4, 1, 3]
pos = (FRAME_WIDTH/2 - 4.25)/2 + 4.25
r1 = Rectangle(height=2.5, width=2.5).shift(pos * LEFT)
r2 = Rectangle(height=2.5, width=2.5).shift(pos * RIGHT)
t1 = TexText("Vivek (Me)").move_to(r1).shift(2 * DOWN)
t2 = TexText("Tony").move_to(r2).shift(2 * DOWN)
s = VGroup()
for i in range(4):
s_i = NumberSquare(
numbers[i], shades[c[i]], side_length=1.5, num_scale=2)
s.add(s_i.shift(2*i * RIGHT))
s.center()
self.play(Write(s), Write(r1), Write(r2), Write(t1), Write(t2))
self.wait()
r = BackgroundRectangle(s[2], buff=0.15)
self.play(ShowCreation(r))
self.wait()
self.play(Uncreate(r))
s2 = VGroup()
for i in range(6):
s_i = NumberSquare(
numbers[i], shades[c[i]], side_length=1, num_scale=1.5)
s2.add(s_i.shift(1.5*i * RIGHT))
s2.center()
b1 = Brace(s2[-2:])
t1 = b1.get_tex("k = 2", tex_to_color_map={
"k": A_YELLOW, "2": A_ORANGE})
t1.scale(1.5)
t1.shift(0.25 * DOWN)
self.play(ReplacementTransform(s, s2[:5]), Write(s2[5:]))
self.wait()
self.play(GrowFromCenter(b1), Write(t1))
self.wait()
r1 = BackgroundRectangle(s2[2], buff=0.15)
r2 = BackgroundRectangle(s2[-1], buff=0.15)
self.play(ShowCreation(r1), ShowCreation(r2))
self.wait()
self.embed()
class RSAppl(Scene):
def construct(self):
title = TexText("Reed-Solomon Codes", color=A_YELLOW)
title.scale(1.5)
title.shift(3 * UP)
self.play(Write(title))
img1 = ImageMobject("img/qr.jpeg")
img1.scale(0.5)
img1.shift(3 * LEFT + UP)
img2 = ImageMobject("img/barcode.png")
img2.scale(0.5)
img2.shift(3 * RIGHT + UP)
img3 = ImageMobject("img/cd.png")
img3.scale(0.75)
img3.shift(2 * DOWN)
self.play(FadeIn(img1), FadeIn(img2), FadeIn(img3))
self.wait()
self.embed()
class ModularIntro(Scene):
def construct(self):
title = TexText("Modular Arithmetic", color=A_YELLOW)
title.shift(3 * UP)
title.scale(1.5)
self.play(Write(title))
self.wait()
s = Text("{0, 1, 2, 3, 4}", t2c={str(i): A_ORANGE for i in range(5)})
s.scale(1.5)
s.move_to(1.5 * UP)
ff = Tex(
r"(+, -, \cross, \olddiv) \mod {p}", tex_to_color_map={"{p}": A_AQUA})
ff.scale(1.5)
self.play(title.shift, 3.5 * LEFT)
ff.next_to(title, RIGHT)
ff.shift(0.5 * RIGHT)
self.play(Write(ff))
self.wait()
eq = Tex(
r"a \equiv b \mod p",
tex_to_color_map={"a": A_ORANGE, "b": A_ORANGE, "p": A_AQUA}
)
eq.scale(2.5)
self.play(Write(eq))
self.wait()
b1 = Brace(eq[1])
b2 = Brace(eq[3:])
t1 = b1.get_text("Congruent")
t2 = b2.get_text("Same Remainder")
t1.shift(0.25 * DOWN)
t2.shift(0.25 * DOWN)
self.play(GrowFromCenter(b1), Write(t1))
self.play(GrowFromCenter(b2), Write(t2))
self.wait()
eq0 = Tex(
r"1 \equiv 6 \mod 5",
tex_to_color_map={"1": A_ORANGE, "6": A_ORANGE, "5": A_AQUA}
)
eq0.scale(2.5)
eq0.shift(0.25 * UP)
self.play(TransformMatchingTex(eq, eq0))
self.wait()
self.play(Uncreate(b1), Uncreate(b2), Uncreate(
t1), Uncreate(t2), Uncreate(eq0))
self.wait()
self.play(Write(s))
self.wait()
eq1 = Tex(
"3 + 3 = 6", tex_to_color_map={"3": A_ORANGE, "6": A_ORANGE, "5": A_AQUA})
eq1.scale(1.5)
eq2 = Tex("3 + 3 \equiv 1 \mod 5",
tex_to_color_map={"3": A_ORANGE, "1": A_ORANGE, "5": A_AQUA})
eq2.scale(1.5)
eq2.move_to(0 * UP + 0.5 * LEFT, RIGHT)
eq1.move_to(eq2, LEFT)
self.play(Write(eq1))
self.wait()
self.play(TransformMatchingTex(eq1, eq2))
self.wait()
self.play(Indicate(eq2[3], scale_factor=1.5))
self.wait()
self.play(s.move_to, 4 * RIGHT + 1.5 * UP)
c = Circle(radius=2, stroke_color=A_GREY)
c.next_to(s, DOWN)
c_nums = VGroup()
for i, t in enumerate(np.linspace(PI/2, PI/2 - 2*PI, 6)[:-1]):
x, y = 1.5 * np.cos(t), 1.5 * np.sin(t)
t = Tex(str(i), color=A_ORANGE)
t.scale(1.5)
t.move_to(c.get_center())
t.shift(x * RIGHT + y * UP)
c_nums.add(t)
self.play(Write(c))
anims = []
s_num = range(1, 14, 3)
for i in range(5):
anims.append(TransformFromCopy(s[s_num[i]], c_nums[i]))
self.play(*anims, run_time=5)
self.wait()
self.play(
Uncreate(s),
ApplyMethod(
VGroup(c, c_nums).move_to, 4 * RIGHT + 0.5 * DOWN)
)
eq3 = Tex(
"4 + 3 \equiv 2 \mod 5",
tex_to_color_map={
"2": A_ORANGE, "3": A_ORANGE,
"4": A_ORANGE, "5": A_AQUA
}
)
eq3.scale(1.5)
eq3.move_to(eq2, LEFT)
self.play(FadeOut(eq2, UP), FadeIn(eq3, UP))
b_rects = VGroup()
for num in c_nums:
b = SurroundingRectangle(num, color="#f5fd62", buff=0.15)
b_rects.add(b)
b1 = SurroundingRectangle(eq3[0], color="#f5fd62", buff=0.15)
self.play(Write(b1))
self.wait()
for i in range(4, 8):
self.play(Transform(b1, b_rects[i % 5]))
self.wait(0.5)
self.wait(1)
eq4 = Tex(
"1 - 3 \equiv 3 \mod 5",
tex_to_color_map={
"1": A_ORANGE, "3": A_ORANGE,
"5": A_AQUA
}
)
eq4.scale(1.5)
eq4.move_to(eq3, LEFT)
self.play(FadeOut(eq3, UP), FadeIn(eq4, UP), Uncreate(b1))
self.wait()
b1 = SurroundingRectangle(eq4[0], color="#f5fd62", buff=0.15)
self.play(Write(b1))
self.wait()
for i in range(6, 2, -1):
self.play(Transform(b1, b_rects[i % 5]))
self.wait(0.5)
self.wait(1)
self.play(Uncreate(b1))
self.wait()
eq5 = Tex(
"1 + (-3) \equiv 3 \mod 5",
tex_to_color_map={
"1": A_ORANGE, "3": A_ORANGE,
"5": A_AQUA
}
)
eq5.scale(1.5)
eq5.move_to(eq4, LEFT)
self.play(TransformMatchingTex(eq4, eq5))
self.wait()
eq6 = Tex("a", "+", "(-a)", "=", "0", tex_to_color_map={"a": A_PINK})
eq6.scale(1.5)
eq6.move_to(eq5, LEFT)
eq6.shift(DOWN)
b = Brace(eq6[2:5])
t_b = b.get_text("Additive inverse", color=A_GREEN)
t_b.scale(1.5)
t_b.shift(0.25 * DOWN)
eq7 = Tex("a", "+", "(-a)", "\equiv", "0", "\mod 5",
tex_to_color_map={"a": A_PINK, "5": A_AQUA})
eq7.scale(1.5)
eq7.move_to(eq6, LEFT)
self.play(ApplyMethod(eq5.shift, UP))
self.play(Write(eq6))
self.wait()
self.play(Write(b), Write(t_b))
self.wait()
self.play(TransformMatchingTex(eq6, eq7))
self.wait()
eq8 = Tex("3", "+", "(-3)", "\equiv", r"0 \mod 5",
tex_to_color_map={"3": A_ORANGE, "5": A_AQUA})
eq8.scale(1.5)
eq8.move_to(eq7, LEFT)
eq9 = Tex("3", "+", "2", "\equiv", r"0 \mod 5",
tex_to_color_map={"3": A_ORANGE, "2": A_ORANGE, "5": A_AQUA})
eq9.scale(1.5)
eq9.move_to(eq7, LEFT)
n = Tex("2", color=A_ORANGE)
n.scale(1.5)
n.move_to([-4.09011112, -0.94835368, 0])
self.play(FadeOut(VGroup(eq7, t_b, b), DOWN))
self.wait()
grp = VGroup(eq5[1][2], eq5[2])
grp2 = VGroup(eq8[2][1], eq8[3])
eq8_2 = eq8[2]
eq8.remove(eq8[2], eq8[3])
self.play(TransformFromCopy(grp, grp2))
self.wait()
self.play(Write(VGroup(eq8_2[0], eq8)))
self.wait()
self.play(Transform(grp2, n))
self.wait()
cp = n.copy()
cp.move_to(grp)
self.play(Transform(grp, cp), TransformFromCopy(n, cp))
self.play(Uncreate(VGroup(eq8, grp2, eq8_2)),
ApplyMethod(VGroup(eq5, cp).shift, DOWN))
self.wait()
self.play(FadeOut(VGroup(eq5, cp), UP))
self.wait()
eq10 = Tex(
"2 \cdot 3 = 6", tex_to_color_map={"2": A_ORANGE, "3": A_ORANGE, "6": A_ORANGE, "5": A_AQUA})
eq10.scale(1.5)
eq11 = Tex("2 \cdot 3 \equiv 1 \mod 5",
tex_to_color_map={"2": A_ORANGE, "3": A_ORANGE, "1": A_ORANGE, "5": A_AQUA})
eq11.scale(1.5)
eq11.move_to(0 * UP + 0.5 * LEFT, RIGHT)
eq10.move_to(eq11, LEFT)
self.play(Write(eq10))
self.wait()
self.play(TransformMatchingTex(eq10, eq11))
self.wait()
eq12 = Tex(r"2+2+2 \equiv 1 \mod 5",
tex_to_color_map={"2": A_ORANGE, "1": A_ORANGE, "5": A_AQUA})
eq12.scale(1.5)
eq12.move_to(eq11, LEFT)
eq12.shift(1.5 * DOWN)
self.play(ApplyMethod(eq11.shift, 0.5 * UP))
self.wait(0.5)
b1 = SurroundingRectangle(eq11[0], color="#f5fd62", buff=0.15)
self.play(TransformFromCopy(eq11[0:3], eq12[:5]), Write(eq12[5:]))
self.wait()
self.play(Write(b1))
self.wait()
self.play(Indicate(eq12[0], scale_factor=1.5))
self.play(Transform(b1, b_rects[2]))
for i in range(1, 3):
self.play(Indicate(eq12[2*(i-1)+1:2*i+1], scale_factor=1.5))
for j in range(3):
self.play(Transform(b1, b_rects[(2*i + j) % 5]))
self.wait()
self.play(Uncreate(VGroup(b1, eq11, eq12)))
self.wait()
eq13 = Tex(r"a", r" \cdot ", r"a^{-1} ", r"\equiv 1 \mod 5",
tex_to_color_map={"a": A_PINK, "5": A_AQUA, "1": A_ORANGE})
eq13.scale(1.5)
eq13.move_to(0.25 * LEFT, RIGHT)
b = Brace(eq13[2:5])
b_t = b.get_text("Multiplicative Inverse", color=A_GREEN)
b_t.scale(1.25)
b_t.shift(0.5 * DOWN + 1.5 * RIGHT)
self.play(Write(eq13))
self.wait()
self.play(Write(b), Write(b_t))
self.wait()
self.play(ApplyMethod(VGroup(eq13, b, b_t).shift, DOWN))
self.wait()
eq14 = Tex("4", r" \olddiv", "2 \equiv 4 \cdot 2^{-1} \mod 5", tex_to_color_map={
"5": A_AQUA, "4": A_ORANGE, "2": A_ORANGE, "1": A_ORANGE})
eq14.scale(1.5)
eq14.move_to(eq13, LEFT)
eq14.shift(2 * UP)
eq15 = Tex("4", r" \olddiv", "2 \equiv 4 \cdot 3 \mod 5", tex_to_color_map={
"5": A_AQUA, "4": A_ORANGE, "2": A_ORANGE, "3": A_ORANGE})
eq15.scale(1.5)
eq15.move_to(eq14, LEFT)
eq15.shift(0.07 * DOWN)
eq16 = Tex("4", r" \olddiv", "2 \equiv 2 \mod 5", tex_to_color_map={
"5": A_AQUA, "4": A_ORANGE, "2": A_ORANGE, "3": A_ORANGE})
eq16.scale(1.5)
eq16.move_to(eq15, LEFT)
self.play(Write(eq14))
self.wait()
self.play(Transform(eq14[5:], eq15[5:]))
self.wait()
self.play(Transform(eq14[4:], eq16[4:]))
self.wait()
self.play(Uncreate(VGroup(c, eq14, b, b_t, eq13, c, c_nums)))
self.wait()
l1 = Line(10 * LEFT, 10 * RIGHT)
l1.move_to(2 * UP)
l2 = Line(2 * UP, 10 * DOWN)
l3 = Line(10 * LEFT, 10 * RIGHT)
l3.move_to(0.75 * UP)
title1 = TexText("Addition", color=A_GREEN)
title1.scale(1.25)
title1.move_to(FRAME_WIDTH/4 * LEFT + 1.375 * UP)
title2 = TexText("Multiplication", color=A_GREEN)
title2.scale(1.25)
title2.move_to(FRAME_WIDTH/4 * RIGHT + 1.3 * UP)
m = {"a": A_PINK, "b": A_PINK, "p": A_AQUA}
def create_eq(eq, n, direction):
return Tex(eq, tex_to_color_map=m if n < 3 else {**m, "c": A_PINK}).shift(FRAME_WIDTH/4 * direction + (n-1) * 0.75 * DOWN)
add_eqs = VGroup()
add_tex = [
"(a + b) + c \equiv a + (b + c)",
"a + b \equiv b + a",
"a(b+c) \equiv ab+ac",
"a + 0 \equiv a \equiv 0 + a",
"a + (-a) \equiv 0 \equiv (-a) + a"
]
mul_eqs = VGroup()
mul_tex = [
"(ab)c \equiv a(bc)",
"ab \equiv ba",
"(a+b)c \equiv ac + bc",
r"a \cdot 1 \equiv a \equiv 1 \cdot a",
r"aa^{-1} \equiv 1 \equiv a^{-1} a; a \neq 0",
]
for i in range(5):
add_eqs.add(create_eq(add_tex[i] + r" \mod p", i+1, LEFT))
mul_eqs.add(create_eq(mul_tex[i] + r" \mod p", i+1, RIGHT))
self.play(Write(l1), Write(l2), Write(l3))
self.play(Write(title1))
self.play(Write(add_eqs))
self.play(Write(title2))
self.play(Write(mul_eqs))
self.wait()
self.embed()
class LagrangeIntro(Scene):
CONFIG = {
"fade_opacity": 0.15,
"curve_opacity": 1
}
def construct(self):
x_vals = [1, 3, 4]
y_vals = [2, 2, -1]
coors = [(x_vals[i], y_vals[i]) for i in range(3)]
colors = [A_RED, A_YELLOW, A_BLUE]
curve_kwargs = {
"stroke_width": 6,
"stroke_opacity": self.curve_opacity
}
axes = Axes(
x_range=(0, 5),
y_range=(-3, 3),
axis_config={"include_tip": False},
x_axis_config={"stroke_width": 6},
y_axis_config={"stroke_width": 6}
)
points = VGroup()
labels = VGroup()
offsets = [0.5 * RIGHT, 0.5 * LEFT, 0.75 * LEFT]
for i, (x, y) in enumerate(coors):
d = Dot(
axes.c2p(x, y),
radius=2 * DEFAULT_DOT_RADIUS, color=A_GREEN
)
t = Tex(f"({x}, {y})")
t.next_to(d, DOWN)
t.shift(offsets[i])
points.add(d)
labels.add(t)
labels_cp = labels.copy()
c = ParametricCurve(
lambda t: axes.c2p(t, self.func(t)),
t_range=(0, 5), stroke_width=6, color=A_GREEN
)
eq = Tex(
"p(x) = -x^2 + 4x - 1",
tex_to_color_map={"p": A_GREEN, "x": A_PINK}
)
eq.scale(1.5)
eq.shift(3.25 * UP)
grp = VGroup(axes, c, points, labels)
self.play(Write(axes), Write(points), Write(labels))
self.wait()
self.play(Write(c))
self.wait()
self.play(ApplyMethod(grp.shift, 0.5 * DOWN))
self.play(Write(eq))
self.wait()
self.play(FadeOut(VGroup(eq, c)), ApplyMethod(grp.shift, 0.5 * UP))
def l1(x): return (x-3)*(x-4)/6
l1_c = ParametricCurve(
lambda t: axes.c2p(t, l1(t)),
t_range=(0, 5), color=colors[0], stroke_width=6,
stroke_opacity=self.curve_opacity
)
l1_cp = l1_c.copy()
l1_cp.set_stroke(opacity=self.fade_opacity)
def l2(x): return (x-1)*(x-4)/-2
l2_c = ParametricCurve(
lambda t: axes.c2p(t, l2(t)),
t_range=(0, 5), color=colors[1], stroke_width=6,
stroke_opacity=self.curve_opacity
)
l2_cp = l2_c.copy()
l2_cp.set_stroke(opacity=self.fade_opacity)
def l3(x): return (x-1)*(x-3)/3
l3_c = ParametricCurve(
lambda t: axes.c2p(t, l3(t)),
t_range=(0, 5), color=colors[2], stroke_width=6,
stroke_opacity=self.curve_opacity
)
l3_cp = l3_c.copy()
l3_cp.set_stroke(opacity=self.fade_opacity)
l_c = VGroup(l1_c, l2_c, l3_c)
l_cp = l_c.deepcopy()
l_faded = VGroup(l1_cp, l2_cp, l3_cp)
self.play(
Write(l_c),
)
self.wait()
def hide_all_but(idx):
anims = []
for i in range(3):
if i != idx:
anims.append(Transform(l_c[i], l_faded[i]))
self.play(*anims)
def show_all():
anims = []
for i in range(3):
anims.append(Transform(l_c[i], l_cp[i]))
self.play(*anims)
def one_hot(x_val, curve_index):
return 1 if x_val == x_vals[curve_index] else 0
def show_curve(curve_index):
points.set_opacity(self.fade_opacity)
hide_all_but(curve_index)
points_0 = VGroup()
for x in x_vals:
points_0.add(
Dot(axes.c2p(x, one_hot(x, curve_index)),
radius=0.1, color=colors[curve_index])
)
labels_0 = VGroup()
for i in range(3):
labels_0.add(
Tex(f"({x_vals[i]}, {one_hot(x_vals[i], curve_index)})"
).next_to(points_0[i], UP)
)
self.play(Write(points_0), Transform(labels, labels_0))
self.wait()
self.play(Indicate(labels[curve_index]))
self.wait()
self.play(Transform(labels, labels_cp),
ApplyMethod(points.set_opacity, 1),
Uncreate(points_0))
show_all()
self.wait()
show_curve(0)
show_curve(1)
show_curve(2)
eq1 = Tex(
"l_1(x)", "=", "(x-3)", "(x-4)",
tex_to_color_map={"x": A_PINK, "l_1": A_RED,
"3": A_ORANGE, "4": A_ORANGE}
)
eq1.scale(1.5)
eq1.shift(3 * UP)
eq2 = Tex(
"l_1(x)", "=", r"{{1} \over {6}}", "(x-3)", "(x-4)",
tex_to_color_map={"x": A_PINK, "l_1": A_RED,
"{1}": A_ORANGE, "{6}": A_ORANGE,
"3": A_ORANGE, "4": A_ORANGE}
)
eq2.scale(1.5)
eq2.shift(3 * UP)
def p(x): return 2*l1(x) + 2*l2(x) - 1*l3(x)
eq3 = Tex(r"p(x) =", r" 2 \cdot l_1(x)", r" + 2 \cdot ", r"l_2(x)", r" + (-1) \cdot ", r"l_3(x)",
tex_to_color_map={"l_1": A_RED, "l_2": A_YELLOW, "l_3": A_BLUE, "x": A_PINK, "p": A_GREEN})
eq3.shift(3.25 * UP)
eq3.scale(1.5)
def func(t):
return axes.c2p(t, p(t))
self.play(Uncreate(l_c))
self.wait()
self.play(Write(eq1))
self.wait()
self.play(TransformMatchingTex(eq1, eq2))
self.wait()
self.play(TransformFromCopy(eq2, l_cp[0]))
self.wait()
self.play(FadeOut(VGroup(eq2[4:], l_cp[0], l_c[0])))
grp2 = VGroup(axes, points, labels)
c.shift(0.5 * DOWN)
l_cp.shift(0.5 * DOWN)
self.play(
ApplyMethod(grp2.shift, 0.5 * DOWN),
ReplacementTransform(eq2[:4], eq3[5:9])
)
def func_lc_1(t):
return axes.c2p(t, 2 * l1(t))
def func_lc_2(t):
return axes.c2p(t, 2*l1(t) + 2*l2(t))
def func_lc_3(t):
return axes.c2p(t, 2*l1(t) + 2*l2(t) + -1*l3(t))
color_avg = rgb_to_hex((hex_to_rgb(A_RED) + hex_to_rgb(A_YELLOW))/2)
lc_1 = ParametricCurve(func_lc_1, t_range=(
0, 5), color=colors[0], **curve_kwargs)
lc_2 = ParametricCurve(func_lc_2, t_range=(
0, 5), color=color_avg, **curve_kwargs)
lc_3 = ParametricCurve(func_lc_3, t_range=(
0, 5), color=A_GREEN, **curve_kwargs)
self.play(Write(eq3[:4]))
self.wait()
self.play(TransformFromCopy(eq3[5:9], l_cp[0]))
self.wait()
self.play(Write(eq3[4]))
self.bring_to_front(eq3[:9], points, labels)
self.play(Indicate(eq3[4]))
self.play(Transform(l_cp[0], lc_1), ApplyMethod(
points[0].set_color, A_RED), run_time=3.5)
self.wait()
self.play(Write(eq3[9:14]))
self.play(Indicate(eq3[9:14]))
self.play(Transform(l_cp[0], lc_2), ApplyMethod(
points[:2].set_color, color_avg), run_time=3.5)
self.wait()
self.play(Write(eq3[14:]))
self.play(Indicate(eq3[14:]))
self.play(Transform(l_cp[0], lc_3), ApplyMethod(
points.set_color, A_GREEN), run_time=3.5)
self.wait()
self.play(TransformMatchingTex(eq3, eq))
self.wait()
self.embed()
def func(self, x):
return -x**2 + 4*x - 1
class LagrangeUnique(Scene):
def construct(self):
dots = Tex(r".\\.\\.\\")
p1 = Tex(
"(x_{1}, y_{1})",
tex_to_color_map={"1": A_ORANGE, "x": A_PINK, "y": A_GREEN}
)
p1.scale(1.5)
p1.shift(1.75 * UP)
p2 = Tex(
"(x_{n}, y_{n})",
tex_to_color_map={"n": A_ORANGE, "x": A_PINK, "y": A_GREEN}
)
p2.scale(1.5)
p2.shift(1.5 * DOWN)
axes = Axes(x_range=(0, 5), y_range=(-3, 3), axis_config={
"include_tip": False}, x_axis_config={"stroke_width": 6}, y_axis_config={"stroke_width": 6})
coors = [[i+np.random.random(), 6*np.random.random()-3]
for i in range(5)]
points = VGroup()
labels = VGroup()
for i, (x, y) in enumerate(coors):
points.add(
Dot(axes.c2p(x, y), color=A_ORANGE, radius=2*DEFAULT_DOT_RADIUS)
)
labels.add(
Tex(f"(x_{{{i+1}}}, y_{{{i+1}}})",
tex_to_color_map={
str(i+1): A_ORANGE,
"x": A_PINK, "y": A_GREEN}
).next_to(points[i], DOWN)
)
self.play(Write(axes))
self.play(Write(points), Write(labels))
self.wait()
grp1 = VGroup(axes, points, labels)
grp2 = VGroup(dots, p1, p2)
grp2.shift(4 * RIGHT)
l = Line(10 * UP, 10 * DOWN)
l.shift(2 * RIGHT)
self.play(Transform(grp1, grp2))
self.play(Write(l))
self.wait()
l1 = Line(10 * LEFT, 2 * RIGHT)
l1.shift(2.25 * UP)
l2 = Line(10 * LEFT, 2 * RIGHT)
l2.shift(0.7 * DOWN)
eq1 = Tex(
"p(x_i)", "=", "q(x_i)", "= y_i",
tex_to_color_map={"p": A_LAVENDER, "q": A_AQUA,
"y": A_GREEN, "i": A_ORANGE, "x": A_PINK}
)
eq1.scale(1.5)
eq1.move_to(2.5 * LEFT + 3 * UP)
eq2 = Tex("r(x)", "=", "p(x)", "-", "q(x)",
tex_to_color_map={"x": A_PINK, "p": A_LAVENDER, "q": A_AQUA, "r": A_RED})
eq2.scale(1.5)
eq2.next_to(eq1, DOWN).shift(0.5 * DOWN)
self.play(Write(eq1))
self.wait()
self.play(TransformFromCopy(eq1[:4], eq2[5:9]), TransformFromCopy(
eq1[6:10], eq2[10:]), Write(eq2[:5]), Write(eq2[9]), run_time=2)
self.play(Write(l1))
self.wait()
eq3 = Tex(r"\mathrm{deg} \ r(x)", r"\leq n-1", tex_to_color_map={"r": A_RED,
"x": A_PINK, r"\mathrm{deg}": A_YELLOW, "n": A_ORANGE, "1": A_ORANGE})
eq3.scale(1.5)
eq3.next_to(eq2, DOWN).shift(0.5 * DOWN)
self.play(TransformFromCopy(eq2[:4], eq3[1:5]), Write(
eq3[0]), Write(eq3[5:]), run_time=2)
self.wait()
eq4 = Tex(r"r(x_i) = p(x_i) - q(x_i)", tex_to_color_map={
"i": A_ORANGE, "p": A_LAVENDER,
"q": A_AQUA, "x": A_PINK, "y": A_GREEN, "r": A_RED}
)
eq4.scale(1.5)
eq4.next_to(eq3, DOWN).shift(0.65 * DOWN)
eq4_1 = Tex(r"r(x_i)", "= 0", tex_to_color_map={
"i": A_ORANGE, "r": A_RED, "x": A_PINK})
eq4_1.scale(1.5)
eq4_1.move_to(eq4, LEFT)
self.play(Write(l2))
self.play(Write(eq4))
self.wait()
self.play(Transform(eq4[5:], eq4_1[5]))
self.play(ApplyMethod(eq4.shift, (eq3.get_center()
[1] - eq4.get_center()[1]) * RIGHT))
self.wait()
eq5 = Tex(r"\mathrm{deg} \ r(x)", r"=n", tex_to_color_map={
"r": A_RED, "x": A_PINK, r"\mathrm{deg}": A_YELLOW, "n": A_ORANGE, "1": A_ORANGE})
eq5.scale(1.5)
eq5.next_to(eq4, DOWN).shift(0.5 * DOWN)
self.play(Write(eq5))
self.wait()
self.embed()