main.py

import math

import numpy as np
import sympy as sy
from manim import *
from manim_slides import Slide
from shapely.geometry import LineString

"""
Some useful function required for the 'simple example'.
"""


def row(*args):
    """Create a symbol row (or col) vector from input arguments."""
    return sy.Matrix(args)


def generate_c(as_gamma=False):
    """Return C(x) and it's derivative."""
    # Unknowns
    s1, s2 = sy.symbols("s_1 s_2", real=True)  # s1, s2 are real values

    # Geometry

    # - Nodes
    BS = row(2, -1)
    UE = row(2, 4)

    # - Interaction points
    X1 = row(s1, s1)
    X2 = row(5, s2)

    # - Surface normals
    n1 = row(1, -1).normalized()
    n2 = row(-1, 0).normalized()

    # - Aliases
    V0 = X1 - BS
    V1 = X2 - X1
    V2 = UE - X2

    if as_gamma:
        g1 = sy.Function(r"\gamma_1", real=True)(s1, s2)
        g2 = sy.Function(r"\gamma_2", real=True)(s1, s2)
    else:
        g1 = V0.norm() / V1.norm()
        g2 = V1.norm() / V2.norm()

    # Write different equations
    eqs = [
        g1 * V1 - (V0 - 2 * V0.dot(n1) * n1),
        g2 * V2 - (V1 - 2 * V1.dot(n2) * n2),
    ]

    F = sy.Matrix.vstack(*eqs)
    f = F.norm() ** 2

    _df = sy.lambdify((s1, s2), row(f.diff(s1), f.diff(s2)))

    def df(x):
        return _df(*x).reshape(-1)

    return sy.lambdify((s1, s2), f), df


def generate_c_ris(phi=0):
    """Return C(x) and it's derivative for a RIS."""
    # Unknowns
    s1, s2 = sy.symbols("s_1 s_2", real=True)  # s1, s2 are real values

    # Geometry

    # - Nodes
    BS = row(2, -1)
    UE = row(2, 4 - 0.5)

    # - Interaction points
    X1 = row(s1, s1)
    X2 = row(5, s2)

    # - Surface normals
    n1 = row(1, -1).normalized()
    n2 = row(-1, 0).normalized()

    # Aliases
    V0 = X1 - BS
    V1 = X2 - X1
    V2 = UE - X2

    cross = n2[0] * V2[1] - n2[1] * V2[0]

    g1 = V0.norm() / V1.norm()

    # Write different equations
    eqs = [
        g1 * V1 - (V0 - 2 * V0.dot(n1) * n1),
        row(cross - sy.sin(phi) * V2.norm()),
    ]

    F = sy.Matrix.vstack(*eqs)
    f = F.norm() ** 2

    _df = sy.lambdify((s1, s2), row(f.diff(s1), f.diff(s2)))

    def df(x):
        return _df(*x).reshape(-1)

    return sy.lambdify((s1, s2), f), df


def gradient_descent(x0, df, tol=1e-12, max_it=100, return_steps=False):
    """Perform a gradient descent using optimal alpha for linear systems."""
    xa = x0
    dfxa = df(xa)
    xb = xa - 0.25 * dfxa  # First step, alpha = .5
    dfxb = df(xb)

    dx = xb - xa
    dfx = dfxb - dfxa

    n_it = 1

    steps = [dx]

    while np.linalg.norm(dx) > tol and n_it < max_it:
        alpha = np.dot(dx, dfx) / np.linalg.norm(dfx) ** 2
        xa, xb = xb, xb - alpha * dfxb
        dfxa, dfxb = dfxb, df(xb)
        dx = xb - xa
        dfx = dfxb - dfxa
        n_it += 1

        if return_steps:
            steps.append(dx)

    if return_steps:
        return steps
    return xb


"""
Here, because I switched the background from black to white,
so I have to make default color for most things to be black (instead of white).
"""


def black(func):
    """Sets default color to black"""

    def wrapper(*args, color=BLACK, **kwargs):
        return func(*args, color=color, **kwargs)

    return wrapper


Tex = black(Tex)
Text = black(Text)
MathTex = black(MathTex)
Line = black(Line)
Dot = black(Dot)
Brace = black(Brace)
Arrow = black(Arrow)
Angle = black(Angle)


"""
Slides generation
"""


class Main(Slide):
    def __init__(self, *args, **kwargs):
        super().__init__(*args, **kwargs)
        self.slide_no = None
        self.slide_text = None

    def write_slide_number(self, inital=1, text=Tex, animation=Write, position=ORIGIN):
        self.slide_no = inital
        self.slide_text = text(str(inital)).shift(position)
        return animation(self.slide_text)

    def update_slide_number(self, text=Tex, animation=Transform):
        self.slide_no += 1
        new_text = text(str(self.slide_no)).move_to(self.slide_text)
        return animation(self.slide_text, new_text)

    def construct(self):
        self.camera.background_color = WHITE
        WALL_COLOR = ORANGE
        BS_COLOR = BLUE
        UE_COLOR = MAROON_D
        GOOD_COLOR = "#28C137"
        BAD_COLOR = "#FF0000"
        IMAGE_COLOR = "#636463"
        X_COLOR = DARK_BROWN

        NW = Dot().to_corner(UL)
        NE = Dot().to_corner(UR)
        SW = Dot().to_corner(DL)
        SE = Dot().to_corner(DR)
        NL = Line(NW.get_center(), NE.get_center()).set_color(WALL_COLOR)
        SL = Line(SW.get_center(), SE.get_center()).set_color(WALL_COLOR)
        WL = Line(NW.get_center(), SW.get_center()).set_color(WALL_COLOR)
        EL = Line(NE.get_center(), SE.get_center()).set_color(WALL_COLOR)

        slide_no_pos = SE.shift(0.15 * RIGHT + 0.2 * DOWN).get_center()

        # TeX Preamble
        tex_template = TexTemplate()
        tex_template.add_to_preamble(
            r"""
\usepackage{fontawesome5}
\usepackage{siunitx}
\DeclareSIQualifier\wattref{W}
\DeclareSIUnit\dbw{\decibel\wattref}
\usepackage{amsmath,amssymb,amsfonts,mathtools}
\newcommand{\bs}{\boldsymbol}
\newcommand{\scp}[3][]{#1\langle #2, #3 #1\rangle}
\newcommand{\bb}{\mathbb}
\newcommand{\cl}{\mathcal}
"""
        )

        # Slide: Title
        title = VGroup(
            Tex(
                r"\textbf{Differentiable Ray Tracing\\ for \\ Telecommunications}",
                font_size=60,
            ),
            Tex("Jérome Eertmans - July 17th, Siegen"),
        ).arrange(DOWN, buff=1)

        self.play(FadeIn(title), self.write_slide_number(position=slide_no_pos))
        self.next_slide()

        # Slide: about the author

        self.play(FadeOut(title), self.update_slide_number())

        about = about_title, profile, interests, claude, laurent = VGroup(
            Tex("About the author").to_corner(UL),
            Tex(
                r"""Profile:
                \begin{itemize}
                \item PhD student at UCLouvain since 09/2021;
                \item Electromechaninal Engineer in mechatronics;
                \item introduced to Ray Tracing (RT)\\during a student job.
                \end{itemize}
                """,
                font_size=30,
                tex_environment=None,
            )
            .to_corner(UL)
            .shift(1.5 * DOWN),
            Tex(
                r"""Interests:
                \begin{itemize}
                \item Programming;
                \item Writing performant solutions;
                \item and open sourcing content\\(jeertmans on \faGithub{} or eertmans.be).
                \end{itemize}
                """,
                font_size=30,
                tex_environment=None,
                tex_template=tex_template,
            )
            .to_corner(UR)
            .shift(1.5 * DOWN),
            Tex("Claude Oestges", font_size=40).to_corner(DL).shift(2 * UP),
            Tex("Laurent Jacques", font_size=40),
        )

        laurent.next_to(claude, 2 * RIGHT)

        claude_image, laurent_image = (
            ImageMobject("claude.jpeg").next_to(claude, DOWN),
            ImageMobject("laurent.jpeg").next_to(laurent, DOWN),
        )

        self.play(FadeIn(about_title))

        self.next_slide()

        self.play(FadeIn(about[1:]), FadeIn(claude_image), FadeIn(laurent_image))

        # Slide: room

        self.next_slide()
        self.play(
            FadeOut(about),
            FadeOut(claude_image),
            FadeOut(laurent_image),
            self.update_slide_number(),
        )

        # Slide: introduction to RT

        self.next_slide()

        BS = speaker = Tex(r"\faWifi", tex_template=tex_template, color=BS_COLOR).shift(
            4 * LEFT
        )
        UE = listener = Tex(
            r"\faPhone", tex_template=tex_template, color=UE_COLOR
        ).shift(3 * RIGHT)

        public = VGroup()

        for i in range(-2, 3):
            for j in range(-2, 3):
                if i != 0 or j != 0:
                    public.add(listener.copy().shift(i * UP + j * LEFT))

        self.play(Write(speaker), Write(listener), Write(public))
        self.next_slide()

        # Slide: show emetting sound waves & received power

        circle = Circle(radius=1, color=BLUE).shift(speaker.get_center())
        self.start_loop()
        self.play(Broadcast(circle, focal_point=speaker.get_center(), run_time=6))
        self.end_loop()

        arrow_end = Dot(public[13].get_center())
        arrow_end.set_opacity(0)
        arrow = always_redraw(
            lambda: Arrow(start=speaker.get_center(), end=arrow_end, color=GOLD)
        )

        brace = always_redraw(lambda: Brace(arrow, UP))

        def fspl(x):
            f = 1e9
            c = 3e8
            return -20 * math.log10(4 * PI * x * f / c)

        def delay_ns(x):
            c = 3e8
            p = 1e9
            return p * x / c

        _, power, _ = power_label = VGroup(
            MathTex("P = "),
            DecimalNumber(
                fspl(arrow.get_length()),
                num_decimal_places=2,
                include_sign=True,
            ),
            Tex(r"\si{\dbw}", tex_template=tex_template),
        )

        _, delay, ns = delay_label = VGroup(
            MathTex(r"\tau = "),
            DecimalNumber(
                delay_ns(arrow.get_length()),
                num_decimal_places=2,
                include_sign=False,
            ),
            Tex(r"\si{\nano\second}", tex_template=tex_template),
        )

        power_label.set_color(BLUE)
        power_label.arrange(RIGHT)
        power_label.next_to(brace, UP)
        always(power_label.next_to, brace, UP)
        f_always(power.set_value, lambda: fspl(arrow.get_length()))

        delay_label.set_color(BLUE)
        delay_label.arrange(RIGHT)
        delay_label.next_to(brace, UP)
        always(delay_label.next_to, brace, UP)
        f_always(delay.set_value, lambda: delay_ns(arrow.get_length()))

        for t in [*power_label, *delay_label]:
            t.align_to(power, DOWN)

        power_label[2].align_to(power, UP)

        self.play(Create(arrow))
        self.play(
            # FadeIn(brace, shift=DOWN),
            FadeIn(power_label, shift=DOWN)
        )

        self.next_slide()

        arrow_end.save_state()

        for pub in [public[4], public[0], public[-5], public[-1]]:
            self.play(
                arrow_end.animate.move_to(pub.get_center()),
                run_time=2,
            )

        self.play(Restore(arrow_end), run_time=4)

        self.next_slide()

        # Slide: transform power in delay
        power_label.save_state()

        self.play(
            Transform(power_label[0], delay_label[0]),
            FadeTransform(power_label[1], delay_label[1]),
            Transform(power_label[2], delay_label[2]),
        )
        self.remove(*power_label)
        self.add(delay_label)
        self.next_slide()

        arrow_end.save_state()

        for pub in [public[4], public[0], public[-5], public[-1]]:
            self.play(
                arrow_end.animate.move_to(pub.get_center()),
                # Broadcast(circle, focal_point=speaker.get_center()),
                run_time=2.5,
            )

        self.play(Restore(arrow_end), run_time=4)

        self.next_slide()

        # Slide: only focus on one listener
        to_fade_out = VGroup(
            public,
            delay_label,
            arrow,
        )
        self.play(
            self.update_slide_number(), FadeOut(to_fade_out, shift=DOWN, run_time=4)
        )

        self.next_slide()
        self.remove(speaker, listener)
        self.add(BS, UE)

        self.play(Create(NL), Create(SL), Create(WL), Create(EL))
        self.next_slide()

        A = BS.copy().shift(0.5 * RIGHT)
        B = UE.copy().shift(0.5 * LEFT)

        LOS = Arrow(
            A.get_center(),
            B.get_center(),
            stroke_width=6,
            buff=0.0,
        )

        self.play(Write(LOS))

        self.next_slide()

        # Slide: multiple paths in indoor environment

        paths = VGroup()

        x = LOS.get_center()[0]
        for wall in [NL, SL]:
            y = wall.get_center()[1]
            middle = [x, y, 0]
            path = VGroup(
                Line(A.get_center(), middle, stroke_width=6),
                Arrow(
                    middle,
                    Dot(UE.get_center()).shift(UP * 0.5 * np.sign(y) + 0.25 * LEFT),
                    stroke_width=6,
                    buff=0.0,
                ),
            )
            path.z_index = -1
            paths.add(path)
            for p in path:
                self.play(Write(p))

            self.wait(0.1)
            self.next_slide()

        channel = MathTex(r"P, \tau, \phi...")
        channel.next_to(UE, UP + RIGHT)

        self.play(Write(channel))
        self.next_slide()

        self.play(FadeOut(paths), FadeOut(channel))

        self.next_slide()

        # Slide: challenge

        self.play(FadeOut(LOS))

        how_to = Tex("How to find all paths?")
        ray_tracing = Tex("Multiple methods exist!")

        group = VGroup(how_to, ray_tracing).arrange(DOWN)

        self.play(FadeIn(how_to))
        self.next_slide()

        self.play(FadeIn(ray_tracing, shift=UP))
        self.next_slide()

        # Slide: outline

        _, sec1, sec2, sec3, sec4, sec5 = outline = VGroup(
            Tex(r"\textbf{Outline:}"),
            Tex("1. RT for Telecom"),
            Tex("2. Image Method"),
            Tex("3. Min-Path-Tracing"),
            Tex("4. About differentiability"),
            Tex("5. Current work"),
        ).arrange(DOWN)

        for t in outline[2:]:
            t.align_to(outline[1], LEFT)

        self.play(FadeOut(group), self.update_slide_number())
        self.play(FadeIn(outline[0]))
        self.next_slide()

        for t in outline[1:]:
            self.play(FadeIn(t, shift=UP))
            self.next_slide()

        # Sec. 1

        # Slide: simple example

        outline -= sec1
        self.play(FadeOut(outline), self.update_slide_number())

        for sec in outline[2:]:
            sec.to_corner(UL)

        BS_dot, I1, I2, UE_dot, W1, W2, X1, X2 = locs = VGroup(
            Dot([2, -1, 0], color=BS_COLOR),
            Dot([-1, 2, 0], color=IMAGE_COLOR),
            Dot([11, 2, 0], color=IMAGE_COLOR),
            Dot([2, 4, 0], color=UE_COLOR),
            Line([3.3, 3.3, 0], [0, 0, 0], color=WALL_COLOR),
            Line([5, 4, 0], [5, 0.5, 0], color=WALL_COLOR),
            Dot([20 / 7, 20 / 7, 0], color=X_COLOR, stroke_width=2, fill_color=WHITE),
            Dot([5, 10 / 3, 0], color=X_COLOR, stroke_width=2, fill_color=WHITE),
        )

        locs.move_to(ORIGIN)

        X_OFFSET, Y_OFFSET, _ = np.array([2, -1, 0]) - BS_dot.get_center()

        self.play(
            sec1.animate.to_corner(UL),
            BS.animate.move_to(locs[0]),
            UE.animate.move_to(locs[3]),
            Transform(WL, W1),
            Transform(EL, W2),
            FadeOut(NL, shift=UP),
            FadeOut(SL, shift=DOWN),
        )

        # Slide: about RT in telecom

        text_in_telecom = Tex(
            r"""Core points:
        \begin{itemize}
            \item used by telecom operators;
            \item for coverage map;
            \item for antennas placement;
            \item using mix of Ray-Launching\\ and Image RT.
        \end{itemize}""",
            font_size=40,
            tex_environment=None,
        ).next_to(W2, RIGHT)

        self.next_slide()
        self.play(FadeIn(text_in_telecom))

        # Slide: Image Method

        # This is an ugly fix...
        sec1.to_corner(UL)
        sec2.to_corner(UL)

        self.next_slide()
        self.play(
            self.update_slide_number(), Transform(sec1, sec2), FadeOut(text_in_telecom)
        )

        self.play(
            Transform(BS, BS_dot),
            Transform(UE, UE_dot),
        )

        self.next_slide()

        LOS = Arrow(BS, UE)

        self.play(Create(LOS))

        self.next_slide()

        self.play(LOS.animate.set_color(BAD_COLOR))

        self.next_slide()

        self.play(FadeOut(LOS))

        self.next_slide()

        arrow_1 = Arrow(BS, I1)
        arrow_2 = Arrow(I1, I2)
        right_angle_1 = RightAngle(arrow_1, W1, color=RED)
        right_angle_2 = RightAngle(arrow_2, W2, color=RED)

        self.play(Create(arrow_1), Create(right_angle_1))
        self.play(FadeIn(I1))

        self.next_slide()

        self.play(FadeOut(arrow_1), FadeOut(right_angle_1))
        self.play(Create(arrow_2), Create(right_angle_2))
        self.play(FadeIn(I2))
        self.play(FadeOut(arrow_2), FadeOut(right_angle_2))

        self.next_slide()

        line1 = Line(UE, I2)
        line2 = Line(X2, I1)

        self.play(Create(line1))

        self.next_slide()

        self.play(FadeIn(X2))

        self.next_slide()

        self.play(FadeOut(line1))

        self.next_slide()

        self.play(Create(line2))
        self.play(FadeIn(X1))
        self.play(FadeOut(line2))

        self.next_slide()

        path = VGroup(
            Line(BS, X1),
            Line(X1, X2),
            Line(X2, UE),
        )

        for p in path:
            self.play(Create(p))

        self.play(path.animate.set_color(GOOD_COLOR))

        self.next_slide()

        # Slide: summary of image RT

        old_objects = [
            mob for mob in self.mobjects if mob not in [self.slide_text, *outline]
        ]

        self.play(self.update_slide_number(), *[FadeOut(mob) for mob in old_objects])
        path.set_color(BLACK)

        pros = VGroup(
            Tex(r"\textbf{Pros}"),
            Tex(r"- Simple"),
            Tex(r"- Fast - $\cl O (n)$", tex_template=tex_template),
        ).arrange(DOWN)

        for pro in pros[1:]:
            pro.align_to(pros[0], LEFT)

        cons = VGroup(
            Tex(r"\textbf{Cons}"),
            Tex(r"- Limited to planar surfaces"),
            Tex(r"- Specular reflection only"),
        ).arrange(DOWN)

        for con in cons[1:]:
            con.align_to(cons[0], LEFT)

        summary = VGroup(
            Tex("Summary:", font_size=60),
            VGroup(pros, cons).arrange(RIGHT, buff=4),
        ).arrange(DOWN, buff=1)

        self.play(FadeIn(summary[0]))

        self.next_slide()

        self.play(FadeIn(summary[1][0]))

        self.next_slide()

        self.play(FadeIn(summary[1][1]))

        # Sec. 2

        # Slide: MPT

        self.next_slide()

        sec2.to_corner(UL)
        self.play(self.update_slide_number(), FadeOut(summary), Transform(sec1, sec3))

        BS_ = BS.copy().move_to(ORIGIN)
        UE_ = UE.copy().move_to(ORIGIN)
        W1_ = Line([-1.5, 0, 0], [1.5, 0, 0], color=WALL_COLOR)
        VGroup(VGroup(BS_, UE_).arrange(RIGHT, buff=5), W1_).arrange(DOWN, buff=3)

        X1_ = X1.copy().move_to(W1_.get_center())

        # Normal vector
        NV_ = always_redraw(lambda: Line(X1_, X1_.get_center() + 3 * UP).add_tip())
        VIN_ = always_redraw(lambda: Line(BS_, X1_))
        VOUT_ = always_redraw(lambda: Line(X1_, UE_))
        AIN_ = Angle(NV_, VIN_.copy().scale(-1), radius=1.01)
        AIN_ = always_redraw(
            lambda: Angle(NV_, VIN_.copy().scale(-1), radius=1.01, color=BS_COLOR)
        )
        AOUT_ = always_redraw(lambda: Angle(VOUT_, NV_, radius=1.01, color=UE_COLOR))
        ain_ = DecimalNumber(AIN_.get_value(degrees=True), unit=r"^{\circ}")
        ain_.next_to(AIN_, 2 * LEFT)
        aout_ = DecimalNumber(AOUT_.get_value(degrees=True), unit=r"^{\circ}")
        aout_.next_to(AOUT_, 2 * RIGHT)

        angle_in_ = VGroup(AIN_, ain_)
        angle_in_.set_color(BS_COLOR)
        ain_.add_updater(
            lambda m: m.set_value(
                Angle(NV_, VIN_.copy().scale(-1)).get_value(degrees=True)
            )
        )
        always(ain_.next_to, AIN_, 2 * LEFT)

        angle_out_ = VGroup(AOUT_, aout_)
        angle_out_.set_color(UE_COLOR)
        aout_.add_updater(
            lambda m: m.set_value(Angle(VOUT_, NV_).get_value(degrees=True))
        )
        always(aout_.next_to, AOUT_, 2 * RIGHT)

        scene_ = VGroup(BS_, UE_, W1_, X1_, NV_, VIN_, VOUT_)
        angles_ = VGroup(angle_in_, angle_out_)

        self.play(FadeIn(scene_))
        self.next_slide()

        self.add(angles_)
        self.wait(0.1)
        self.next_slide()

        def I_(BS, X1, UE):
            vin = X1.get_center() - BS.get_center()
            vout = UE.get_center() - X1.get_center()
            n = np.array([0, 1, 0])
            vin /= np.linalg.norm(vin)
            vout /= np.linalg.norm(vout)
            error = vout - (vin - 2 * np.dot(vin, n) * n)

            return np.linalg.norm(error) ** 2

        def C_(X1):
            line_y = W1_.get_center()[1]
            y = X1.get_center()[1]

            return (y - line_y) ** 2

        self.play(X1_.animate.move_to(W1_.get_start()))
        self.play(X1_.animate.move_to(W1_.get_end()))
        self.play(X1_.animate.move_to(W1_.get_center()))
        self.wait(0.1)
        self.next_slide()

        cost, i_number, plus, c_number = cost_label = (
            VGroup(
                MathTex(r"\mathcal{C} =", tex_template=tex_template),
                DecimalNumber(I_(BS_, X1_, UE_)),
                MathTex("+"),
                DecimalNumber(C_(X1_)),
            )
            .arrange(RIGHT)
            .next_to(W1_, 2 * DOWN)
            .set_color(BLUE)
        )

        def label_constructor(*args, **kwargs):
            return MathTex(*args, tex_template=tex_template, **kwargs)

        i_brace, c_brace = braces = VGroup(
            BraceLabel(i_number, r"\cl I", label_constructor=label_constructor),
            BraceLabel(c_number, r"\cl F", label_constructor=label_constructor),
        ).set_color(BLUE)

        i_number.add_updater(lambda m: m.set_value(I_(BS_, X1_, UE_)))
        c_number.add_updater(lambda m: m.set_value(C_(X1_)))

        self.play(FadeIn(cost), FadeIn(i_number), FadeIn(i_brace))
        self.next_slide()

        self.play(X1_.animate.move_to(W1_.get_start()))
        self.play(X1_.animate.move_to(W1_.get_end()))
        self.play(X1_.animate.move_to(W1_.get_center()))
        self.wait(0.1)
        self.next_slide()

        self.play(X1_.animate.shift(UP))
        self.wait(0.1)
        self.next_slide()

        self.play(FadeIn(plus, c_number, c_brace))
        self.next_slide()

        self.play(X1_.animate.move_to(W1_.get_center()))
        self.wait(0.1)

        # Slide: any reflection
        self.next_slide()

        arc_ = Arc(
            radius=1.5,
            arc_center=X1_.copy().shift(1.5 * DOWN).get_center(),
            color=WALL_COLOR,
            start_angle=PI,
            angle=-PI,
        )

        interaction = Tex("Reflection")
        interaction.next_to(NV_, UP)

        interaction_eq = MathTex(
            r"\cl I \sim \hat{\bs r} = \hat{\bs \imath} - 2 \scp{\hat{\bs \imath}}{\hat{\bs n}}\hat{\bs n}",
            tex_template=tex_template,
        )
        interaction_eq.to_corner(UR)

        self.play(
            FadeOut(cost_label),
            FadeOut(braces),
            FadeIn(interaction),
            FadeIn(interaction_eq),
        )
        self.next_slide()

        # Diffraction (setup)

        DIFF_W1_A = Polygon(
            W1_.get_start(),
            W1_.get_end(),
            W1_.get_end() + DOWN + 0.25 * LEFT,
            W1_.get_start() + DOWN + 0.25 * LEFT,
            stroke_opacity=0,
            fill_color=WALL_COLOR,
            fill_opacity=0.7,
        )

        DIFF_W1_B = Polygon(
            W1_.get_start(),
            W1_.get_end(),
            W1_.get_end() + 0.8 * DOWN + 0.25 * RIGHT,
            W1_.get_start() + 0.8 * DOWN + 0.25 * RIGHT,
            stroke_opacity=0,
            fill_color=WALL_COLOR,
            fill_opacity=0.5,
        )

        D_NV_ = Line(X1_, X1_.get_center() + RIGHT * 3).add_tip()
        D_AIN_ = Angle(
            D_NV_.copy().scale(-1),
            VIN_.copy().scale(-1),
            radius=1.01,
            other_angle=True,
            color=BS_COLOR,
        )
        D_AOUT_ = Angle(VOUT_, D_NV_, radius=1.01, other_angle=True, color=UE_COLOR)
        D_ain_ = DecimalNumber(
            D_AIN_.get_value(degrees=True), unit=r"^{\circ}", color=BS_COLOR
        )
        D_ain_.next_to(D_AIN_, 2 * LEFT)
        D_aout_ = DecimalNumber(
            D_AOUT_.get_value(degrees=True), unit=r"^{\circ}", color=UE_COLOR
        )
        D_aout_.next_to(D_AOUT_, 2 * RIGHT)

        # Slide: reflection on sphere

        W1_.save_state()
        self.play(Transform(W1_, arc_))
        self.next_slide()

        # Slide: reflection on metasurface

        UE_.save_state()

        phi = MathTex(r"\phi", color=UE_COLOR).move_to(aout_.get_center())

        self.play(
            Restore(W1_),
            UE_.animate.shift(RIGHT),
            FadeTransform(aout_, phi),
            Transform(
                interaction, Tex("Reflection on metasurfaces").move_to(interaction)
            ),
            Transform(
                interaction_eq,
                MathTex(
                    r"\cl I \sim \bs r = f(\hat{\bs n}, \phi)",
                    tex_template=tex_template,
                ).to_corner(UR),
            ),
        )

        self.next_slide()

        # Slide: diffraction

        refl_config = VGroup(NV_, AIN_, AOUT_, ain_, aout_)
        diff_config = VGroup(D_NV_, D_AIN_, D_AOUT_, D_ain_, D_aout_)
        refl_config.save_state()

        self.play(
            *[
                Transform(refl, diff)
                if not isinstance(refl, DecimalNumber)
                else FadeTransform(refl, diff)
                for refl, diff in zip(refl_config, diff_config)
            ],
            Restore(W1_),
            Restore(UE_),
            FadeOut(phi),
            FadeIn(DIFF_W1_B),
            FadeIn(DIFF_W1_A),
            Transform(interaction, Tex("Diffraction").move_to(interaction)),
            Transform(
                interaction_eq,
                MathTex(
                    r"\cl I \sim \frac{\scp{\bs i}{\hat{\bs e}}}{\| \bs i \|} =  \frac{\scp{\bs d}{\hat{\bs e}}}{\|\bs d\|}",
                    tex_template=tex_template,
                ).to_corner(UR),
            ),
        )
        self.remove(*refl_config)
        self.add(*diff_config)
        self.next_slide()

        # Slide: refraction

        (UE_.shift(DOWN * 4),)

        R_NV_ = Line(X1_, X1_.get_center() + UP * 3).add_tip()
        R_AIN_ = Angle(
            R_NV_,
            VIN_.copy().scale(-1),
            radius=1.01,
            color=BS_COLOR,
        )
        R_AOUT_ = Angle(
            R_NV_.copy().scale(-1), Line(X1_, UE_), radius=1.01, color=UE_COLOR
        )
        R_ain_ = DecimalNumber(
            R_AIN_.get_value(degrees=True), unit=r"^{\circ}", color=BS_COLOR
        )
        R_ain_.next_to(R_AIN_, 2 * LEFT)
        R_aout_ = DecimalNumber(
            R_AOUT_.get_value(degrees=True), unit=r"^{\circ}", color=UE_COLOR
        )
        R_aout_.next_to(R_AOUT_, DR + RIGHT)

        refr_config = VGroup(R_NV_, R_AIN_, R_AOUT_, R_ain_, R_aout_)

        dashed = DashedLine(X1_, X1_.get_center() + 2 * DOWN, color=GRAY)

        self.play(
            Write(dashed),
            FadeOut(DIFF_W1_A),
            FadeOut(DIFF_W1_B),
            *[
                Transform(refl, diff)
                if not isinstance(refl, DecimalNumber)
                else FadeTransform(refl, diff)
                for refl, diff in zip(diff_config, refr_config)
            ],
            Transform(interaction, Tex("Refraction").move_to(interaction)),
            Transform(
                interaction_eq,
                MathTex(
                    r"\cl I \sim v_1 \sin(\theta_2) = v_2 \sin(\theta_1)",
                    tex_template=tex_template,
                ).to_corner(UR),
            ),
        )

        self.remove(*diff_config)
        self.add(*refr_config)

        self.next_slide()

        self.play(
            FadeOut(dashed),
            FadeOut(refr_config),
            FadeOut(scene_),
            FadeOut(interaction),
            FadeOut(interaction_eq),
            self.update_slide_number(),
        )
        self.next_slide()

        minimize_eq = Tex(
            r"\[\underset{\bs{\cl X} \in \bb R^{n_t}}{\text{minimize}}\ \cl C(\bs{\cl X}) := \|\cl I(\bs{\cl X})\|^2 + \|\cl F(\bs{\cl X})\|^2\]",
            tex_template=tex_template,
        )
        nt_eq = Tex("where $n_t$ is the total number of unknowns").shift(DOWN)
        constraint_eq = MathTex(
            r"\cl C(\bs{\cl X})", r"= 0", tex_template=tex_template
        ).shift(2 * DOWN)
        constraint_eq_relaxed = MathTex(
            r"\cl C(\bs{\cl X})", r"\le \epsilon", tex_template=tex_template
        ).shift(2 * DOWN)

        self.play(FadeIn(minimize_eq))

        self.next_slide()

        self.play(FadeIn(nt_eq, shift=DOWN))

        self.next_slide()

        self.play(FadeIn(constraint_eq))

        self.next_slide()

        self.play(Transform(constraint_eq, constraint_eq_relaxed))

        self.next_slide()

        if_we_know = Tex(
            r"If we know a mapping s.t. $(x_k, y_k) \leftrightarrow t_k$"
        ).shift(UP)

        self.play(
            FadeIn(if_we_know),
        )

        self.next_slide()

        self.play(
            Transform(
                minimize_eq,
                Tex(
                    r"\[\underset{\bs{\cl T} \in \bb R^{n_r}}{\text{minimize}}\ \cl C(\bs{\cl X}(\bs{\cl T})) := \|\cl I(\bs{\cl X }(\bs{\cl T}))\|^2\]",
                    tex_template=tex_template,
                ).move_to(minimize_eq),
            ),
            Transform(
                nt_eq,
                Tex("where $n_r$ is the total number of (2d) reflections").move_to(
                    nt_eq
                ),
            ),
            Transform(
                constraint_eq,
                MathTex(
                    r"\cl C(\bs{\cl X(\cl T)})",
                    r"\le \epsilon",
                    tex_template=tex_template,
                ).move_to(constraint_eq),
            ),
        )

        self.next_slide()

        # Slide: gradient descent on simple example using MPT method

        self.play(
            FadeOut(if_we_know),
            FadeOut(minimize_eq),
            FadeOut(nt_eq),
            FadeOut(constraint_eq),
            self.update_slide_number(),
        )

        X1.move_to(W1.get_center())
        X2.move_to(W2.get_center())

        def intersects(l1, l2):
            l1 = LineString([l1.get_start()[:-1], l1.get_end()[:-1]])
            l2 = LineString([l2.get_start()[:-1], l2.get_end()[:-1]])
            return l1.intersects(l2)

        old_objects.remove(I1)
        old_objects.remove(I2)
        path.remove(*path)
        path.add(
            always_redraw(lambda: Line(BS, X1)),
            always_redraw(lambda: Line(X1, X2)),
            always_redraw(
                lambda: Line(
                    X2, UE, color=BAD_COLOR if intersects(Line(X2, UE), W1) else BLACK
                )
            ),
        )
        self.play(*[FadeIn(mob) for mob in old_objects if mob != sec1])

        self.next_slide()

        # Slide: animate actual gradient descent

        f, df = generate_c()

        def remap(X1, X2):
            s1 = X1.get_center()[0]
            s2 = X2.get_center()[1]
            return s1 + X_OFFSET, s2 + Y_OFFSET

        cost_label, f_number = f_label = VGroup(
            MathTex(r"\mathcal{C} = ", tex_template=tex_template),
            DecimalNumber(
                f(*remap(X1, X2)),  # f(s1, s2)
                num_decimal_places=2,
                include_sign=False,
            ),
        )

        f_label.set_color(BLUE)
        f_label.arrange(RIGHT)
        f_label.next_to(W2, RIGHT)
        always(f_label.next_to, W2, RIGHT)
        f_always(f_number.set_value, lambda: f(*remap(X1, X2)))

        cost_eq = MathTex(
            r"\|[\cl I_1(t_1, t_2); \cl I_2(t_1,t_2)] \|^2",
            tex_template=tex_template,
            color=BLUE,
            font_size=40,
        ).next_to(cost_label, RIGHT)

        cost_eq_full = MathTex(
            r"&\left(- t_1 + t_2 + \frac{\left(t_2 - 4\right) \sqrt{\left(t_1 - 5\right)^{2} + \left(t_1 - t_2\right)^{2}}}{\sqrt{\left(t_2 - 4\right)^{2} + 9}}\right)^{2} \\+& \left(t_1 + \frac{3 \sqrt{\left(t_1 - 5\right)^{2} + \left(t_1 - t_2\right)^{2}}}{\sqrt{\left(t_2 - 4\right)^{2} + 9}} - 5\right)^{2} \\+& \Bigg|{t_1 + \frac{\left(t_1 - 5\right) \sqrt{\left(t_1 - 2\right)^{2} + \left(t_1 + 1\right)^{2}}}{\sqrt{\left(t_1 - 5\right)^{2} + \left(t_1 - t_2\right)^{2}}} - \frac{\sqrt{2} \left(\sqrt{2} \left(t_1 - 2\right) - \sqrt{2} \left(t_1 + 1\right)\right)}{2} - 2}\Bigg|^{2} \\+& \left|{t_1 + \frac{\left(t_1 - t_2\right) \sqrt{\left(t_1 - 2\right)^{2} + \left(t_1 + 1\right)^{2}}}{\sqrt{\left(t_1 - 5\right)^{2} + \left(t_1 - t_2\right)^{2}}} + \frac{\sqrt{2} \left(\sqrt{2} \left(t_1 - 2\right) - \sqrt{2} \left(t_1 + 1\right)\right)}{2} + 1}\right|^{2}",
            color=BLUE,
            font_size=16,
        ).next_to(cost_label, RIGHT)

        self.play(FadeIn(cost_label), FadeIn(cost_eq))

        self.next_slide()

        self.play(Transform(cost_eq, cost_eq_full))

        self.next_slide()

        self.play(FadeTransform(cost_eq, f_number))
        self.next_slide()

        x0 = remap(X1, X2)

        for ds1, ds2 in gradient_descent(x0, df, return_steps=True):
            self.play(
                X1.animate.shift([ds1, ds1, 0]),
                X2.animate.shift([0, ds2, 0]),
                run_time=1.3,
            )

        self.wait(0.1)

        self.next_slide()

        # Slide: what if METASURFACE?

        what_if_ms_text = Tex("What if we had a metasurface?").to_corner(UR)
        ms_text = Tex(r"$\phi=0$", color=GREEN).next_to(W2).shift(DOWN)
        self.play(FadeIn(what_if_ms_text))
        self.next_slide()

        self.play(UE.animate.shift(np.array([0, -0.5, 0])))
        self.play(EL.animate.set_color(GREEN))  # EL is W2
        self.play(FadeIn(ms_text))

        f, df = generate_c_ris()

        _, f_number = f_label = VGroup(
            MathTex(r"\mathcal{C} = ", tex_template=tex_template),
            DecimalNumber(
                f(*remap(X1, X2)),  # f(s1, s2)
                num_decimal_places=2,
                include_sign=False,
            ),
        )

        f_label.set_color(BLUE)
        f_label.arrange(RIGHT)
        f_label.next_to(W2, RIGHT)
        always(f_label.next_to, W2, RIGHT)
        f_always(f_number.set_value, lambda: f(*remap(X1, X2)))

        self.play(FadeIn(f_label, shift=UP))
        self.next_slide()

        x0 = remap(X1, X2)

        for ds1, ds2 in gradient_descent(x0, df, return_steps=True):
            self.play(
                X1.animate.shift([ds1, ds1, 0]),
                X2.animate.shift([0, ds2, 0]),
                run_time=0.6,
            )

        self.next_slide()

        # Sec. 3

        sec3.to_corner(UL)
        self.play(*[FadeOut(mob) for mob in self.mobjects])
        self.play(self.update_slide_number(), Transform(sec1, sec3))
        self.next_slide()

        geom = SVGMobject("geometry.svg").scale(5)
        tabl = Tex(
            r"""
\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r}
        Number of interactions & \multicolumn{1}{r}{1}  & \multicolumn{3}{r}{2} & \multicolumn{7}{r}{3} \\
        \hline\\
        Interactions list & D & RD & DR & DD & RRD & RDR & RDD & DRR & DRD & DDR & DDD \\
        $E/E_\text{LOS}$ (\si{\decibel}) & \textbf{-32} & -236 & -242 & \textbf{-44} & -231 & -246 & \textbf{-69} & -212 & \textbf{-72} & -81 & \textbf{-60} \\
\end{tabular}
""",
            tex_template=tex_template,
        )
        results = VGroup(geom, tabl).arrange(DOWN, buff=2).scale(0.4)

        self.play(FadeIn(geom))
        self.next_slide()

        self.play(FadeIn(tabl))
        self.next_slide()

        # Slide: summary of MPT method
        self.play(FadeOut(results), self.update_slide_number())

        pros = (
            VGroup(
                Tex(r"\textbf{Pros}"),
                Tex(r"- Any geometry (but requires more info.)"),
                Tex(r"- Any \# of reflect., diff., and refract."),
                Tex(r"- Allows for multiple solutions"),
                Tex(r"- Optimizer can be chosen"),
            )
            .scale(0.5)
            .arrange(DOWN)
        )

        for pro in pros[1:]:
            pro.align_to(pros[0], LEFT)

        cons = (
            VGroup(
                Tex(r"\textbf{Cons}"),
                Tex(r"- In general, problem is not convex"),
                Tex(r"- Slower - $\cl O (k \cdot n)$", tex_template=tex_template),
            )
            .scale(0.5)
            .arrange(DOWN)
        )

        for con in cons[1:]:
            con.align_to(cons[0], LEFT)

        summary = VGroup(
            Tex("Summary:", font_size=60),
            VGroup(pros, cons).arrange(RIGHT, buff=2),
        ).arrange(DOWN, buff=1)

        cons.align_to(pros, UP)

        self.play(FadeIn(summary[0]))

        self.next_slide()

        self.play(FadeIn(summary[1][0]))

        self.next_slide()

        self.play(FadeIn(summary[1][1]))

        # Slide: comparison with FPT

        self.next_slide()

        comparison_with_fermat = Tex(
            r"""
            MPT $\ne$ \textit{Fermat}'s path:
            \[\underset{\bs{\cl X} \in \bb R^{n_t}}{\text{minimize}}\ \cl L(\bs{\cl X}) := \sum_i \|\bs{\cl X_i} - \bs{\cl X_{i-1}}\|\]
            """,
            tex_template=tex_template,
        )

        fermat_paper = ImageMobject("paper-fermat-path.png")

        fermat_paper.next_to(comparison_with_fermat, 2 * DOWN)

        self.play(FadeOut(summary), self.update_slide_number())
        self.play(FadeIn(comparison_with_fermat))
        self.next_slide()
        self.play(FadeIn(fermat_paper))
        self.next_slide()

        # Slide: about differentiabiliy

        self.play(
            self.update_slide_number(),
            Transform(sec1, sec4),
            FadeOut(comparison_with_fermat),
            FadeOut(fermat_paper),
        )
        self.next_slide()

        rt_eq = MathTex(r"I = \sum\limits_{p}^\text{paths} f(p, \theta)")
        self.play(FadeIn(rt_eq))
        self.next_slide()

        los_image = ImageMobject("los.png")
        self.play(FadeOut(rt_eq))
        self.play(FadeIn(los_image))
        self.next_slide()

        # Slide: current work

        self.play(
            self.update_slide_number(),
            Transform(sec1, sec5),
            los_image.animate.shift(3 * LEFT),
        )
        self.next_slide()

        los_approx_image = ImageMobject("los_approx.png").shift(3 * RIGHT)
        self.play(FadeIn(los_approx_image))

        self.next_slide()
        los_approx_soft_image = ImageMobject("los_approx_soft.png").shift(3 * RIGHT)
        self.play(Transform(los_approx_image, los_approx_soft_image))

        self.next_slide()
        power_image = ImageMobject("power.png")
        self.play(FadeOut(los_image), FadeOut(los_approx_image))
        self.play(FadeIn(power_image))

        self.next_slide()

        # Slide: fade out everything and thanks

        self.play(*[FadeOut(mob) for mob in self.mobjects])

        thanks = Tex("Thanks for listening!").scale(2)

        self.play(FadeIn(thanks))

        self.wait()
        self.next_slide()