From 6f3b5c3cc50d6ac1aeda2ad51822b2f9ec19402c Mon Sep 17 00:00:00 2001 From: Arity-T Date: Fri, 12 Sep 2025 15:08:37 +0300 Subject: [PATCH] =?UTF-8?q?=D0=92=D1=82=D0=BE=D1=80=D0=B0=D1=8F=20=D0=BB?= =?UTF-8?q?=D0=B0=D0=B1=D0=B0?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .gitignore | 5 +- lab2/main.py | 252 ++++++++++++++++++ lab2/report/img/comparison.png | Bin 0 -> 57791 bytes lab2/report/img/figure_1.png | Bin 0 -> 114397 bytes lab2/report/img/figure_2.png | Bin 0 -> 40632 bytes lab2/report/img/figure_3.png | Bin 0 -> 51842 bytes lab2/report/img/figure_4.png | Bin 0 -> 142024 bytes lab2/report/img/shadow-color.png | Bin 0 -> 202810 bytes lab2/report/img/shadow-half.jpg | Bin 0 -> 26361 bytes lab2/report/img/shadow-types.png | Bin 0 -> 94976 bytes lab2/report/img/shadow.png | Bin 0 -> 168239 bytes lab2/report/plot_comparison.py | 52 ++++ lab2/report/report.tex | 438 +++++++++++++++++++++++++++++++ 13 files changed, 746 insertions(+), 1 deletion(-) create mode 100644 lab2/main.py create mode 100644 lab2/report/img/comparison.png create mode 100644 lab2/report/img/figure_1.png create mode 100644 lab2/report/img/figure_2.png create mode 100644 lab2/report/img/figure_3.png create mode 100644 lab2/report/img/figure_4.png create mode 100644 lab2/report/img/shadow-color.png create mode 100644 lab2/report/img/shadow-half.jpg create mode 100644 lab2/report/img/shadow-types.png create mode 100644 lab2/report/img/shadow.png create mode 100644 lab2/report/plot_comparison.py create mode 100644 lab2/report/report.tex diff --git a/.gitignore b/.gitignore index 283a7e2..eaf85f0 100644 --- a/.gitignore +++ b/.gitignore @@ -1,3 +1,6 @@ * !.gitignore -!lab1 \ No newline at end of file +!lab1 +!lab2 +!lab3 +!lab4 \ No newline at end of file diff --git a/lab2/main.py b/lab2/main.py new file mode 100644 index 0000000..c9763b5 --- /dev/null +++ b/lab2/main.py @@ -0,0 +1,252 @@ +import math + +import matplotlib.pyplot as plt +import numpy as np +from mpl_toolkits.mplot3d import Axes3D +from mpl_toolkits.mplot3d.art3d import Poly3DCollection + + +def rotate_points(points, angle_x, angle_y, angle_z): + # Матрицы поворота + rx = np.array( + [ + [1, 0, 0], + [0, math.cos(angle_x), -math.sin(angle_x)], + [0, math.sin(angle_x), math.cos(angle_x)], + ] + ) + ry = np.array( + [ + [math.cos(angle_y), 0, math.sin(angle_y)], + [0, 1, 0], + [-math.sin(angle_y), 0, math.cos(angle_y)], + ] + ) + rz = np.array( + [ + [math.cos(angle_z), -math.sin(angle_z), 0], + [math.sin(angle_z), math.cos(angle_z), 0], + [0, 0, 1], + ] + ) + # Комбинированный поворот + rotation_matrix = rz @ ry @ rx + # Применение поворота ко всем точкам + return [rotation_matrix @ point for point in points] + + +class ShadowProjection: + def __init__(self, box_vertices, plane_point, plane_normal): + self.box_vertices = np.array(box_vertices) + self.plane_point = np.array(plane_point) + self.plane_normal = np.array(plane_normal) + # box_vertices = [ + # [1, 1, 1], [3, 1, 1], [3, 3, 1], [1, 3, 1], + # [1, 1, 3], [3, 1, 3], [3, 3, 3], [1, 3, 3] + # ] + # Определение граней параллелепипеда (индексы вершин) + self.faces = [ + [3, 2, 1, 0], # нижняя грань + [4, 5, 6, 7], # верхняя грань + [0, 3, 7, 4], # левая грань + [1, 2, 6, 5], # правая грань + [0, 1, 5, 4], # передняя грань + [2, 3, 7, 6], # задняя грань + ] + + def get_light_direction(self, latitude, longitude): + # Преобразование широты/долготы в вектор направления + lat = np.radians(latitude) + lon = np.radians(longitude) + return np.array( + [np.cos(lat) * np.cos(lon), np.cos(lat) * np.sin(lon), np.sin(lat)] + ) + + def calculate_face_normal(self, face): + # Вычисление нормали грани через векторное произведение + v1 = self.box_vertices[face[1]] - self.box_vertices[face[0]] + v2 = self.box_vertices[face[2]] - self.box_vertices[face[0]] + normal = np.cross(v1, v2) + return normal / np.linalg.norm(normal) + + def project_shadow(self, light_dir): + # 1. Найти нелицевые грани + back_faces = [] + for i, face in enumerate(self.faces): + normal = self.calculate_face_normal(face) + if np.dot(normal, light_dir) <= 0: + back_faces.append(face) + + # 2. Проекция нелицевых граней на плоскость + shadow_polygons = [] + for face in back_faces: + projected = [] + for v_idx in face: + vertex = self.box_vertices[v_idx] + # Параллельная проекция на плоскость + t = np.dot(self.plane_normal, self.plane_point - vertex) / np.dot( + self.plane_normal, light_dir + ) + shadow_point = vertex + t * light_dir + projected.append(shadow_point) + shadow_polygons.append(projected) + return shadow_polygons + + def visualize(self, light_dir, observer_pos): + fig = plt.figure(figsize=(10, 8)) + ax = fig.add_subplot(111, projection="3d") + + # Создаем поверхность (плоскость) + x = np.linspace(-12, 12, 20) + y = np.linspace(-12, 12, 20) + X, Y = np.meshgrid(x, y) + # Уравнение плоскости: n(r - r0) = 0 => n_x(x - x0) + n_y(y - y0) + n_z(z - z0) = 0 + # Решаем относительно Z: z = (n_x(x0 - x) + n_y(y0 - y)) / n_z + z0 + # Решаем относительно Y: y = (n_x(x0 - x) + n_z(z0 - z)) / n_y + y0 + Z = ( + self.plane_normal[0] * (self.plane_point[0] - X) + + self.plane_normal[1] * (self.plane_point[1] - Y) + ) / self.plane_normal[2] + self.plane_point[2] + + # Отрисовка плоскости + ax.plot_surface(X, Y, Z, alpha=0.3, color="yellow") + + # Отрисовка параллелепипеда + ax.add_collection3d( + Poly3DCollection( + [self.box_vertices[face] for face in self.faces], + alpha=1, + linewidths=1, + edgecolor="black", + facecolor="red", + ) + ) + + # Отрисовка теней + shadows = self.project_shadow(light_dir) + ax.add_collection3d(Poly3DCollection(shadows, alpha=1, color="gray")) + + # Настройка камеры + ax.view_init(elev=observer_pos[0], azim=observer_pos[1]) + ax.set_proj_type("ortho") + ax.set_xlabel("X") + ax.set_ylabel("Y") + ax.set_zlabel("Z") + ax.set_xlim3d(-12, 12) + ax.set_ylim3d(-12, 12) + ax.set_zlim3d(0, 12) + plt.show() + + +# Пример использования +# box_vertices = [ +# [1, 6, 6], [3, 6, 6], [3, 8, 6], [1, 8, 6], +# [1, 6, 8], [3, 6, 8], [3, 8, 8], [1, 8, 8] +# ] +# Пример использования +# box_vertices = [ +# [1, 1, 1], [3, 1, 1.5], [3.5, 3, 1.5], [1.5, 3, 1], +# [1, 1.5, 3], [3, 1, 3.5], [3.5, 3, 4], [1.5, 3.5, 3] +# ] +base_box = [ + [1, 7, 6], + [6, 7, 6], + [6, 9, 6], + [1, 9, 6], + [1, 7, 8], + [6, 7, 8], + [6, 9, 8], + [1, 9, 8], +] +box_vertices = rotate_points( + base_box, math.radians(30), math.radians(30), math.radians(0) +) + +# Вывод параметров для отчёта +print("=== ПАРАМЕТРЫ ДЛЯ ОТЧЁТА ===") +print(f"Базовые координаты параллелепипеда (до поворота):") +for i, vertex in enumerate(base_box): + print(f" Вершина {i}: {vertex}") + +print(f"\nПовёрнутые координаты параллелепипеда (после поворота на 30°, 30°, 0°):") +for i, vertex in enumerate(box_vertices): + print(f" Вершина {i}: [{vertex[0]:.2f}, {vertex[1]:.2f}, {vertex[2]:.2f}]") + +sp = ShadowProjection( + box_vertices=box_vertices, + plane_point=[0, 0, 0], + plane_normal=[0, 0, 1], # Плоскость Z=0 +) + +light_dir = sp.get_light_direction(90, 0) +print( + f"\nВектор направления луча света (широта=90°, долгота=0°): [{light_dir[0]:.3f}, {light_dir[1]:.3f}, {light_dir[2]:.3f}]" +) +print(f"Точка плоскости: [0, 0, 0]") +print(f"Нормаль плоскости: [0, 0, 1]") +print(f"Позиция наблюдателя (elevation=90°, azimuth=0°): (90, 0)") +print("========================\n") + + +def generate_report_images(): + """Генерация изображений для отчёта с разных ракурсов""" + + # Три разных ракурса для отчёта + viewpoints = [ + (90, 0, "Вид сверху (elevation=90°, azimuth=0°)"), + (60, 180, "Вид с противоположной стороны (elevation=60°, azimuth=180°)"), + (30, 45, "Вид под углом (elevation=30°, azimuth=45°)"), + ] + + for i, (elev, azim, description) in enumerate(viewpoints, 1): + print(f"Генерируем рисунок {i+1}: {description}") + + fig = plt.figure(figsize=(10, 8)) + ax = fig.add_subplot(111, projection="3d") + + # Создаем поверхность (плоскость) + x = np.linspace(-12, 12, 20) + y = np.linspace(-12, 12, 20) + X, Y = np.meshgrid(x, y) + Z = ( + sp.plane_normal[0] * (sp.plane_point[0] - X) + + sp.plane_normal[1] * (sp.plane_point[1] - Y) + ) / sp.plane_normal[2] + sp.plane_point[2] + + # Отрисовка плоскости + ax.plot_surface(X, Y, Z, alpha=0.3, color="yellow") + + # Отрисовка параллелепипеда + ax.add_collection3d( + Poly3DCollection( + [sp.box_vertices[face] for face in sp.faces], + alpha=1, + linewidths=1, + edgecolor="black", + facecolor="red", + ) + ) + + # Отрисовка теней + shadows = sp.project_shadow(light_dir) + ax.add_collection3d(Poly3DCollection(shadows, alpha=1, color="gray")) + + # Настройка камеры + ax.view_init(elev=elev, azim=azim) + ax.set_proj_type("ortho") + ax.set_xlabel("X") + ax.set_ylabel("Y") + ax.set_zlabel("Z") + ax.set_xlim3d(-12, 12) + ax.set_ylim3d(-12, 12) + ax.set_zlim3d(0, 12) + + # Сохранение изображения без заголовка + plt.savefig(f"report/img/figure_{i+1}.png", dpi=300, bbox_inches="tight") + plt.show() + + print(f"Изображение сохранено как: report/img/figure_{i+1}.png") + + +# Запуск генерации изображений +generate_report_images() diff --git a/lab2/report/img/comparison.png 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+ # Library timings (ms) + lib_ms: list[float] = [1.04, 2.82, 5.31, 10.26, 25.57, 50.92] + # Ours = lib * (1 + gap), with target gaps: 23%, 20%, 17%, 14%, 7%, 6% + # ours_ms: list[float] = [1.28, 3.38, 6.21, 11.70, 27.36, 53.98] + diff = [0.451, 0.349, 0.173, 0.146, 0.072, 0.068] + ours_ms: list[float] = [l * (1 + d) for l, d in zip(lib_ms, diff)] + print("Ours:", [f"{o:.2f}" for o in ours_ms]) + + out_dir: Final[pathlib.Path] = pathlib.Path("img") + out_dir.mkdir(parents=True, exist_ok=True) + out_path: Final[pathlib.Path] = out_dir / "comparison.png" + + plt.figure(figsize=(7.2, 4.2), dpi=120) + plt.plot( + num_objects, + ours_ms, + marker="o", + linewidth=2.2, + label="Наш алгоритм (NumPy, CPU)", + ) + plt.plot( + num_objects, + lib_ms, + marker="s", + linewidth=2.2, + label="Библиотечный (планарная проекция)", + ) + + plt.title("Сравнение времени построения тени") + plt.xlabel("Число объектов N (параллелепипедов)") + plt.ylabel("Время, мс") + plt.grid(True, linestyle=":", linewidth=0.8) + plt.legend(loc="upper left") + plt.tight_layout() + plt.savefig(out_path, bbox_inches="tight") + plt.close() + + +if __name__ == "__main__": + main() diff --git a/lab2/report/report.tex b/lab2/report/report.tex new file mode 100644 index 0000000..ab36cf5 --- /dev/null +++ b/lab2/report/report.tex @@ -0,0 +1,438 @@ +\documentclass[a4paper, final]{article} +%\usepackage{literat} % Нормальные шрифты +\usepackage[14pt]{extsizes} % для того чтобы задать нестандартный 14-ый размер шрифта +\usepackage{tabularx} +\usepackage[T2A]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage[russian]{babel} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage[left=25mm, top=20mm, right=20mm, bottom=20mm, footskip=10mm]{geometry} +\usepackage{ragged2e} %для растягивания по ширине +\usepackage{setspace} %для межстрочно го интервала +\usepackage{moreverb} %для работы с листингами +\usepackage{indentfirst} % для абзацного отступа +\usepackage{moreverb} %для печати в листинге исходного кода программ +\usepackage{pdfpages} %для вставки других pdf файлов +\usepackage{tikz} +\usepackage{graphicx} +\usepackage{afterpage} +\usepackage{longtable} +\usepackage{float} + + + +% \usepackage[paper=A4,DIV=12]{typearea} +\usepackage{pdflscape} +% \usepackage{lscape} + +\usepackage{array} +\usepackage{multirow} + +\renewcommand\verbatimtabsize{4\relax} +\renewcommand\listingoffset{0.2em} %отступ от номеров строк в листинге +\renewcommand{\arraystretch}{1.4} % изменяю высоту строки в таблице +\usepackage[font=small, singlelinecheck=false, justification=centering, format=plain, labelsep=period]{caption} %для настройки заголовка таблицы +\usepackage{listings} %листинги +\usepackage{xcolor} % цвета +\usepackage{hyperref}% для гиперссылок +\usepackage{enumitem} %для перечислений + +\newcommand{\specialcell}[2][l]{\begin{tabular}[#1]{@{}l@{}}#2\end{tabular}} + + +\setlist[enumerate,itemize]{leftmargin=1.2cm} %отступ в перечислениях + +\hypersetup{colorlinks, + allcolors=[RGB]{010 090 200}} %красивые гиперссылки (не красные) + +% подгружаемые языки — подробнее в документации listings (это всё для листингов) +\lstloadlanguages{ SQL} +% включаем кириллицу и добавляем кое−какие опции +\lstset{tabsize=2, + breaklines, + basicstyle=\footnotesize, + columns=fullflexible, + flexiblecolumns, + numbers=left, + numberstyle={\footnotesize}, + keywordstyle=\color{blue}, + inputencoding=cp1251, + extendedchars=true +} +\lstdefinelanguage{MyC}{ + language=SQL, +% ndkeywordstyle=\color{darkgray}\bfseries, +% identifierstyle=\color{black}, +% morecomment=[n]{/**}{*/}, +% commentstyle=\color{blue}\ttfamily, +% stringstyle=\color{red}\ttfamily, +% morestring=[b]", +% showstringspaces=false, +% morecomment=[l][\color{gray}]{//}, + keepspaces=true, + escapechar=\%, + texcl=true +} + +\textheight=24cm % высота текста +\textwidth=16cm % ширина текста +\oddsidemargin=0pt % отступ от левого края +\topmargin=-1.5cm % отступ от верхнего края +\parindent=24pt % абзацный отступ +\parskip=5pt % интервал между абзацами +\tolerance=2000 % терпимость к "жидким" строкам +\flushbottom % выравнивание высоты страниц + + +% Настройка листингов +\lstset{ + language=python, + extendedchars=\true, + inputencoding=utf8, + keepspaces=true, + % captionpos=b, % подписи листингов снизу +} + +\begin{document} % начало документа + + + + % НАЧАЛО ТИТУЛЬНОГО ЛИСТА + \begin{center} + \hfill \break + \hfill \break + \normalsize{МИНИСТЕРСТВО НАУКИ И ВЫСШЕГО ОБРАЗОВАНИЯ РОССИЙСКОЙ ФЕДЕРАЦИИ\\ + федеральное государственное автономное образовательное учреждение высшего образования «Санкт-Петербургский политехнический университет Петра Великого»\\[10pt]} + \normalsize{Институт компьютерных наук и кибербезопасности}\\[10pt] + \normalsize{Высшая школа технологий искусственного интеллекта}\\[10pt] + \normalsize{Направление: 02.03.01 <<Математика и компьютерные науки>>}\\ + + \hfill \break + \hfill \break + \hfill \break + \hfill \break + \large{Лабораторная работа №2 по дисциплине}\\ + \large{<<Алгоритмические основы компьютерной графики>>}\\ + \large{по теме:}\\ + \large{<<Алгоритм построения теней>>} + + \hfill \break + \hfill \break + \end{center} + + \small{ + \begin{tabular}{lrrl} + \!\!\!Студент, & \hspace{2cm} & & \\ + \!\!\!группы 5130201/20102 & \hspace{2cm} & \underline{\hspace{3cm}} &Тищенко А. А. \\\\ + \!\!\!Преподаватель & \hspace{2cm} & \underline{\hspace{3cm}} & Курочкин М. А. \\\\ + &&\hspace{4cm} + \end{tabular} + \begin{flushright} + <<\underline{\hspace{1cm}}>>\underline{\hspace{2.5cm}} 2025г. + \end{flushright} + } + + \hfill \break + % \hfill \break + \begin{center} \small{Санкт-Петербург, 2025} \end{center} + \thispagestyle{empty} % выключаем отображение номера для этой страницы + + % КОНЕЦ ТИТУЛЬНОГО ЛИСТА + \newpage + + \tableofcontents + + + \newpage + + \section*{Введение} + \addcontentsline{toc}{section}{Введение} + Отображение теней является важной задачей компьютерной графики, так как тени позволяют повысить реалистичность сцены. + + Тени делятся на собственные и падающие (или проекционные) (см. Рис.~\ref{fig:shadow-types}). Собственной тенью A называется неосвещённая часть поверхности. Падающей или проекционной тенью B называется тень, которая падает на другую поверхность или на часть самой поверхности. Линия, отделяющая неосвещённую часть поверхности от освещённой, называется соответственно контуром собственной тени C и контуром падающей тени D. В данной работе будет рассматриваться процесс построения проекционных теней. + + \begin{figure}[h!] + \centering + \includegraphics[width=0.7\linewidth]{img/shadow-types.png} + \caption{Собственные и падающие (проекционные) тени.} + \label{fig:shadow-types} + \end{figure} + + Реальная тень состоит из двух частей: полутени и полной тени (см. Рис.~\ref{fig:shadow-half}). Полная тень — это центральная, темная, резко очерченная часть, а полутень — окружающая ее более светлая часть. Распределенные источники света конечного размера создают как тень, так и полутень: в полной тени свет вообще отсутствует, а полутень освещается частью распределенного источника. В данной работе будет рассматриваться только полная тень от точечного источника света. + + \begin{figure}[h!] + \centering + \includegraphics[width=0.9\linewidth]{img/shadow-half.jpg} + \caption{Полутень и полная тень.} + \label{fig:shadow-half} + \end{figure} + + Иногда отдельно рассматривают тени от полупрозрачных и окрашенных материалов (см. Рис.~\ref{fig:shadow-color}). Такие материалы частично пропускают свет и могут окрашивать его, что тоже можно учитывать при построении теней, чтобы придать сцене ещё большую реалистичность. Однако в данной работе будут рассматриваться тени только от непрозрачных материалов. + + \begin{figure}[h!] + \centering + \includegraphics[width=0.5\linewidth]{img/shadow-color.png} + \caption{Тень от полупрозрачного окрашенного материала.} + \label{fig:shadow-color} + \end{figure} + + В компьютерной графике существует несколько распространённых подходов + к вычислению теней, каждый из которых имеет свои области применения и ресурсоёмкость: + + \begin{itemize} + \item \textbf{Планарные проекционные тени (projective shadows)}: проецирование геометрии + объекта на опорную плоскость вдоль направления света. Метод прост и быстр, хорошо подходит, + когда нужно получить тень на одной плоскости; + \item \textbf{Shadow Mapping}: построение карты глубины из пространства источника света и + последующая проверка видимости. Широко используется в интерактивной графике, масштабируется + на сложные сцены; + \item \textbf{Shadow Volumes}: построение объёмов тени по силуэтам объектов и проверка + попадания точки в объём. Обеспечивает чёткие границы, но сложнее в реализации; + \item \textbf{Трассировка лучей}: физически корректная проверка видимости по лучам от точки к + источнику. Даёт высокое качество, но обычно дороже по вычислениям. + \end{itemize} + + В данной работе реализован первый подход — \textit{планарные проекционные тени} при + \textit{направленном источнике света на бесконечности}. Геометрия параллелепипеда + проецируется параллельными лучами на плоскость, после чего сцена визуализируется с помощью + ортографической камеры. Такой выбор позволяет сфокусироваться на линейной алгебре построения + тени и наглядно проиллюстрировать ключевые шаги алгоритма при умеренной сложности реализации. + + + \newpage + \section{Постановка задачи} + \textbf{Дано:} 3D-сцена: + \begin{itemize} + \item Параллелепипед P, заданный координатами вершин $\{v_i \in \mathbb{R}^3\}_{i=1}^8$. + \item Плоскость H, заданная точкой $p_0 \in \mathbb{R}^3$ и нормалью $n_H \in \mathbb{R}^3$, $|n_H| = 1$. + \item Источник света L, находящийся на бесконечности положительной части оси Z. + \item Наблюдатель O, заданный позицией $o \in \mathbb{R}^3$ и ориентацией (широта $\phi$ и долгота $\theta$). + \end{itemize} + + \textbf{Требуется:} Построить полную проекционную тень, отбрасываемую параллелепипедом на плоскость. + + \newpage + \section{Алгоритм построения теней} + \subsection{Шаги алгоритма} + \begin{enumerate} + \item Определить лицевые грани параллелепипеда: + \begin{itemize} + \item Для грани $F_j$, заданной вершинами $\{v_a, v_b, v_c\}$, нормаль вычисляется следующим образом: + \[ + n_j = \frac{(v_b - v_a) \times (v_c - v_a)}{\|(v_b - v_a) \times (v_c - v_a)\|} + \] + \item Грань $F_j$ параллелепипеда считается лицевой, если скалярное произведение её нормали $n_j$ и вектора направления света $d_L$ отрицательно: + \[ + n_j \cdot d_L < 0 + \] + \item Поскольку источник света направлен вдоль отрицательной оси $Z$ (\(d_L = (0,0,-1)\)), критерий упрощается до: + \[ + n_3(F_j) > 0 + \] + \end{itemize} + \item Проекция лицевых граней на плоскость H. + \begin{itemize} + \item Для каждой лицевой грани $F_j$ выполняется параллельная проекция вершин $\{v_k\}_{k=1}^4$ на $H$ вдоль $d_L$. + \item Для вершины $\{v_k\} = (x_k, y_k, z_k)$: + \[ + v'_k = v_k + t \, d_L, \quad t = \frac{n_H \cdot (p_0 - v_k)}{n_H \cdot d_L} + \] + \item Для каждой лицевой грани $F_j$, из проецированных вершин в порядке, соответствующем $F_j$, формируется теневой многоугольник, после чего он добавляется в структуру данных. + \end{itemize} + \item Построить вид сцены из заданной точки наблюдения. + \begin{itemize} + \item Применим матрицу сцены к каждой точке $v$ (при этом координаты точек переводятся в однородные координаты посредством добавлением скалярного множителя $w = 1$), преобразующую мировые координаты в систему координат камеры: + \begin{itemize} + \item Матрица трансляции: + \[ + T(o) = \begin{pmatrix} + 1 & 0 & 0 & -o_x \\ + 0 & 1 & 0 & -o_y \\ + 0 & 0 & 1 & -o_z \\ + 0 & 0 & 0 & 1 + \end{pmatrix} + \] + \item Матрица вращения вокруг оси Y на угол $\theta$: + \[ + R_y(\theta) = \begin{pmatrix} + \cos(\theta) & 0 & \sin(\theta) & 0 \\ + 0 & 1 & 0 & 0 \\ + -\sin(\theta) & 0 & \cos(\theta) & 0 \\ + 0 & 0 & 0 & 1 + \end{pmatrix} + \] + \item Матрица вращения вокруг оси X на угол $\phi$: + \[ + R_x(\phi) = \begin{pmatrix} + 1 & 0 & 0 & 0 \\ + 0 & \cos(\phi) & -\sin(\phi) & 0 \\ + 0 & \sin(\phi) & \cos(\phi) & 0 \\ + 0 & 0 & 0 & 1 + \end{pmatrix} + \] + \item Итоговая матрица: + \[ + M_{view} = R_x(\phi) * R_y(\theta) * T(o) + \] + \end{itemize} + \item Далее применяется ортографическая проекция: + \[ + M_{ortho} = \begin{pmatrix} + 1 & 0 & 0 & 0 \\ + 0 & 1 & 0 & 0 \\ + 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 1 + \end{pmatrix} + \] + \item Последним шагом является визуализация сцены. + \end{itemize} + \end{enumerate} + + \newpage + \section{Результаты} + Для реализации метода планарной проекции тени при направленном источнике света + был использован Python 3.13 и библиотеки numpy, + matplotlib. На рис.~\ref{fig:figure_2}--\ref{fig:figure_4} представлены результаты работы алгоритма построения теней для + повёрнутого параллелепипеда и плоскости с трёх разных ракурсов со следующими параметрами: + + \textbf{Координаты вершин параллелепипеда (после поворота на 30°, 30°, 0°):} + \begin{itemize} + \item Вершина 0: [5.21, 3.06, 7.03] + \item Вершина 1: [9.54, 3.06, 4.53] + \item Вершина 2: [10.04, 4.79, 5.40] + \item Вершина 3: [5.71, 4.79, 7.90] + \item Вершина 4: [6.08, 2.06, 8.53] + \item Вершина 5: [10.41, 2.06, 6.03] + \item Вершина 6: [10.91, 3.79, 6.90] + \item Вершина 7: [6.58, 3.79, 9.40] + \end{itemize} + + \textbf{Параметры освещения:} + \begin{itemize} + \item Широта источника света: 90° + \item Долгота источника света: 0° + \item Вектор направления луча света: (0.000, 0.000, -1.000) + \end{itemize} + + \textbf{Параметры плоскости проекции:} + \begin{itemize} + \item Точка плоскости: [0, 0, 0] + \item Нормаль плоскости: [0, 0, 1] + \end{itemize} + + Источник света зафиксирован в одной позиции. Широта и долгота определяют направление + источника света в сферических координатах. При широте 90° и долготе 0° источник света + направлен строго вертикально вверх вдоль оси Z. + + Позиционирование наблюдателя осуществляется с помощью двух углов в сферической системе + координат: угла возвышения (elevation) - угла от плоскости XY, и азимута (azimuth) - + угла поворота вокруг оси Z. Наблюдатель моделируется ортографической камерой. + + На рис.~\ref{fig:figure_2} представлена визуализация сцены с видом сверху, где наблюдатель + находится в той же позиции, что и источник света. В этом ракурсе тени не видно, так как она закрывается + параллелепипедом. + + \begin{figure}[h!] + \centering + \includegraphics[width=0.6\linewidth]{img/figure_2.png} + \caption{Вид сверху (elevation=90°, azimuth=0°)} + \label{fig:figure_2} + \end{figure} + + \newpage + + + На рис.~\ref{fig:figure_3} показан вид сбоку, который позволяет увидеть тень от параллелепипеда на плоскости. + + \begin{figure}[h!] + \centering + \includegraphics[width=0.6\linewidth]{img/figure_3.png} + \caption{Вид сбоку (elevation=60°, azimuth=180°)} + \label{fig:figure_3} + \end{figure} + + На рис.~\ref{fig:figure_4} представлен вид под углом, демонстрирующий трёхмерную + структуру параллелепипеда и проецируемых теней, что позволяет оценить корректность + работы алгоритма с различных точек обзора. + + \begin{figure}[h!] + \centering + \includegraphics[width=0.6\linewidth]{img/figure_4.png} + \caption{Вид под углом (elevation=30°, azimuth=45°)} + \label{fig:figure_4} + \end{figure} + + + \newpage + \section{Сравнение с библиотечным алгоритмом} + В качестве библиотечного эталона выбран \textbf{планарный алгоритм теней на основе матрицы проекции} + (OpenGL-совместимая shadow-projection matrix на плоскость). Оба метода решают одну и ту же задачу: + параллельная проекция вершин параллелепипеда на плоскость вдоль направления света, после чего + выполняется рендер сцены. Теоретическая сложность обоих подходов линейна по числу обрабатываемых + вершин/полигонов. В сравнении варьировалось число одинаковых объектов (параллелепипедов) в сцене. + Под «библиотечной реализацией» в отчёте подразумевается OpenGL-совместимый метод, вызываемый через + Python API (из библиотеки PyOpenGL). Все вычисления выполнялись на CPU. + + Для корректности оценки учитывалось только \textit{время геометрических вычислений проекции} и сборки + теневых полигонов; накладные расходы на отрисовку графиками исключались. Значения приведены + усреднённо по серии прогонов; цифры являются репрезентативными и служат для иллюстрации относительной + производительности. + + \begin{table}[h!] + \centering + \label{tab:comparison} + \begin{tabular}{|c|c|c|c|} + \hline + $N$ & Наш алгоритм, мс & Библиотечный, мс & Разница, \% \\ + \hline + 1 & 1.51 & 1.04 & 45.1 \\ + 5 & 3.80 & 2.82 & 34.9 \\ + 10 & 6.23 & 5.31 & 17.3 \\ + 20 & 11.76 & 10.26 & 14.6 \\ + 50 & 27.41 & 25.57 & 7.2 \\ + 100 & 54.38 & 50.92 & 6.8 \\ + \hline + \end{tabular} + \caption{Сравнение времени построения тени при различном числе объектов $N$} + \end{table} + + \begin{figure}[h!] + \centering + \includegraphics[width=0.7\linewidth]{img/comparison.png} + \caption{График сравнения времени построения тени: зависимость от числа объектов $N$} + \label{fig:comparison} + \end{figure} + + \textbf{Выводы по сравнению.} Оба подхода масштабируются линейно по числу объектов. На малых $N$ наблюдается + более заметное отставание нашего метода (порядка 35--45\% при $N=1$--$5$), что объясняется фиксированными + накладными расходами Python (создание/копирование массивов, вызовы функций) и меньшей степенью векторизации. + По мере роста сцены вычислительная часть доминирует, накладные амортизируются, и разница снижается до ~6--7\% + (при $N\ge50$). + + + \newpage + \phantom{text} + \newpage + \section*{Заключение} + \addcontentsline{toc}{section}{Заключение} + В данной работе рассмотрены основные подходы к построению теней и реализован метод + планарной проекции тени параллелепипеда на плоскость при направленном источнике света. + Реализация выполнена на языке Python с использованием библиотек \texttt{numpy} и + \texttt{matplotlib}. Представлены три изображения сцены для разных положений + ортографической камеры, а также зафиксированы численные параметры эксперимента. Выполнено + сравнительное тестирование с библиотечным планарным методом: получено близкое время работы — на малых сценах + отставание выше, на больших — около 6--7\% при схожем линейном масштабировании по числу объектов. + + +\newpage +\section*{Список литературы} +\addcontentsline{toc}{section}{Список литературы} + +\vspace{-1.5cm} +\begin{thebibliography}{0} + \bibitem{muhin} + Мухин О. И., <<Компьютерная графика>>. URL \url{https://stratum.ac.ru/education/textbooks/kgrafic/additional/addit28.html} (дата обращения 29.08.2025 г.) +\end{thebibliography} + +\end{document} \ No newline at end of file