Donut in C++

This project implements a rotating 3D torus (donut) displayed in the console using C++ and Windows API.

Table of Contents

• Installation
• Mathematical Explanation

Installation

Prerequisites

• Windows operating system
• C++ compiler (e.g., MSVC, MinGW)

Build Instructions

1. Clone the repository:

   git clone https://github.com/yourusername/donut-cpp.git
   cd donut-cpp

2. Compile the code:

   g++ -o donut main.cpp -lgdi32

3. Run compiled file in a teminal window:

   ./donut.exe

Mathematical Explanation

The 3D torus (donut) visualization involves several mathematical concepts:

1. Torus Parametric Equations

A torus can be described parametrically with two angles, ( \theta ) and ( \phi ), representing the position on the torus:

$$ \begin{align*} x &= (R + r \cos(\phi)) \cos(\theta) \\ y &= (R + r \cos(\phi)) \sin(\theta) \\ z &= r \sin(\phi) \end{align*} $$

Here:

• $R$ is the major radius (distance from the center of the tube to the center of the torus).
• $r$ is the minor radius (radius of the tube).

2. Rotation Matrices

To rotate the torus, we use rotation matrices for the X, Y, and Z axes. These matrices allow us to rotate points in 3D space.

Rotation around X-axis:

$$ \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos(\alpha) & -\sin(\alpha) \\ 0 & \sin(\alpha) & \cos(\alpha) \end{bmatrix} $$

Rotation around Y-axis:

$$ \begin{bmatrix} \cos(\beta) & 0 & \sin(\beta) \\ 0 & 1 & 0 \\ -\sin(\beta) & 0 & \cos(\beta) \end{bmatrix} $$

Rotation around Z-axis:

$$ \begin{bmatrix} \cos(\gamma) & -\sin(\gamma) & 0 \\ \sin(\gamma) & \cos(\gamma) & 0 \\ 0 & 0 & 1 \end{bmatrix} $$

3. Projection onto 2D Plane

To display the 3D torus on a 2D console, we project 3D points onto a 2D plane using the perspective projection formula:

$$ \begin{align*} x_{\text{2D}} &= \frac{f \cdot x}{z} \\ y_{\text{2D}} &= \frac{f \cdot y}{z} \end{align*} $$

Here, $f$ is the focal length, representing the distance from the viewer to the projection plane. This projection simulates depth by scaling the $x$ and $y$ coordinates based on their distance $z$ from the viewer.

4. Frame Rate Control

The torus rotation and drawing are controlled to achieve a consistent frame rate. This involves calculating the time taken to process each frame and adjusting the sleep time to maintain a stable frame rate, typically 60 frames per second.

By combining these mathematical concepts, we achieve a rotating 3D torus that is projected and visualized in the console.

Contacts

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