+ 
+
In the [Poincaré disk](https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model), the entire infinite hyperbolic plane is
compressed into the interior of a circle. But there’s a catch—the edge of the circle represents points at infinity. Just
diff --git a/docs/I/isometries.md b/docs/I/isometries.md
index 19b57a8..a170660 100644
--- a/docs/I/isometries.md
+++ b/docs/I/isometries.md
@@ -2,7 +2,7 @@
## Euclidean Isometry
-A **Euclidean [isometry](https://en.wikipedia.org/wiki/Isometry)** is a transformation of the Euclidean space that preserves distances between all points. In other words, if you have a map $f: \mathbb{R}^n \to \mathbb{R}^n$, it is an isometry if, for every pair of points $x$ and $y$ in the space, the distance between $x$ and $y$ is exactly the same as the distance between $f(x)$ and $f(y)$; mathematically,
+A **Euclidean [isometry](https://en.wikipedia.org/wiki/Isometry)** is a transformation of the Euclidean space that preserves distances between all points. In other words, if you have a map %%f: \mathbb{R}^n \to \mathbb{R}^n%%, it is an isometry if, for every pair of points %%x%% and %%y%% in the space, the distance between %%x%% and %%y%% is exactly the same as the distance between %%f(x)%% and %%f(y)%%; mathematically,
$$d(f(x), f(y)) = d(x, y).$$
@@ -12,31 +12,39 @@ The consequence of this property is that the overall shape, size, and angles of
- **[Rotations](
+
+
+[![Contributors][contributors-shield]][contributors-url]
+[![Forks][forks-shield]][forks-url]
+[![Stargazers][stars-shield]][stars-url]
+[![Issues][issues-shield]][issues-url]
+[![License][license-shield]][license-url]
+
+
+
+
++
+ 
+
+
+---
+
+**Hyper** is an experimental engine for visualizing **non-Euclidean geometry**, starting with the **Hyperbolic 2D Plane** and extending into hybrid 3D space. It renders an infinite world, offering an educational, interactive view of **hyperbolic tilings** and **maze algorithms**.
+
+## Quickstart
+
+Clone the repository
+
+```bash
+git clone https://github.com/cocosol007/hyper.git
+cd hyper
+```
+
+### 3D
+
+To run the 3D view, run the command
+
+```bash
+./gradlew run
+```
+
+
+
+[Source](https://github.com/user-attachments/assets/5156987d-9099-42a6-9bd4-1a86abcd819e)
+
+#### 3D Klein model
+
+To run the 3D Klein model view, run the command
+
+```bash
+./gradlew run --args="klein"
+```
+
+
+
+[Source](https://github.com/user-attachments/assets/b18b1862-54cd-4e70-bc3e-eb668aba2c67)
+
+#### 3D Gnomonic model
+
+To run the 3D Gnomonic model view, run the command
+
+```bash
+./gradlew run --args="gnomonic"
+```
+
+
+
+[Source](https://github.com/user-attachments/assets/bc754d07-b1c2-435d-b459-7e679d5f27de)
+
+#### Ray-Caster Maze
+
+To run the RayCaster maze, run the command
+
+```bash
+./gradlew runRayCaster
+```
+
+
+
+[Source](https://github.com/user-attachments/assets/cd564d2c-b59d-4107-a9d4-a5095251b5a3)
+
+### Poincaré and Beltrami–Klein disk model
+
+To run the 2D Poicaré disk model, run the command
+
+```bash
+./gradlew run2D --args='poincare'
+```
+
+To run the 2D Klein model, run the command
+
+```bash
+./gradlew run2D --args='Klein'
+```
+
+
+
+[Source](https://github.com/user-attachments/assets/cd1cf6c3-3550-4b40-a5ad-6ffeced9316c)
+
+### Gnomonic projection
+
+To run the 2D Gnomonic projection model, run the command
+
+```bash
+./gradlew run2D --args='gnomonic'
+```
+
+
+
+[Source](https://github.com/user-attachments/assets/23b0f337-2429-4373-aad7-e0cc986aa4b7)
+
+## Documentation
+
+### 0. Project Setup
+
+- A. [How to Run the Project](running-the-project.md)
+- B. [How to contribute](.github/CONTRIBUTING.md)
+- C. [What for?](what_for.md)
+
+### I. Foundations in Hyperbolic Geometry
+
+- A. [Introduction](I/introduction.md)
+- B. [Poincaré Disk Model](I/basic-mathematics-in-the-poincare-disk-model.md)
+- C. [Isometries](I/isometries.md)
+
+### II. Tessellations
+
+- A. [Introduction to Tessellation](II/introduction_to_tesslation.md)
+- B. [Coordinate System](II/coordinates_system.md)
+- C. [Implementation](II/implementation.md)
+
+### III. 3D Rendering
+
+- A. Ray-Caster Maze (WIP)
+- B. [Visual effects](III/3d-effects.md)
+
+### IV. In-Depth Explorations
+
+- A. [Other Models](IV/other_models.md)
+
+## Licence
+
+See all about our license [here](LICENCE.md)
+
+
+
+## Contributors
+
+[](https://github.com/cocosol007/hyper/graphs/contributors)
+
+[contributors-shield]: https://img.shields.io/github/contributors/cocosol007/hyper.svg?style=for-the-badge
+[contributors-url]: https://github.com/cocosol007/hyper/graphs/contributors
+[forks-shield]: https://img.shields.io/github/forks/cocosol007/hyper.svg?style=for-the-badge
+[forks-url]: https://github.com/cocosol007/hyper/network/members
+[stars-shield]: https://img.shields.io/github/stars/cocosol007/hyper.svg?style=for-the-badge
+[stars-url]: https://github.com/cocosol007/hyper/stargazers
+[issues-shield]: https://img.shields.io/github/issues/cocosol007/hyper.svg?style=for-the-badge
+[issues-url]: https://github.com/cocosol007/hyper/issues
+[license-shield]: https://img.shields.io/github/license/cocosol007/hyper.svg?style=for-the-badge
+[license-url]: https://github.com/cocosol007/hyper/blob/main/LICENCE
diff --git a/docs/_coverpage.md b/docs/_coverpage.md
new file mode 100644
index 0000000..0707d4d
--- /dev/null
+++ b/docs/_coverpage.md
@@ -0,0 +1,4 @@
+
+
+
+ A Non-Euclidean Rendering Engine.
+
+
+> A Non-Euclidean Rendering Engine.
+
diff --git a/docs/_navbar.md b/docs/_navbar.md
new file mode 100644
index 0000000..27441b3
--- /dev/null
+++ b/docs/_navbar.md
@@ -0,0 +1,28 @@
+
+
+* [Home](README.md)
+* [O. Project Setup](running-the-project.md)
+
+ * [A. How to Run the Project](running-the-project.md)
+ * [B. How to contribute](.github/CONTRIBUTING.md)
+ * [C. What for?](what_for.md)
+
+* [I. Foundations in Hyperbolic Geometry](I/introduction.md)
+
+ * [A. Introduction](I/introduction.md)
+ * [B. Poincaré Disk Model](I/basic-mathematics-in-the-poincare-disk-model.md)
+ * [C. Isometries](I/isometries.md)
+
+* [II. Tessellations](II/introduction_to_tesslation.md)
+
+ * [A. Introduction to Tessellation](II/introduction_to_tesslation.md)
+ * [B. Coordinate System](II/coordinates_system.md)
+ * [C. Implementation](II/implementation.md)
+
+* [III. 3D Rendering](III/3d-effects.md)
+
+ * [B. Visual effects](III/3d-effects.md)
+
+* [IV. In-Depth Explorations](IV/other_models.md)
+
+ * [A. Other Models](IV/other_models.md)
\ No newline at end of file
diff --git a/docs/complex_numbers.md b/docs/complex_numbers.md
index 1fa5e56..9e9500b 100644
--- a/docs/complex_numbers.md
+++ b/docs/complex_numbers.md
@@ -4,28 +4,28 @@ Complex numbers are a fundamental tool in mathematics, allowing us to extend the
## Definition
-A complex number is a number of the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, which satisfies $i^2 = -1$. The real part of the complex number is $a$, and the imaginary part is $b$.
+A complex number is a number of the form %%a + bi%%, where %%a%% and %%b%% are real numbers and %%i%% is the imaginary unit, which satisfies %%i^2 = -1%%. The real part of the complex number is %%a%%, and the imaginary part is %%b%%.
## Properties
Complex numbers have many useful properties, including:
-- **Addition**: Complex numbers can be added by adding the real and imaginary parts separately: $(a + bi) + (c + di) = (a + c) + (b + d)i$.
-- **Multiplication**: Complex numbers can be multiplied by multiplying the real and imaginary parts separately, and then combining the results: $(a + bi) \times (c + di) = (ac - bd) + (ad + bc)i$.
-- **Conjugation**: The complex conjugate of a complex number $a + bi$ is $a - bi$, which is obtained by changing the sign of the imaginary part.
+- **Addition**: Complex numbers can be added by adding the real and imaginary parts separately: %%(a + bi) + (c + di) = (a + c) + (b + d)i%%.
+- **Multiplication**: Complex numbers can be multiplied by multiplying the real and imaginary parts separately, and then combining the results: %%(a + bi) \times (c + di) = (ac - bd) + (ad + bc)i%%.
+- **Conjugation**: The complex conjugate of a complex number %%a + bi%% is %%a - bi%%, which is obtained by changing the sign of the imaginary part.
## Geometry
Complex numbers have a natural connection to geometry, as they can be used to represent points and vectors in the complex plane. In particular, complex numbers can be used to:
-- **Represent points**: A point in the complex plane can be represented by a complex number $a + bi$, where $a$ is the x-coordinate and $b$ is the y-coordinate.
+- **Represent points**: A point in the complex plane can be represented by a complex number %%a + bi%%, where %%a%% is the x-coordinate and %%b%% is the y-coordinate.
- **Perform rotations**: Multiplying a complex number by a complex number of modulus 1 (i.e. a unit complex number) rotates the point in the complex plane by the angle of the unit complex number.
- **Perform translations**: Adding a complex number to another complex number translates the point in the complex plane by the amount of the added complex number.
Two other properties of complex numbers are useful in geometry:
-- **Modulus**: The modulus of a complex number $a + bi$ is its distance from the origin in the complex plane, and is given by $|a + bi| = \sqrt{a^2 + b^2}$.
-- **Argument**: The argument of a complex number $a + bi$ is the angle between the real and imaginary parts, and is given by $\arg(a + bi) = \tan^{-1}(\frac{b}{a})$.
+- **Modulus**: The modulus of a complex number %%a + bi%% is its distance from the origin in the complex plane, and is given by %%|a + bi| = \sqrt{a^2 + b^2}%%.
+- **Argument**: The argument of a complex number %%a + bi%% is the angle between the real and imaginary parts, and is given by %%\arg(a + bi) = \tan^{-1}(\frac{b}{a})%%.
-> [!note]
+> Note:
> For a better representation of complex numbers, see a well explained [youtube video](https://www.youtube.com/watch?v=HrvEkftPZpw).
diff --git a/docs/index.html b/docs/index.html
new file mode 100644
index 0000000..65cb51f
--- /dev/null
+++ b/docs/index.html
@@ -0,0 +1,50 @@
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
diff --git a/docs/running-the-project.md b/docs/running-the-project.md
index 2529828..fee116a 100644
--- a/docs/running-the-project.md
+++ b/docs/running-the-project.md
@@ -10,12 +10,12 @@ Before starting, make sure you have:
## Installation
1. **Clone the project**
- ```sh
+ ```bash
git clone https://github.com/CoCoSol007/hyper.git
cd hyper
```
2. **Grant execution permission to the Gradle wrapper (Linux/Mac)**
- ```sh
+ ```bash
chmod +x gradlew
```
@@ -24,7 +24,7 @@ Before starting, make sure you have:
### Using the Gradle Wrapper (recommended)
- **On Linux/Mac**:
- ```sh
+ ```bash
./gradlew run
```
- **On Windows**:
@@ -36,21 +36,21 @@ Before starting, make sure you have:
You can use the command:
-```sh
+```bash
gradle run
```
## Other Useful Commands
- **Build the project**:
- ```sh
+ ```bash
./gradlew build
```
- **Clean generated files**:
- ```sh
+ ```bash
./gradlew clean
```
- **Run tests**:
- ```sh
+ ```bash
./gradlew test
```