Docs: Add detailed mathematical explanation of mouse movement algorithm

- Added deep-dive documentation in `docs/move_naturally_full_detailed_explanation.md` covering Bezier curves and Quadratic Ease-In-Out acceleration.
- Added visual charts (`chart.png` and `chart2.png`) to illustrate the coordinate system and acceleration profiles.
- Updated `README.md` with a high-level summary of the "Human Movement" algorithm and a link to the full technical reference.

Author: Deadpool2000

README.md

@@ -262,6 +262,21 @@ The software thinks you're at your desk. You're at the coffee machine. Everyone
 
 **Remember**: WASP is for reasonable breaks, not for gaming all day!
 
+---
+## 🧠 The Science of "Human" Movement
+
+WASP doesn't just "jump" the mouse from point A to point B. That would be an instant red flag for tracking algorithms. Instead, we realized that human hand movement follows specific laws of physics.
+
+### The Algorithm
+We implemented a **Cubic Bezier Curve** algorithm with **Quadratic Ease-In-Out** acceleration.
+
+1.  **Curvature**: Real hands move in arcs, not straight lines. We calculate a random control point offset to create a natural curve for every movement.
+2.  **Acceleration**: Humans have mass. We take time to speed up and slow down. WASP mathematically simulates this inertia, starting slow, accelerating, and decelerating on arrival.
+3.  **Micro-Jitter**: We add imperceptible randomness to the path, just like the micro-tremors in a human hand.
+
+[**📚 Read the Full Mathematical Explanation**](docs/move_naturally_full_detailed_explanation.md)
+*Includes XY graphs, Bezier formulas, and acceleration calculus.*
+
 ---
 
 ## 🤝 Legal Disclaimer

core/__pycache__/__init__.cpython-313.pyc

@@ -0,0 +1,266 @@
+# Mathematical Explanation of Human-Like Mouse Movement Algorithm
+
+This document provides a **complete, step-by-step mathematical and
+visual explanation** of how the `move_naturally` function computes
+**Bezier curve control points** and moves the mouse in a human-like
+manner.
+
+Everything explained here corresponds directly to the code logic.
+
+------------------------------------------------------------------------
+
+## 1. What Problem Are We Solving?
+
+Robotic mouse movement looks like this:
+
+-   Constant speed
+-   Straight lines
+-   No curvature
+
+Humans move a mouse like this:
+
+-   Accelerate → cruise → decelerate
+-   Slight curves
+-   Small randomness
+
+The solution used here is a **Cubic Bezier Curve**.
+
+------------------------------------------------------------------------
+
+## 2. XY Coordinate System
+
+We use a standard XY graph:
+
+-   X axis → horizontal
+-   Y axis → vertical
+
+Example values used throughout this explanation:
+
+  Point   Meaning   Coordinates
+  ------- --------- -------------
+  S       Start     (100, 100)
+  E       End       (400, 300)
+
+------------------------------------------------------------------------
+
+## 3. Step 1 -- Direction Vector (Straight Line)
+
+From the code:
+
+``` python
+dx = x - start_x
+dy = y - start_y
+```
+
+Calculation:
+
+    dx = 400 - 100 = 300
+    dy = 300 - 100 = 200
+
+So the straight-line direction vector is:
+
+    SE = (300, 200)
+
+Distance between points:
+
+    distance = sqrt(dx² + dy²)
+             = sqrt(300² + 200²)
+             = sqrt(130000)
+             ≈ 360.56
+
+------------------------------------------------------------------------
+
+## 4. Step 2 -- Perpendicular Vector (Creates Curvature)
+
+To avoid straight movement, a perpendicular vector is created:
+
+``` python
+px = -dy
+py = dx
+```
+
+Calculation:
+
+    px = -200
+    py = 300
+
+This vector points sideways relative to the direct path.
+
+------------------------------------------------------------------------
+
+## 5. Step 3 -- Normalize the Perpendicular Vector
+
+Normalize to length 1:
+
+    length = sqrt(px² + py²)
+           = sqrt(200² + 300²)
+           = sqrt(130000)
+           ≈ 360.56
+
+Normalized vector:
+
+    px = -200 / 360.56 ≈ -0.555
+    py =  300 / 360.56 ≈  0.832
+
+------------------------------------------------------------------------
+
+## 6. Step 4 -- Random Arc Offset
+
+From the code:
+
+``` python
+arc_scale = random.uniform(-0.2, 0.2)
+offset_x = px * distance * arc_scale
+offset_y = py * distance * arc_scale
+```
+
+Assume:
+
+    arc_scale = 0.1
+
+Then:
+
+    offset_x = -0.555 * 360.56 * 0.1 ≈ -20
+    offset_y =  0.832 * 360.56 * 0.1 ≈  30
+
+This offset bends the path.
+
+------------------------------------------------------------------------
+
+## 7. Step 5 -- Compute Bezier Control Points
+
+Control points are placed **along the path** but shifted sideways.
+
+### Control Point C1 (25% of the path)
+
+    C1_x = start_x + 0.25*dx + offset_x
+         = 100 + 75 - 20
+         = 155
+
+    C1_y = start_y + 0.25*dy + offset_y
+         = 100 + 50 + 30
+         = 180
+
+    C1 = (155, 180)
+
+------------------------------------------------------------------------
+
+### Control Point C2 (75% of the path)
+
+    C2_x = start_x + 0.75*dx + offset_x
+         = 100 + 225 - 20
+         = 305
+
+    C2_y = start_y + 0.75*dy + offset_y
+         = 100 + 150 + 30
+         = 280
+
+    C2 = (305, 280)
+
+------------------------------------------------------------------------
+
+## 8. XY Graph Diagram
+
+![Coordinate System Diagram](chart.png)
+
+*   **Dashed line**: Robotic straight path
+*   **Curved line**: Human-like Bezier movement
+*   **Start/End**: The mouse journey
+*   **C1/C2**: Invisible control points pulling the curve
+
+------------------------------------------------------------------------
+
+## 9. Bezier Curve Mathematical Formula
+
+Cubic Bezier equation:
+
+    B(t) = (1−t)³S
+         + 3(1−t)²tC1
+         + 3(1−t)t²C2
+         + t³E
+
+Where: - `t ∈ [0, 1]` - `B(0) = Start` - `B(1) = End`
+
+------------------------------------------------------------------------
+
+## 10. Example: Bezier Point at t = 0.5
+
+### X coordinate:
+
+    B_x(0.5) =
+    (0.5³ * 100)
+    + 3*(0.5²*0.5*155)
+    + 3*(0.5*0.5²*305)
+    + (0.5³ * 400)
+
+    ≈ 235
+
+### Y coordinate:
+
+    B_y(0.5) =
+    (0.5³ * 100)
+    + 3*(0.5²*0.5*180)
+    + 3*(0.5*0.5²*280)
+    + (0.5³ * 300)
+
+    ≈ 223
+
+So the midpoint lies at:
+
+    (235, 223)
+
+------------------------------------------------------------------------
+
+## 11. Acceleration (Ease-In-Out)
+
+The function applies an **Ease-In-Out Quadratic** easing to the time variable `t`. This transforms linear time (constant speed) into "perceived" time `alpha` (variable speed).
+
+![Acceleration Graph](chart2.png)
+
+### The Math
+
+We use a piecewise function based on the progress `t` (from 0 to 1):
+
+1. **Acceleration Phase** (Start to Halfway, `t < 0.5`):
+   ```math
+   alpha = 2 * t^2
+   ```
+   *   At `t=0`, speed is 0.
+   *   Velocity increases linearly.
+
+2. **Deceleration Phase** (Halfway to End, `t >= 0.5`):
+   ```math
+   alpha = -1 + (4 - 2 * t) * t
+   ```
+   *   This is an inverted parabola.
+   *   Velocity decreases linearly to 0 at the end.
+
+### Why this specific formula?
+
+At the midpoint `t = 0.5`:
+*   **Formula 1:** $2 * (0.5)^2 = 0.5$
+*   **Formula 2:** $-1 + (4 - 1) * 0.5 = 0.5$
+
+The values match perfectly, and more importantly, the **derivative (speed)** matches, ensuring a perfectly smooth transition with no "jerk" at the halfway point. This mimics the physics of a hand starting a movement, reaching peak speed, and slowing down to stop.
+
+## 12. Why This Looks Human
+
+  Feature                Effect
+  ---------------------- ---------------
+  Perpendicular offset   Curved motion
+  Random arc             No repetition
+  Bezier curve           Smooth path
+  Ease-in-out            Natural speed
+
+------------------------------------------------------------------------
+
+## 13. Final Result
+
+✔ Natural-looking mouse movement\
+✔ Hard to distinguish from real user\
+✔ Smooth curves and timing\
+✔ No sharp jumps
+
+------------------------------------------------------------------------
+
+### End of Detailed Explanation