# Technical Document Extraction: Horizon Length vs. Step Accuracy
## 1. Image Overview
This image is a technical line chart illustrating the relationship between **Step Accuracy ($p$)** and **Horizon Length**. It features five distinct data series, each representing a different threshold or parameter denoted as $H_{\text{value}}$. The chart uses a logarithmic-like growth pattern where the Horizon Length increases exponentially as Step Accuracy approaches 1.
---
## 2. Component Isolation
### A. Header / Legend
* **Location:** Top-left quadrant of the chart area.
* **Content:** A white box with a light gray border containing five entries.
* **Legend Items (Color-Coded):**
1. **Yellow Line:** $H_{0.10}$
2. **Orange Line:** $H_{0.25}$
3. **Pink/Magenta Line:** $H_{0.50}$
4. **Purple Line:** $H_{0.75}$
5. **Dark Indigo/Violet Line:** $H_{0.90}$
### B. Main Chart Area (Axes and Grid)
* **Y-Axis (Vertical):**
* **Label:** "Horizon Length"
* **Scale:** Linear, ranging from 0 to 100.
* **Major Markers:** 20, 40, 60, 80, 100.
* **X-Axis (Horizontal):**
* **Label:** "Step Accuracy (p)"
* **Scale:** Linear, ranging from 0.8 to 1.
* **Major Markers:** 0.8, 0.85, 0.9, 0.95, 1.
* **Grid:** Dashed gray lines corresponding to the major markers on both axes.
---
## 3. Data Series Analysis and Trend Verification
All five series exhibit a **positive exponential growth trend**. As Step Accuracy ($p$) moves from 0.8 toward 1.0, the Horizon Length increases. The rate of increase is highest for the $H_{0.10}$ series and lowest for the $H_{0.90}$ series at any given point on the x-axis.
### Detailed Series Extraction
| Series Label | Color | Visual Trend Description | Approx. Value at $p=0.8$ | Approx. Value at $p=0.9$ | Vertical Asymptote Behavior |
| :--- | :--- | :--- | :--- | :--- | :--- |
| **$H_{0.10}$** | Yellow | Steepest upward curve; reaches the y-limit (100) earliest. | ~11 | ~22 | Crosses $y=100$ at $p \approx 0.975$ |
| **$H_{0.25}$** | Orange | Moderate-steep upward curve. | ~7 | ~13 | Crosses $y=100$ at $p \approx 0.985$ |
| **$H_{0.50}$** | Pink | Moderate upward curve. | ~3 | ~6 | Crosses $y=100$ at $p \approx 0.993$ |
| **$H_{0.75}$** | Purple | Shallow curve until $p > 0.95$, then sharp rise. | ~2 | ~3 | Crosses $y=100$ at $p \approx 0.998$ |
| **$H_{0.90}$** | Indigo | Shallowest curve; remains near $y=0$ until $p > 0.98$. | ~1 | ~1.5 | Crosses $y=100$ at $p \approx 0.999$ |
---
## 4. Key Observations and Data Patterns
1. **Inverse Relationship of Subscripts:** There is an inverse relationship between the subscript value in $H$ and the Horizon Length. For a fixed Step Accuracy (e.g., $p=0.9$), $H_{0.10}$ has the highest value (~22) while $H_{0.90}$ has the lowest (~1.5).
2. **Convergence at $p=1$:** All lines appear to be asymptotic to the vertical line $p=1$. As accuracy reaches perfection (1.0), the Horizon Length theoretically approaches infinity for all series.
3. **Sensitivity:** The "Horizon Length" is extremely sensitive to small changes in "Step Accuracy" once $p$ exceeds 0.95. For example, in the $H_{0.10}$ (yellow) series, the length doubles from ~20 to ~40 between $p=0.88$ and $p=0.94$, but then jumps from 40 to 100 between $p=0.94$ and $p=0.975$.
---
## 5. Text Transcription
* **Y-Axis Label:** Horizon Length
* **X-Axis Label:** Step Accuracy (p)
* **Legend Text:**
* $H_{0.10}$
* $H_{0.25}$
* $H_{0.50}$
* $H_{0.75}$
* $H_{0.90}$
* **Numerical Markers (Y):** 20, 40, 60, 80, 100
* **Numerical Markers (X):** 0.8, 0.85, 0.9, 0.95, 1