# 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

# Technical Document Extraction: Line Chart Analysis ## Chart Overview The image depicts a line chart illustrating the relationship between **Step Accuracy (p)** and **Horizon Length** across five distinct parameter values (H). The chart uses color-coded curves to represent different H values, with a legend in the top-left corner for reference. --- ### **Axis Labels and Scales** - **X-Axis (Horizontal):** - Title: **Step Accuracy (p)** - Range: **0.8** to **1** - Tick Marks: **0.8, 0.85, 0.9, 0.95, 1** - Scale: Linear - **Y-Axis (Vertical):** - Title: **Horizon Length** - Range: **0** to **100** - Tick Marks: **20, 40, 60, 80, 100** - Scale: Linear --- ### **Legend** - **Location:** Top-left corner (inside a gray box). - **Labels and Colors:** | H Value | Color | |---------|------------| | H₀.₁₀ | Yellow | | H₀.₂₅ | Orange | | H₀.₅₀ | Red | | H₀.₇₅ | Purple | | H₀.₉₀ | Dark Blue | --- ### **Data Series and Trends** 1. **H₀.₁₀ (Yellow Line):** - **Trend:** Starts at ~10 Horizon Length when p=0.8, curves steeply upward, reaching ~95 at p=1. - **Key Points:** - p=0.8 → ~10 - p=0.9 → ~30 - p=0.95 → ~70 - p=1 → ~95 2. **H₀.₂₅ (Orange Line):** - **Trend:** Starts at ~5 Horizon Length when p=0.8, curves upward, reaching ~85 at p=1. - **Key Points:** - p=0.8 → ~5 - p=0.9 → ~25 - p=0.95 → ~60 - p=1 → ~85 3. **H₀.₅₀ (Red Line):** - **Trend:** Starts at ~2 Horizon Length when p=0.8, curves upward, reaching ~90 at p=1. - **Key Points:** - p=0.8 → ~2 - p=0.9 → ~15 - p=0.95 → ~50 - p=1 → ~90 4. **H₀.₇₅ (Purple Line):** - **Trend:** Starts at ~1 Horizon Length when p=0.8, curves upward, reaching ~98 at p=1. - **Key Points:** - p=0.8 → ~1 - p=0.9 → ~10 - p=0.95 → ~75 - p=1 → ~98 5. **H₀.₉₀ (Dark Blue Line):** - **Trend:** Nearly flat line starting at ~0.5 Horizon Length when p=0.8, rising sharply near p=1 to ~99. - **Key Points:** - p=0.8 → ~0.5 - p=0.9 → ~2 - p=0.95 → ~50 - p=1 → ~99 --- ### **Key Observations** - **Inverse Relationship:** Higher H values (e.g., H₀.₉₀) exhibit steeper curves, indicating rapid increases in Horizon Length as Step Accuracy approaches 1. - **Convergence:** All lines converge near p=1, with Horizon Length approaching 100 for all H values. - **Divergence:** Lower H values (e.g., H₀.₁₀) show gradual increases, while higher H values (e.g., H₀.₉₀) exhibit near-vertical rises. --- ### **Spatial Grounding** - **Legend Placement:** Top-left corner (x=0.05, y=0.95 relative to chart boundaries). - **Line Placement:** - Yellow (H₀.₁₀): Topmost curve. - Orange (H₀.₂₅): Second-highest curve. - Red (H₀.₅₀): Third-highest curve. - Purple (H₀.₇₅): Fourth-highest curve. - Dark Blue (H₀.₉₀): Bottommost curve. --- ### **Conclusion** The chart demonstrates that **Higher H values correlate with steeper increases in Horizon Length as Step Accuracy improves**. This suggests a parameter-dependent sensitivity in the modeled system, where higher H values amplify the relationship between accuracy and horizon length.