## Heatmap: Wave Patterns with Varying Parameters ### Overview The image presents four heatmaps, each displaying a wave pattern. The x and y axes represent spatial dimensions in meters (m), ranging from approximately -0.45 to 0.45. The color gradient, ranging from blue to red, represents the wave amplitude, with blue indicating a value of -1 and red indicating a value of 1. Each heatmap is associated with a vector η, which likely represents parameters influencing the wave pattern. ### Components/Axes * **X-axis:** x (m), ranging from -0.4 to 0.4 * **Y-axis:** y (m), ranging from -0.4 to 0.4 * **Colorbar:** Ranges from -1 (blue) to 1 (red), with 0.5 and 0 marked as intermediate values. * **Parameter Vector:** Each heatmap is labeled with a vector η. ### Detailed Analysis **Heatmap 1:** * **Parameter Vector:** η = \[1.0, 0.0, 0.0]^T * **Wave Pattern:** Vertical stripes alternating between red (positive amplitude) and blue (negative amplitude). The stripes are oriented parallel to the y-axis. * **Observations:** The wave pattern is uniform across the y-axis. The wavelength appears consistent. **Heatmap 2:** * **Parameter Vector:** η = \[0.7, 0.7, 0.0]^T * **Wave Pattern:** Diagonal stripes alternating between red and blue. The stripes are oriented at an angle relative to both the x and y axes. * **Observations:** The wave pattern is uniform. The wavelength appears consistent. **Heatmap 3:** * **Parameter Vector:** η = \[0.0, 1.0, 0.0]^T * **Wave Pattern:** Horizontal stripes alternating between red and blue. The stripes are oriented parallel to the x-axis. * **Observations:** The wave pattern is uniform across the x-axis. The wavelength appears consistent. **Heatmap 4:** * **Parameter Vector:** η = \[-0.7, 0.7, 0.0]^T * **Wave Pattern:** Diagonal stripes alternating between red and blue. The stripes are oriented at an angle, opposite to the angle in Heatmap 2. * **Observations:** The wave pattern is uniform. The wavelength appears consistent. ### Key Observations * The parameter vector η seems to control the orientation of the wave pattern. * The first component of η appears to influence the x-axis orientation, while the second component influences the y-axis orientation. * The third component of η is consistently 0.0 across all heatmaps. ### Interpretation The heatmaps visualize wave patterns with varying orientations, controlled by the parameter vector η. The components of η likely represent weights or coefficients that determine the direction and possibly the frequency of the wave. The consistent value of 0.0 for the third component suggests it might be related to a parameter not visualized in these specific plots, such as amplitude or phase shift. The data suggests a relationship between the vector η and the angle of the wave propagation. A higher first component in η results in waves that are more vertically oriented, while a higher second component results in waves that are more horizontally oriented. The negative value in the first component of η in the fourth heatmap results in a wave pattern with an opposite diagonal orientation compared to the second heatmap.

\n ## Heatmaps: Parameter Variation Study ### Overview The image presents four heatmaps, arranged horizontally. Each heatmap visualizes a two-dimensional distribution, likely representing a function's value across a coordinate plane. The heatmaps appear to be illustrating the effect of varying a parameter vector 'η' on the distribution. The x and y axes are labeled in meters (m). Each heatmap is labeled with a specific value for the parameter vector η. ### Components/Axes Each heatmap shares the following components: * **X-axis:** Labeled "x (m)", ranging from approximately -0.4 to 0.4 meters. * **Y-axis:** Labeled "y (m)", ranging from approximately -0.4 to 0.4 meters. * **Color Scale:** A continuous color scale ranging from approximately -1 (blue) to 1 (red), with 0 represented by white. The scale is positioned to the right of each heatmap. * **Parameter Label:** Below each heatmap, a label indicates the value of the parameter vector η. ### Detailed Analysis or Content Details **Heatmap 1:** η = [1.0, 0.0, 0.0]^T The heatmap displays vertical stripes. The color transitions from red to blue and back to red as you move from left to right. The stripes are approximately 0.1 meters wide. The values appear to oscillate between approximately 1 and -1. **Heatmap 2:** η = [0.7, 0.7, 0.0]^T This heatmap shows diagonal stripes, running from the bottom-left to the top-right. The color transitions along these diagonals, alternating between red and blue. The stripes are approximately 0.15 meters wide. The values oscillate between approximately 1 and -1. **Heatmap 3:** η = [0.0, 1.0, 0.0]^T This heatmap displays horizontal stripes. The color transitions from red to blue and back to red as you move from top to bottom. The stripes are approximately 0.1 meters wide. The values appear to oscillate between approximately 1 and -1. **Heatmap 4:** η = [-0.7, 0.7, 0.0]^T This heatmap shows diagonal stripes, running from the top-left to the bottom-right. The color transitions along these diagonals, alternating between red and blue. The stripes are approximately 0.15 meters wide. The values oscillate between approximately 1 and -1. ### Key Observations * The parameter vector η significantly alters the orientation of the stripes in the heatmaps. * When the first element of η is non-zero, the stripes are vertical. * When the second element of η is non-zero, the stripes are horizontal. * When the first and second elements of η are both non-zero, the stripes are diagonal. * The magnitude of the elements in η appears to influence the width of the stripes. * The color scale consistently represents values between -1 and 1 across all heatmaps. ### Interpretation The data suggests that the parameter vector η controls the directionality of a periodic pattern. The heatmaps likely represent a solution to a partial differential equation or a wave phenomenon where the parameter η influences the wave's propagation direction. The different components of η seem to correspond to different spatial frequencies or wave vectors. The consistent color scale indicates that the underlying function's range remains constant, while the parameter η modulates its spatial distribution. The change in stripe orientation with different η values suggests a linear relationship between the parameter components and the pattern's direction. The observed patterns could represent interference phenomena or the behavior of a system under varying external forces. The fact that the values oscillate between -1 and 1 suggests a normalized or bounded quantity, potentially a correlation coefficient or a normalized amplitude.

## Heatmap Visualization: Directional Pattern Analysis ### Overview The image displays four heatmaps arranged horizontally, each representing a directional pattern influenced by a 3D vector η. The patterns transition from vertical/horizontal stripes to diagonal configurations, with color gradients ranging from blue (-1) to red (1). Each heatmap is labeled with a distinct η vector configuration. ### Components/Axes - **X-axis**: Labeled "x (m)", spanning -0.4 to 0.4 meters. - **Y-axis**: Labeled "y (m)", spanning -0.4 to 0.4 meters. - **Color Bar**: Vertical gradient from blue (-1) to red (1), positioned on the right of each heatmap. - **Labels**: Each heatmap includes a vector η = [a, b, c]^T, where a, b, c are scalar values. ### Detailed Analysis 1. **First Heatmap (η = [1.0, 0.0, 0.0]^T)**: - **Pattern**: Vertical red/blue stripes aligned with the y-axis. - **Color Distribution**: Alternating red (1) and blue (-1) bands along the x-axis. - **Spatial Grounding**: Legend (color bar) matches stripe intensity; red corresponds to x = ±0.4, blue to x = ±0.2. 2. **Second Heatmap (η = [0.7, 0.7, 0.0]^T)**: - **Pattern**: Diagonal stripes sloping from bottom-left to top-right. - **Color Distribution**: Red (1) dominates upper-right regions; blue (-1) in lower-left. - **Spatial Grounding**: Diagonal bands align with η's equal x/y components (0.7, 0.7). 3. **Third Heatmap (η = [0.0, 1.0, 0.0]^T)**: - **Pattern**: Horizontal red/blue stripes aligned with the x-axis. - **Color Distribution**: Red (1) at y = ±0.4; blue (-1) at y = ±0.2. - **Spatial Grounding**: Horizontal bands reflect η's y-component dominance (1.0). 4. **Fourth Heatmap (η = [-0.7, 0.7, 0.0]^T)**: - **Pattern**: Diagonal stripes sloping from top-left to bottom-right. - **Color Distribution**: Red (1) in lower-right; blue (-1) in upper-left. - **Spatial Grounding**: Negative x-component (-0.7) inverts the diagonal direction compared to the second heatmap. ### Key Observations - **Directional Correlation**: η's components directly influence stripe orientation. Equal x/y magnitudes (e.g., 0.7, 0.7) produce 45° diagonal patterns. - **Sign Sensitivity**: Negative x-components (e.g., -0.7) reverse diagonal slope direction. - **Axis Alignment**: η components with zero values (e.g., [1.0, 0.0, 0.0]^T) produce axis-aligned patterns. ### Interpretation The heatmaps visualize a directional field where η's components modulate spatial patterns. The first and third heatmaps demonstrate axis-aligned responses to single-axis dominance (x or y), while the second and fourth illustrate combined x/y influences. The color gradient (-1 to 1) quantifies the magnitude of the directional effect, with red/blue representing opposing polarities. This suggests a system where vector inputs (η) control spatial modulation, potentially modeling wave interference, stress distribution, or fluid dynamics. The consistent scaling across heatmaps implies a normalized response metric.