Contour Plot

A contour plot displays three-dimensional data in two dimensions by using contour lines (isolines) that connect points of equal value. Similar to a topographic map showing elevation, a contour plot shows how a Z value varies across an X-Y plane. Closely spaced contour lines indicate steep changes, while widely spaced lines show gradual variation. Contour plots are essential for visualizing continuous functions, response surfaces, and spatial distributions.

Best used for:

• Visualizing three-dimensional relationships in 2D
• Response surface analysis and optimization
• Geographic elevation and topographic mapping
• Identifying peaks, valleys, and saddle points
• Understanding how a dependent variable changes with two independent variables
• Spatial interpolation and gradient visualization

Common Use Cases

Engineering & Optimization

• Response surface methodology (RSM)
• Design of experiments (DOE) analysis
• Multi-parameter optimization
• Process parameter tuning
• Performance mapping (efficiency, power, speed)

Scientific Research

• Geographic and topographic mapping
• Temperature or pressure distributions
• Electromagnetic field visualization
• Molecular orbital surfaces
• Population density mapping

Machine Learning & Statistics

• Decision boundary visualization
• Loss function landscapes
• Parameter space exploration
• Probability density functions
• Classification regions

Business Analytics

• Customer segmentation (two dimensions)
• Pricing optimization (price vs demand)
• Risk mapping (two-factor risk analysis)
• Market positioning analysis

Options

X Axis

Required - Horizontal dimension.

Select the numerical column for the X-axis. This represents one of the independent variables.

Y Axis

Required - Vertical dimension.

Select the numerical column for the Y-axis. This represents the second independent variable.

Z Value

Required - Height/value dimension.

Select the numerical column for the Z values. This is the dependent variable whose contours will be plotted. Z values determine the contour line positions and colors.

Settings

Hide Empty Values

Optional - Exclude points with no data.

When enabled, missing or null Z values are excluded from interpolation and rendering.

Understanding Contour Plots

Contour Lines

• Isoline: Line connecting points of equal Z value
• Contour interval: Difference in Z value between adjacent lines
• Closely spaced lines: Steep gradient (rapid change)
• Widely spaced lines: Gentle gradient (gradual change)
• Closed loops: Local maxima or minima

Color Mapping

• Color intensity: Represents Z value magnitude
• Color scale: Legend showing Z value to color mapping
• Smooth gradients: Continuous Z variation
• Distinct bands: Discrete Z levels

Key Features

• Peaks: Local maxima (highest Z values)
• Valleys: Local minima (lowest Z values)
• Saddle points: Inflection points (curves in opposite directions)
• Ridges: Linear high points
• Contour labels: Show exact Z values on lines

Topographic Analogy

Think of a contour plot like a topographic map:

• Elevation = Z value
• Steep hills = Closely spaced contours
• Flat plains = Widely spaced contours
• Mountain peaks = Closed high-value contours
• Valleys = Closed low-value contours

Tips for Effective Contour Plots

1. Data Requirements:

   • Need X, Y, Z values for each observation
   • Data should cover the region of interest adequately
   • More data points = smoother, more accurate contours
   • Consider regular grid for best results (though not required)

2. Interpolation Considerations:

   • Contour plots interpolate between data points
   • Sparse data leads to uncertain contours
   • Avoid extrapolation beyond data boundaries
   • Use sufficient density in areas of interest

3. Contour Interval Selection:

   • Too few contours: Loss of detail
   • Too many contours: Cluttered and hard to read
   • Start with 10-20 contour levels
   • Focus on meaningful Z value ranges

4. Color Scale Choice:

   • Sequential: For continuous data (low to high)
   • Diverging: For data with meaningful zero or midpoint
   • Perceptually uniform: Viridis, Plasma for accurate perception
   • Ensure colorblind accessibility

5. Identifying Features:

   • Look for closed contours (peaks/valleys)
   • Follow contour spacing (gradient magnitude)
   • Identify flat regions (constant Z)
   • Locate optimal points or sweet spots

6. Combining Visualizations:

   • Add contour labels for exact values
   • Overlay data points to show sample locations
   • Use filled contours (contourf) for clearer areas
   • Add surface plot for 3D perspective

Contour Plot vs Related Visualizations

vs Heatmap

• Contour Plot: Shows continuous isolines, better for gradients
• Heatmap: Shows discrete cells, better for matrices
• Contour advantage: Clearer gradient visualization

vs 3D Surface Plot

• Contour Plot: 2D projection, easier to read exact values
• 3D Surface: Shows height directly, more intuitive for some
• Contour advantage: No occlusion, precise value reading

vs Density Heatmap

• Contour Plot: Smooth continuous function
• Density Heatmap: Binned data, shows point concentrations
• Different purposes: Contour for continuous Z(x,y), density for point distribution

Example Scenarios

Response Surface Analysis

Optimization showing temperature vs pressure with yield as Z value. Peak indicates optimal conditions.

Topographic Map

Geographic elevation data showing mountains, valleys, and terrain features.

Decision Boundary

Machine learning classification boundary between two features, with probability as Z.

Temperature Distribution

Spatial temperature map showing hot and cold zones across a surface.

Interpreting Contour Plots

Reading Contours

1. Identify contour values: Check labels or legend
2. Follow lines: Trace paths of equal value
3. Assess spacing: Determine gradient steepness
4. Locate extrema: Find highest and lowest values
5. Understand trends: See how Z changes across X-Y space

Common Patterns

• Bowl shape: Single minimum (optimization target)
• Peak: Single maximum (avoid or target)
• Ridge: Linear high zone
• Valley: Linear low zone
• Saddle: Mixed curvature (optimization challenge)
• Flat: Insensitive region (parameters don't matter)

Optimization Context

• Global optimum: Highest/lowest point in entire domain
• Local optimum: Highest/lowest in local region
• Sweet spot: Desirable value region
• Constraint regions: Areas to avoid or maintain

Troubleshooting

Issue: Contours are jagged or irregular

• Solution: Increase data density, use more observations, smooth interpolation settings, or check for outliers causing distortion.

Issue: No contours are visible

• Solution: Check Z value range (may be too narrow), verify data is numeric, ensure sufficient data points, check color scale range matches data.

Issue: Contours don't make physical sense

• Solution: Verify X, Y, Z columns are correctly assigned, check for data entry errors, ensure correct units, validate interpolation method.

Issue: Can't identify peaks or valleys

• Solution: Increase number of contour levels, use contour labels, add color fill, adjust Z-axis range to focus on relevant area.

Issue: Plot is too crowded with lines

• Solution: Reduce number of contour levels, increase contour interval, use filled contours with fewer lines, or focus on subset of Z range.

Issue: Interpolation artifacts between sparse data

• Solution: Add more data points in sparse regions, use simpler interpolation, show data point locations, or acknowledge uncertainty.

Issue: Can't read contour values

• Solution: Enable contour labels, increase font size, show legend with color-to-value mapping, reduce number of contour lines.

Issue: Colors don't show variation

• Solution: Check Z value range, adjust color scale limits, use diverging palette if data crosses zero, remove outliers compressing scale.