Constructing Accurate Ternary Diagrams: Tools and Techniques

Constructing Accurate Ternary Diagrams: Tools and TechniquesA ternary diagram (ternary plot, triangle plot) is a two-dimensional graphic representation of the proportions of three variables that sum to a constant — typically 100% or 1. Ternary diagrams are widely used in geology, materials science, chemistry, soil science, petrology, and many applied fields to visualize compositional data and to reveal relationships, trends, mixing lines, and end-member contributions. Constructing an accurate ternary diagram requires careful attention to the underlying data, coordinate transformations, plotting conventions, labeling, and the choice of tools. This article provides a practical, step-by-step guide to building reliable ternary plots, discusses common pitfalls, and surveys software and programmatic options for producing publication-quality figures.

1. Principles of ternary plotting

• Composition constraint: Each data point is a triplet (A, B, C) with A + B + C = constant (often 100 or 1). Only two independent degrees of freedom exist; the ternary diagram maps these to a 2D triangle.
• Geometric interpretation: Each apex corresponds to 100% of one component and 0% of the others. Lines parallel to a side indicate constant fraction of the opposite component.
• Coordinate transforms: To plot a composition (A, B, C) one converts it to cartesian coordinates (x, y) inside an equilateral triangle using standard transformations described below.

Coordinate conversion (assuming components expressed as fractions that sum to 1 and an equilateral triangle of side length 1): Let A, B, C be fractions summing to 1. One common transform to Cartesian coordinates is: x = 0.5 * (2*B + C) y = (sqrt(3)/2) * C (Other equivalent transforms exist depending on triangle orientation and scaling.)

Note: If your data sum to 100, divide by 100 first. If the sum differs, normalize by total for each sample: A’ = A/(A+B+C), etc.

2. Data preparation and normalization

• Check completeness: Remove or handle samples with missing components. A ternary plot requires three values per sample; if a fourth or more components exist, consider grouping or using compositional data analysis methods (see Section 6).
• Normalize: Convert raw component values to proportions that sum to 1. Use: A’ = A / (A + B + C), etc.
• Handle zeros and detection limits: Zeros cause issues in log-ratio transforms and some plotting strategies. Strategies:
  • Leave as zero and plot at the boundary (valid but may be sensitive).
  • Replace with a small positive value (careful — introduces bias).
  • Use compositional data methods (e.g., replacement algorithms like multiplicative replacement) if statistical treatment is required.
• Units and consistency: Ensure all three components are comparable (mass fraction, mole fraction, volume fraction). Mixing different units without conversion yields meaningless plots.

3. Coordinate transformation and plotting geometry

• Standard equilateral-triangle geometry: Place vertices at (0,0), (1,0), and (0.5, sqrt(3)/2) for convenience, or scale to desired figure size.
• Example transform (for A at left apex, B at right apex, C at top apex), with A + B + C = 1: x = 0.5 * (2*B + C) y = (sqrt(3)/2) * C
• Alternative mapping (frequently used in geology where points are plotted relative to base AB): x = 0.5*(2*A + B) y = (sqrt(3)/2) * B Always verify which component aligns with which apex in your plotting library or custom script.
• Gridlines / isopleths: To show constant values of one component, draw lines parallel to the opposite side at intervals (e.g., 10% steps). Compute endpoints by converting edge-intersection compositions to (x,y) and drawing straight lines.

4. Visual design and best practices

• Aspect ratio: Preserve equal scaling in x and y so the triangle remains equilateral on the final figure. Disable automatic axis autoscaling that distorts aspect ratio.
• Labels and tick marks: Label each apex with component name and units. Add tick marks along triangle edges indicating percent/composition steps. Place tick labels clearly, rotated or offset as necessary.
• Color, symbols, and layering:
  • Use color scales only if representing an additional variable (e.g., temperature, age). Avoid encoding component values redundantly with color that could confuse interpretation.
  • Choose marker shapes and sizes appropriate for data density. Use transparency (alpha) for overlapping points.
  • For scientific figures, use color-blind-friendly palettes (e.g., ColorBrewer).
• Annotations: Mark end-members, mixing lines, and sample groups. Include legend and concise caption explaining normalization and any data treatments.
• Error bars: If component uncertainties are available, propagate them into triangular coordinate space (convert upper/lower bounds to (x,y)) and plot error bars or uncertainty polygons.

5. Common pitfalls and how to avoid them

• Failing to normalize: Plotting raw counts or unequal-sum triplets will misplace points. Always normalize.
• Distorted triangle: Letting plotting software rescale axes independently will distort geometry. Fix aspect ratio.
• Mislabeling axes: Ternary plots have no orthogonal axes like Cartesian plots; labeling must clearly indicate which apex corresponds to which component.
• Mixing units: Converting mass fractions and mole fractions inconsistently leads to wrong interpretations—convert first.
• Ignoring zeros and detection limits: Treat zeros thoughtfully; report replacement strategies if used.
• Overplotting and poor color choices: Make dense data legible with transparency, jitter, or density contours.

6. Statistical considerations for compositional data

Compositional data lie in the simplex, a constrained sample space; standard statistical methods on raw proportions can be misleading. Consider using compositional data analysis (CoDA) techniques:

• Aitchison geometry and log-ratio transforms:
  • Additive log-ratio (alr), centered log-ratio (clr), and isometric log-ratio (ilr) transforms map compositions to real Euclidean space for standard statistical techniques.
  • Example: alr(A,B,C) = (ln(A/C), ln(B/C)) for a chosen reference part C.
• Use CoDA for clustering, principal component analysis, regression, and hypothesis testing on compositional datasets.
• Back-transform results to the simplex for plotting and interpretation.

7. Tools and software for constructing ternary diagrams

Below are practical options from simple plotting to advanced compositional analysis:

• Excel / spreadsheet add-ins:
  • Simple ternary plots can be created with built-in scatter plots after coordinate conversion, or via templates and add-ins. Limited styling and isopleth drawing.
• R:
  • packages: ggtern (extension of ggplot2 for ternary plots), compositions (CoDA workflows), robCompositions.
  • Advantages: full control, publication-quality output, CoDA integration, scripting for reproducibility.
  • Example: ggtern easily creates isopleths, contours, classifications, and supports theme customization.
• Python:
  • packages: python-ternary (simple and flexible), plotly (ternary support, interactive), matplotlib with custom transforms, scikit-bio (some compositional tools).
  • Advantages: integration with scientific workflows, interactivity via Plotly, and programmatic control.
• MATLAB:
  • File Exchange functions and some toolboxes provide ternary plotting capabilities.
• GIS and specialized packages:
  • Some geochemical and petrological packages include ternary plotting tailored to domain conventions.

8. Example workflows

1. Quick exploratory plot (Python + python-ternary)

• Normalize data to fractions.
• Use python-ternary to draw the triangle, add gridlines at 10% intervals, plot points with transparency, add legends.

1. Publication-quality figure with statistical rigor (R + ggtern + compositions)

• Normalize and examine zeros.
• Apply clr or ilr transform for multivariate analyses.
• Use ggtern for plotting raw compositions with annotated mixing lines and confidence regions derived from transformed-space statistics.

9. Advanced topics

• Contour and density plots: Kernel density estimates or hex-binning adapted to the simplex can show sample density; compute in (alr/ilr) space or directly on transformed (x,y) coordinates, then map back.
• Ternary thermodynamics diagrams: Incorporate phase fields, tie-lines, and invariant points; ensure correct projection of thermodynamic model outputs onto compositional space.
• Higher-dimensional compositional visualization: For >3 components, use pairwise ternary plots, radial plots, or dimensionality reduction after CoDA transforms.

10. Quick reference: step-by-step checklist

1. Verify components are comparable and measured in consistent units.
2. Handle missing values and zeros appropriately.
3. Normalize each sample so A + B + C = 1 (or 100).
4. Convert to Cartesian coordinates for your chosen triangle orientation.
5. Plot with fixed aspect ratio and add gridlines/isopleths.
6. Label apices, add tick marks, and use clear legends.
7. Use CoDA methods if performing statistical analysis.
8. Export high-resolution figure with preserved aspect ratio for publication.

Constructing accurate ternary diagrams combines careful data handling, correct geometric transforms, thoughtful visual design, and—when analyses are required—appropriate compositional statistics. Choosing the right tools (ggtern, python-ternary, Plotly, MATLAB, or Excel for quick checks) and following the checklist above will help you produce clear, accurate, and reproducible ternary plots suitable for exploration, publication, and decision-making.