# Interpolating and contouring

#### Martin R. Smith

#### 2026-02-03

> Before you start, have you read the introduction to the Ternary package?

A ternary plot can be shaded and contoured according to a known function:

    library("Ternary")
    par(mar = rep(0.2, 4))

    FunctionToContour <- function(a, b, c) {
      a - c + (4 * a * b) + (27 * a * b * c)
    }

    values <- TernaryPointValues(FunctionToContour, resolution = 24L)
    TernaryPlot(alab = "a", blab = "b", clab = "c",
                # Place an opaque fill behind grid lines:
                panel.first = ColourTernary(values, spectrum = hcl.colors(256)))
    TernaryContour(FunctionToContour, resolution = 36L, legend = TRUE,
                   legend... = list(bty = "n"))

If it is not computationally feasible to execute a function at every point, it is possible to interpolate between known values.

Inverse distance weighting is a simple approach:

    # If using your own data, set
    # abc <- [Three-column matrix containing a, b, c coordinates of points]
    # response <- [Vector of values at the points specified in abc]

    # We'll use random data for this example.

    # First we'll generate some points at which to sample:
    set.seed(0)
    nPts <- 50
    a <- runif(nPts)
    b <- runif(nPts) * (1 - a)
    c <- 1 - a - b
    abc <- rbind(a, b, c)

    # Then we'll measure the response variable at each of those points:
    response <- FunctionToContour(a, b, c)

    # Now we must start a plot, to define the coordinate system
    par(mar = rep(0.2, 4))
    TernaryPlot(alab = "a", blab = "b", clab = "c")

    # Convert measured points to XY
    xy <- TernaryToXY(abc)

    # Use an inverse distance weighting to interpolate between measured points
    Predict <- function(predXY) {
      Distance <- function(a, b) {
        apply(a, 2, function(pt) sqrt(colSums((pt - b) ^ 2)))
      }
      dists <- Distance(xy, predXY)
      id <- 1 / dists
      idw <- id / rowSums(id)

      # Return:
      colSums(response * t(idw))
    }

    # Predict at triangle centres
    tri <- TriangleCentres(resolution = 12L)
    # Adjust the resolution to suit your own dataset

    # Now we interpolate between our known values to generate a colour for each
    # of our tiles
    predicted <- Predict(tri[1:2, ])
    map <- rbind(x = tri["x", ], y = tri["y", ], z = predicted,
                 down = tri["triDown", ])

    # Place a semitransparent colour fill over grid lines:
    ColourTernary(map)

    # Calculate contours
    PredictABC <- function(a, b, c) Predict(TernaryToXY(rbind(a, b, c)))
    TernaryContour(PredictABC, resolution = 36L, legend = 6,
                   legend... = list(bty = "n"))

    # Mark the points at which we took measurements
    TernaryPoints(abc, pch = 3, col = "#cc3333")

More sophisticated interpolation approaches are possible, e.g. kriging; these may help to alleviate artefacts of sampling intensity as observed towards the bottom of the example plot above.

# Interpolation within a sampled region

We might also encounter a situation where we have only sampled a small region of ternary space and do not wish to extrapolate beyond this region.

    # Select some points at which to sample
    set.seed(0)
    nPts <- 50
    a <- runif(nPts, 0.3, 0.7)
    b <- 0.15 + runif(nPts, 0, 0.7 - a)
    c <- 1 - a - b
    abc <- rbind(a, b, c)

    # Measure the response variable at each of those points:
    response <- FunctionToContour(a, b, c)

    # Convert measured points to XY
    xy <- TernaryToXY(abc, direction = 1)

    # Start a plot, to define the coordinate system
    par(mar = rep(0.2, 4))
    TernaryPlot(alab = "a", blab = "b", clab = "c",
                # Only plot the sampled region:
                xlim = range(xy["x", ]), ylim = range(xy["y", ])
    )

    # Get centres of tiles to plot
    tri <- TriangleCentres(resolution = 36L)
    # Adjust the resolution to suit your own dataset

    # Calculate which triangular tiles are within the boundaries of our data
    inHull <- TriangleInHull(tri, abc)

    # Pre-generate an empty list of predictions
    predicted <- rep(NA, ncol(tri))

    # Now we interpolate between our known values to generate a colour for each
    # of our tiles
    predicted[inHull$inside] <- Predict(tri[1:2, inHull$inside])
    map <- rbind(tri, z = predicted)

    # Colour only those tiles where we have made a prediction
    ColourTernary(map)

    # Calculate contours
    PredictABC <- function(a, b, c) Predict(TernaryToXY(rbind(a, b, c)))
    TernaryContour(PredictABC, resolution = 36L,
                   # `within` controls where contours are calculated
                   within = inHull$hull)

    # Mark the points at which we took measurements
    TernaryPoints(abc, pch = 3, col = "#cc3333")

    # Mark the region in which we sampled
    polygon(t(xy)[chull(t(xy)), ], border = "#cc3333")

# Where next

For other examples of usage, see the accompanying vignettes:

- Introduction for new R users

- Getting started guide

- Annotation of points

- Plotting Holdridge climate zones