Curve Interpolation in Multiple Dimensions

This documentation was generated with the assistance of AI. Please report any inaccuracies.

A different kind of multidimensional interpolation problem arises when an ordered sequence of tabulated points in dimensions is known to lie along a one-dimensional curve, and the goal is to interpolate other points along that curve — as opposed to interpolating a scalar function of scattered coordinates. It is described in Press et al., 2007, Section 3.7.5 (page 147), and implemented as CurveInterpolator in the com.irurueta.numerical.interpolation package.

Parameterizing the curve by arc length

The approach is straightforward given the tools already available: approximate the distance traveled along the curve by the sum of chord lengths between consecutive tabulated points, then build one cubic spline per coordinate ( ) as a function of that arc-length parameter , rescaled to lie in . Since no single coordinate’s derivative with respect to arc length can exceed 1, the resulting splines are guaranteed to behave well, even where the curve doubles back on itself in ordinary coordinates.

Two boundary conditions on the curve are supported:

  • Open curve — and are genuine endpoints. Interpolating with a parameter outside extrapolates (and, as with any extrapolation, this is best avoided).

  • Closed curve — an implied segment connects back to . The parameter is periodic with period 1. The trick used to build this correctly is to duplicate half a period’s worth of points at the beginning and end of the working array before constructing the splines, then use only the middle half of the result — which sidesteps the discontinuity a naive wraparound would otherwise introduce at the seam.

Endpoint derivatives for each per-coordinate spline are estimated (when at least 4 points are available) from four consecutive tabulated abscissa/ordinate pairs via a finite-difference formula derived by power-series expansion, rather than left to default to a natural-spline boundary condition.

flowchart TD A["Ordered points x_0 .. x_N-1 in n dimensions"] --> B["Accumulate chord-length distances\nbetween consecutive points"] B --> C["Rescale accumulated arc length to parameter s in [0, 1]\n(for closed curves: duplicate half a period at each end)"] C --> D["Estimate endpoint derivatives per coordinate\nfrom 4 nearest tabulated points"] D --> E["Build one CubicSplineInterpolator per coordinate,\nparameterized by s"] E --> F["interpolate(t): evaluate every per-coordinate spline at t\n(wrapped into [0,1) first, if closed)"]

Library implementation

CurveInterpolator wraps CubicSplineInterpolator instances, one per coordinate, all sharing the same arc-length parameterization:

  • CurveInterpolator(Matrix ptsin, boolean close) — rows are the tabulated points, columns their coordinates; close selects an open (false) or closed (true) curve.

  • CurveInterpolator(Matrix ptsin) — shorthand for an open curve.

  • interpolate(double t) — returns a coordinate array for the point at parameter t. For an open curve, values of t outside [0, 1] extrapolate; for a closed curve, t is treated as periodic with period 1.

For a closed curve, do not duplicate the first point as the last one in the input — the constructor already connects back to itself.

Example

final var interpolator = new CurveInterpolator(points, false);
final var pointAtHalfway = interpolator.interpolate(0.5); // coordinate array