Working definition
Interpolation constructs an approximating function that agrees with known samples and estimates values between them. Polynomial, spline, rational, and trigonometric methods make different smoothness and stability assumptions; interpolation should not be silently extended into extrapolation.
Notation
p(xᵢ) = yᵢf(x) = p(x) + R(x)Assumptions
- The target lies inside the supported sample range.
- Sampling resolves relevant variation.
- The selected interpolant matches local smoothness.
Invariants
- The interpolant reproduces declared nodes within tolerance.
- Units are preserved.
- Method and node set determine the result.
Reproducible procedure
- Select bracketing samples.
- Choose a stable interpolation family.
- Estimate or test residual error against denser references.
Error and boundary controls
- High-degree equispaced polynomials can oscillate.
- Sparse nodes miss sharp changes.
- Interpolation error is separate from source-data error.
What this does not establish
Interpolation fills a numerical gap under a model; it does not create observations or justify extrapolated predictions.
Explicit applications
1 cross-domain bridges
Ephemeris state interpolation
Estimate a body state between tabulated or integrated ephemeris samples.
Inputs
- bracketing state vectors
- target epoch
- interpolation method
Outputs
- state estimate
- method identifier
- residual bound
Transformation: Interpolate within the supported interval and compare against denser reference output.
Limit: Interpolation cannot repair an incorrect ephemeris, timescale, or observer origin.
Open connected system →Authoritative references
- [1]DLMF Chapter 3: Numerical Methods · National Institute of Standards and Technology
Establishes: Reference definitions, algorithms, convergence conditions, and error terms for interpolation, quadrature, differentiation, and nonlinear equation solving.
Boundary: A numerical method is reliable only under its stated regularity, conditioning, precision, and convergence assumptions; the reference does not validate any domain interpretation.
- [2]JPL Horizons System · NASA Jet Propulsion Laboratory
Establishes: Ephemeris products and documented observer, target, timescale, coordinate, and output conventions for reproducible Solar System state and observable calculations.
Boundary: Ephemeris agreement validates positions under declared conventions; it does not validate downstream symbolic classifications or interpretations.
Direct answer
- Interpolation constructs an approximating function that agrees with known samples and estimates values between them. Polynomial, spline, rational, and trigonometric methods make different smoothness and stability assumptions; interpolation should not be silently extended into extrapolation.
Mechanism and method
- Select bracketing samples.
- Choose a stable interpolation family.
- Estimate or test residual error against denser references.
What is measured
- The interpolant reproduces declared nodes within tolerance.
- Units are preserved.
- Method and node set determine the result.
Limitations
- High-degree equispaced polynomials can oscillate.
- Sparse nodes miss sharp changes.
- Interpolation error is separate from source-data error.
- The target lies inside the supported sample range.
- Sampling resolves relevant variation.
- The selected interpolant matches local smoothness.
What this does not establish
- Interpolation fills a numerical gap under a model; it does not create observations or justify extrapolated predictions.
Bridge: Ephemeris state interpolation
- Estimate a body state between tabulated or integrated ephemeris samples.
- Input: bracketing state vectors
- Input: target epoch
- Input: interpolation method
- Output: state estimate
- Output: method identifier
- Output: residual bound
- Limit: Interpolation cannot repair an incorrect ephemeris, timescale, or observer origin.