Mathematics & Conversions

Linear Interpolation Calculator with Slope, Interpolation Ratio & Validation

Calculate interpolated value using x<sub>1</sub>, x<sub>2</sub> and query point based on linear interpolation.

Unit-aware inputs Deterministic calculation Engineering interpretation
Calculation workspace

Enter the known values and review the calculated result

Deterministic calculation
01
Parameters

Input parameters

Use consistent values and select the intended engineering units.

Interpolation points

Interpolation query

Neighbor points [Optional]

Options

02
Output

Results

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Engineering reference

Method, application and limitations

Review the calculation method, intended application and engineering assumptions before using the result in a design decision.

01
Method

Formula and calculation method

Interpolated value formula:

If x2 < x1, the calculator swaps the two interpolation points:

x1 ↔ x2, y1 ↔ y2

Δx = x2 − x1

Δy = y2 − y1

k = Δy / Δx

λ = (x − x1) / Δx

y = y1 + λ · Δy

Advanced normalization mode:

xmid = 0.5 · (x1 + x2)

ymid = 0.5 · (y1 + y2)

xscale = max(|x1 − x2|, 1)

yscale = max(|y1 − y2|, 1)

x1,n = (x1 − xmid) / xscale

x2,n = (x2 − xmid) / xscale

y1,n = (y1 − ymid) / yscale

y2,n = (y2 − ymid) / yscale

xn = (x − xmid) / xscale

Δxn = x2,n − x1,n

Δyn = y2,n − y1,n

kn = Δyn / Δxn

λ = (xn − x1,n) / Δxn

yn = y1,n + λ · Δyn

y = yn · yscale + ymid

k = kn · (yscale / xscale)

where:

  • y — interpolated value (-)
  • k — slope dy/dx (-)
  • λ — interpolation ratio (-)
  • x1 — first interpolation point x-coordinate (-)
  • y1 — first interpolation point value (-)
  • x2 — second interpolation point x-coordinate (-)
  • y2 — second interpolation point value (-)
  • x — query point (-)
02
Application

When to use this calculator

When to use this calculator:

  • Estimate a value between two known data points using a linear segment.
  • Check whether the query point is inside the interpolation segment or outside it.
  • Calculate the segment slope between two measured or tabulated points.
  • Evaluate the normalized position of the query point between the two interpolation points.
  • Validate neighboring points for monotonicity and local slope consistency.
  • Detect large local changes in slope when previous and next points are provided.
03
Decision support

How to interpret the result

Interpolated value is defined as the value on the straight line between two endpoint values at the selected query point.

The result depends on endpoint values, endpoint spacing and query point position. Increasing the query point toward the second endpoint increases the interpolation ratio. Increasing the second endpoint value increases the interpolated value when the query point remains between the endpoints.

  • Safe — the query point is inside the endpoint range, no monotonicity violation is detected, local linearity error is not greater than 0.2, normalized gradient jump is not greater than 2, and no invalid condition is triggered.
  • Warning — the query point is outside the endpoint range, or neighboring slopes change sign, or local linearity error is greater than 0.2 and not greater than 0.5, or normalized gradient jump is greater than 2.
  • Unsafe — local linearity error is greater than 0.5, or the query point is outside the endpoint range with |λ| greater than 2.
  • Invalid — a required input is not finite, the endpoint x-coordinates are equal within tolerance, the interpolation ratio magnitude is greater than 5, the full series lengths differ, the full series x-values are not strictly increasing, or enforced monotonicity fails.

The result is used to evaluate the interpolated value, segment slope and query position validity for a local linear data segment.

04
Worked case

Calculation example

Example:

A user has two tabulated values and wants to estimate the value at a query point located between them.

  • x1 = 10
  • y1 = 100
  • x2 = 20
  • y2 = 160
  • x = 15

y = 130, k = 6, λ = 0.5

05
Model boundaries

Assumptions and limitations

  • The value between the two endpoints is represented by a straight line segment.
  • The two endpoint x-coordinates must not be equal within the adaptive numerical tolerance.
  • If a full data series is provided, x-values must be strictly increasing.
  • Neighbor analysis is applied only when previous or next neighbor coordinates and values are provided.
  • Advanced normalization changes the internal calculation scale and then converts the interpolated value and slope back to the original scale.
06
Questions

Frequently asked questions

How to calculate interpolated value?
Interpolated value is calculated using the linear interpolation formula y = y1 + λ · (y2 − y1). It depends on the two endpoint values and the query point position between the endpoint x-coordinates. Moving the query point closer to the second endpoint moves the interpolated value toward the second endpoint value.
What affects interpolated value the most?
The result depends on the endpoint values, endpoint spacing and query point position. A larger difference between endpoint values increases the slope magnitude. A query point closer to the second endpoint increases the interpolation ratio.
When is the interpolated value formula not valid?
The formula is not valid when any required input is not a finite number, when the two endpoint x-coordinates are equal within the adaptive tolerance, when the interpolation ratio magnitude is greater than 5, or when monotonicity enforcement is enabled and neighboring slopes change sign.
Can this calculator be used for extrapolation?
It can calculate a result outside the endpoint range when the interpolation ratio is below 0 or above 1. The result is marked as warning outside the segment and unsafe when the point is outside the segment with |λ| greater than 2.
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Engineering notice. Verify input data, assumptions, applicable standards and required safety factors before using results for design release or safety-critical decisions.