Linear Interpolation Calculator
Free Linear interpolation Calculator for coordinate geometry. Enter values to get step-by-step solutions with formulas and graphs.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Linear Interpolation Calculator
Calculator
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Formula: y = y₁ + (x - x₁) × (y₂ - y₁) / (x₂ - x₁)
Worked example — Temperature at 10:00 AM: 70°F | 50% along interval | Rate: 5°F/hour
Formula
y = y₁ + (x - x₁) × (y₂ - y₁) / (x₂ - x₁)
Given two known points (x₁, y₁) and (x₂, y₂), the interpolated value y at position x is computed by proportionally blending between y₁ and y₂ based on where x falls relative to x₁ and x₂.
Worked Examples
Example 1: Temperature Estimation
Problem:At 8:00 AM the temperature was 60°F and at 12:00 PM it was 80°F. Estimate the temperature at 10:00 AM.
Solution:x1 = 8, y1 = 60, x2 = 12, y2 = 80, x = 10 y = 60 + (10 - 8) * (80 - 60) / (12 - 8) y = 60 + 2 * 20 / 4 y = 60 + 10 = 70°F Fraction along: (10-8)/(12-8) = 0.50 (50% between points) Slope: 5°F per hour
Result:Temperature at 10:00 AM: 70°F | 50% along interval | Rate: 5°F/hour
Example 2: Engineering Table Lookup
Problem:Steel yield strength is 250 MPa at 20°C and 200 MPa at 400°C. Find strength at 150°C.
Solution:x1 = 20, y1 = 250, x2 = 400, y2 = 200, x = 150 y = 250 + (150 - 20) * (200 - 250) / (400 - 20) y = 250 + 130 * (-50) / 380 y = 250 - 17.105 = 232.895 MPa Fraction along: (150-20)/(400-20) = 0.3421 (34.21%)
Result:Yield strength at 150°C: 232.9 MPa | 34.2% along interval
Frequently Asked Questions
What is the difference between interpolation and extrapolation?
Interpolation estimates values within the range of known data points (between x1 and x2), while extrapolation estimates values outside that range. Interpolation is generally reliable because the data points bound the estimate and the assumption of linearity is more reasonable over small intervals. Extrapolation, however, becomes increasingly unreliable as you move further from the known data because the linear assumption may not hold. A temperature measured between two weather stations can be interpolated with reasonable accuracy, but predicting temperature far beyond the station network is extrapolation and may produce unrealistic results. Linear Interpolation Calculator indicates whether your query falls within or outside the known range.
Where is linear interpolation used in practice?
Linear interpolation has widespread applications across many fields. In engineering, it is used to read between values in steam tables, material property tables, and calibration curves. In finance, it interpolates interest rates between benchmark maturities to construct yield curves. In computer graphics, it blends colors, positions, and textures between vertices (often called lerp). In scientific research, it estimates measurements between recorded observations. Weather services interpolate temperature, pressure, and humidity between weather stations. Manufacturing uses it for CNC machine tool paths between programmed points. Even everyday digital audio and image resizing rely heavily on linear interpolation.
How accurate is linear interpolation compared to other methods?
Linear interpolation is exact when the underlying relationship is truly linear, but introduces errors when the relationship is curved. The error is proportional to the square of the interval width and the magnitude of the second derivative (curvature) of the actual function. For smooth functions with small intervals between data points, linear interpolation is quite accurate. For curved functions, higher-order methods like cubic spline interpolation or polynomial interpolation provide better accuracy by capturing curvature. However, higher-order methods require more data points and can exhibit oscillation (Runge phenomenon) near the edges. For most practical engineering tables with reasonably spaced entries, linear interpolation achieves accuracy within 1-2% of the true value.
What is the lerp function in programming?
Lerp (short for linear interpolation) is a fundamental function in programming, particularly in game development and computer graphics. The standard form is lerp(a, b, t) = a + t * (b - a), where a and b are the start and end values, and t is a parameter from 0 to 1. When t = 0, the result is a; when t = 1, the result is b; when t = 0.5, the result is the midpoint. Lerp is used for smooth camera movements, animation blending, color gradients, UI transitions, and physics simulations. Many game engines provide built-in lerp functions for scalars, vectors, quaternions, and colors. The concept extends to bilinear interpolation (2D) and trilinear interpolation (3D) for more complex spatial data.
Can linear interpolation be used in multiple dimensions?
Yes, linear interpolation extends naturally to multiple dimensions. Bilinear interpolation works in two dimensions by performing three linear interpolations: first in the x-direction at two y-levels, then in the y-direction between those intermediate results. Trilinear interpolation extends this to three dimensions using seven linear interpolations. These multidimensional versions are essential in image processing (resizing and rotating images), 3D graphics (texture mapping), meteorology (spatial weather data), and scientific computing (interpolating simulation results on grids). Each additional dimension doubles the number of required data points: 2 for 1D, 4 for 2D, and 8 for 3D. The computational simplicity and reasonable accuracy make multidimensional linear interpolation the default choice in many applications.
How do you handle multiple interpolation intervals?
When you have more than two data points and need to interpolate across the full range, you use piecewise linear interpolation. This creates a connected series of straight line segments, each defined between consecutive data points. For a query point x, you first identify which interval it falls in (between which pair of consecutive x values), then apply the linear interpolation formula for that specific interval. This approach ensures continuity (no jumps) at the data points but may have slope discontinuities (kinks) where segments meet. Piecewise linear interpolation is used in lookup tables, data visualization, and as a baseline for comparing more sophisticated methods like cubic splines.
What is inverse linear interpolation?
Inverse linear interpolation solves the reverse problem: given a target y value, find the corresponding x value. Using the same two known points, the formula is x = x1 + (y_target - y1) * (x2 - x1) / (y2 - y1). This is equivalent to swapping the roles of x and y in the standard formula. Inverse interpolation is useful when you need to find the input that produces a desired output, such as finding the temperature at which a material reaches a specific strength, or the concentration at which a reaction achieves a target yield. The method requires that y1 and y2 are different; otherwise, the inverse mapping is undefined because a horizontal line corresponds to any x value.
What are the limitations of linear interpolation?
Linear interpolation has several important limitations. It assumes a constant rate of change between data points, which is inaccurate for nonlinear relationships like exponential growth, oscillations, or step functions. The slope changes abruptly at data points, creating corners (C0 continuity only), which can cause problems in applications requiring smooth derivatives like robotics path planning. It cannot capture local maxima or minima between data points. For unevenly spaced data, the quality of interpolation varies across the domain. When data contains noise or measurement errors, linear interpolation reproduces these errors exactly rather than smoothing them out. Despite these limitations, its simplicity and computational efficiency make it appropriate for the vast majority of practical interpolation needs.
How does linear interpolation relate to weighted averages?
Linear interpolation is mathematically equivalent to a weighted average of the two endpoint values. The interpolated value y = y1 * (1 - t) + y2 * t, where t = (x - x1) / (x2 - x1), is a weighted combination where the weights (1-t) and t sum to 1. When x is closer to x1, the weight on y1 is larger; when x is closer to x2, the weight on y2 dominates. This weighted average perspective is particularly intuitive and useful in computer graphics and signal processing. It also connects linear interpolation to the broader concept of convex combinations, which are fundamental in geometry, optimization, and probability theory. Any point on the line segment between two points can be expressed as their convex combination.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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