Linear Interpolation

Linear interpolation is the simplest form of Interpolation, connecting two data points with a straight line.

Formula

Given two points $(x_0,y_0)$ and $(x_1,y_1)$, the interpolated value at $x$ is:

$$y=y_0+(x-x_0)\frac{y_1-y_0}{x_1-x_0}$$

Alternatively, using the geometric interpretation (equal slopes):

$$\frac{y-y_0}{x-x_0}=\frac{y_1-y_0}{x_1-x_0}$$

Which can be rearranged to:

$$y=\frac{y_0(x_1-x)+y_1(x-x_0)}{x_1-x_0}$$

Terminology

Linear interpolation is sometimes called “lerp” (a portmanteau of “linear” and “interpolation”), particularly in computer graphics and game development.

Connection to Taylor Series

Linear interpolation corresponds to the first-order Taylor series approximation. It captures the local linear behaviour of a function but ignores higher-order terms (curvature, etc.).

Accuracy

The accuracy of linear interpolation depends on:

• Point spacing: Closer points → better approximation
• Function linearity: More linear functions → better fit
• Curvature: High curvature between points leads to larger errors

For a non-linear function, linear interpolation is an approximation that improves as the distance $|x_1-x_0|$ decreases.

Error Analysis

For a twice-differentiable function, the maximum interpolation error can be bounded by:

$$|f(x)-p(x)|\le \frac{1}{8}{h}^2\underset{x_0\le \xi \le x_1}{\max }|f^{\prime \prime }(\xi )|$$

Where $h=x_1-x_0$ and $p(x)$ is the linear interpolant.

Advantages

• Simple: Easy to understand and implement
• Fast: Minimal computation required
• Stable: No numerical instability issues

Limitations

• Low accuracy for non-linear functions
• Non-smooth: Piecewise linear interpolation has discontinuous derivatives
• Only uses two points: Ignores information from nearby data

Python Implementation

def linear_interpolation(x, x0, y0, x1, y1):
    """
    Perform linear interpolation.
    
    Parameters
    ----------
    x : float
        Point at which to interpolate
    x0, y0 : float
        First data point
    x1, y1 : float
        Second data point
        
    Returns
    -------
    float
        Interpolated value at x
    """
    return y0 + (x - x0) * (y1 - y0) / (x1 - x0)

Applications

• Quick estimates between known values
• Computer graphics (texture mapping, animation)
• Signal processing
• Data visualization (connecting discrete points)

Related Methods

• Bilinear Interpolation: Linear interpolation in 2D
• Spline Interpolation: Piecewise smooth alternative
• Polynomial Interpolation: Higher-order alternative
• Newton Divided Difference: Generalization to higher orders