Linear Interpolation

📐 Definition

For $a,b \in \mathbb{R}$ (or vectors) and $\alpha \in \mathbb{R}$,

$$\operatorname{lerp}(a,b;\alpha) = (1-\alpha)a + \alpha b$$

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Domain and Codomain

Applies to scalars or vectors in a common linear space. Output lies on the affine segment between $a$ and $b$ for $\alpha \in [0,1]$.

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⚙️ Key Properties

Linear in each argument; satisfies $\operatorname{lerp}(a,b;0)=a$, $\operatorname{lerp}(a,b;1)=b$. For uniform grids, piecewise lerp yields first-order accurate interpolation.

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Extrapolation occurs for $\alpha \notin [0,1]$. Symmetry: $\operatorname{lerp}(a,b;\alpha) = \operatorname{lerp}(b,a;1-\alpha)$.

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🎯 Special Cases and Limits

• $\alpha\in[0,1]$ yields convex combinations; outside this range extrapolates.
• Symmetry: swapping endpoints corresponds to $\alpha\mapsto 1-\alpha$.

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🔗 Related Functions

Clamping bounds $\alpha$; normalization can be applied before or after blending.

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Usage in Oakfield

Oakfield uses linear blending in a few internal “glue” paths:

• Warp limited continuity blends nearby valid samples (effectively a simple linear average) when a warp gradient is undefined at the probe point.
• UI/runtime parameter flows often apply explicit a + t(b-a) style blending when smoothing or staging parameter updates (even when not calling a dedicated lerp helper).
• Math helpers in core/math_utils.h provide sim_lerp_double / sim_lerp_complex for kernels that want a standard lerp.

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Historical Foundations

📜 Affine Combinations

Linear interpolation is the simplest affine blending operation and a building block for piecewise-linear approximation on grids.

🌍 Modern Perspective

It is a ubiquitous primitive in numerical methods, geometry, and graphics for blending and sampling.

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📚 References

• Quarteroni, Sacco, Saleri, Numerical Mathematics