LinearGridSurface Class

public sealed class LinearGridSurface : GridSurface

Public NotInheritable Class LinearGridSurface
	Inherits GridSurface

public ref class LinearGridSurface sealed : public GridSurface

[<SealedAttribute>]
type LinearGridSurface = 
    class
        inherit GridSurface
    end

Inheritance

Object  →  Surface  →  GridSurface  →  LinearGridSurface

This surface performs multi-linear interpolation within each grid cell. For 2D grids this is bilinear interpolation, for 3D it is trilinear, and for N-D it generalizes to 2^N corner interpolation using weighted averaging of cell vertices.

Characteristics:

• Continuity: C⁰ (continuous everywhere)
• Smoothness: First derivatives are discontinuous at cell boundaries
• Performance: Fast evaluation, O(2^D) operations per query where D is dimension
• Gradient: Piecewise constant within each cell, discontinuous at boundaries

When to use:

• General-purpose interpolation: Best default choice for most continuous data. Provides good balance between smoothness, accuracy, and computational cost.
• Real-time applications: When smooth interpolation is needed but cubic methods are too expensive (e.g., game graphics, interactive visualization).
• Large datasets: For high-dimensional or large grids where cubic methods would require excessive memory or computation time.
• Non-smooth data: When the underlying function has discontinuities or sharp features that cubic methods might overshoot (ringing artifacts).

Alternatives:

• NearestGridSurface: Use for categorical data or when maximum speed is required and continuity is not necessary.
• CubicGridSurface2D or CubicGridSurface3D: Use for 2D/3D data when C² smoothness is required (smooth first and second derivatives). Provides better visual quality and derivative accuracy at higher computational cost.
• TensorSplineGridSurface: Use for N-dimensional smooth interpolation with C¹ continuity. Better than linear for smooth functions but more expensive than specialized 2D/3D cubic methods.

Applications:

• Image and texture resampling (bilinear filtering)
• 3D volume rendering and medical imaging
• Physical simulation lookup tables (temperature, pressure fields)
• Geographic data interpolation (elevation, climate data)
• Computer graphics and real-time rendering
• N-dimensional parameter space exploration in optimization

Technical notes:

Multi-linear interpolation linearly interpolates along each dimension in sequence. For a 2D cell with corners (f₀₀, f₀₁, f₁₀, f₁₁) at normalized coordinates (s, t) ∈ [0,1]², the interpolated value is:

f(s,t) = (1-s)(1-t)f₀₀ + s(1-t)f₁₀ + (1-s)t f₀₁ + st f₁₁

This generalizes naturally to higher dimensions using tensor products of 1D linear interpolants.

Creating instances: Use GridLinearInterpolator(IReadOnlyList<IReadOnlyList<Double>>, IReadOnlyList<Double>, ExtrapolationMode) factory methods to create instances of this class.

Reference