Matrix Interpolation (LERP) 4×4

Linear interpolation between two 4x4 matrices

Matrix Interpolation Calculator

Instructions

Enter the values of both matrices (A and B) and the weighting value t (0.0 to 1.0). The result will be a linear interpolation: A × (1-t) + B × t. Empty fields are counted as zero.

Matrix A (Start)
| |
| |
| |
| |
Matrix B (End)
| |
| |
| |
| |
Interpolation Parameter
t=0: Result = A | t=0.5: Midpoint | t=1: Result = B
Result: lerp(A, B, t)
| |
| |
| |
| |

Matrix Interpolation - Overview

What is Matrix Interpolation?

Linear interpolation (LERP) creates a smooth transition between two matrices. The weighting parameter t determines how much of each matrix contributes to the result.

Interpolation Formula

The linear interpolation (LERP) formula for matrices:

\(\text{lerp}(A, B, t) = A \cdot (1-t) + B \cdot t\)

or equivalently: \(A + t \cdot (B - A)\)

Weighting Values
  • t = 0.0: Result is identical to matrix A
  • t = 0.5: Result is the midpoint (average) of A and B
  • t = 1.0: Result is identical to matrix B
  • 0 < t < 1: Result lies between A and B
  • t < 0 or t > 1: Extrapolation beyond the range
Properties
  • Element-wise: Each matrix element interpolated separately
  • Linear: Straight-line path from A to B
  • Symmetric: lerp(A, B, t) = lerp(B, A, 1-t)
  • Commutative weights: t and (1-t) sum to 1


Description of Matrix Interpolation

Fundamentals

A linear interpolation (LERP) is performed from the first to a second matrix based on a weighting value t that indicates how much of the second matrix to include in the result.

Mathematical Formula:

\(M_{\text{result}} = (1-t) \cdot A + t \cdot B\)

For each element (i,j):

\(M_{ij} = (1-t) \cdot A_{ij} + t \cdot B_{ij}\)

How It Works
  1. Start matrix A: The initial state (t=0)
  2. End matrix B: The final state (t=1)
  3. Weighting t: Controls the blend (0 to 1)
  4. Calculate: For each element, compute weighted sum
  5. Result: A smooth transition between A and B

Example Calculation

Simple 2×2 Example

Given matrices A and B, with t = 0.5:

\(A = \begin{bmatrix}10 & 20\\30 & 40\end{bmatrix}, \quad B = \begin{bmatrix}0 & 10\\20 & 30\end{bmatrix}\)

\(\text{lerp}(A,B,0.5) = 0.5 \cdot A + 0.5 \cdot B = \begin{bmatrix}5 & 15\\25 & 35\end{bmatrix}\)

Step-by-Step:

For element (1,1) with t=0.5:

\(M_{11} = (1-0.5) \cdot 10 + 0.5 \cdot 0 = 5\)

For element (1,2) with t=0.5:

\(M_{12} = (1-0.5) \cdot 20 + 0.5 \cdot 10 = 15\)

Weighting Behavior
  • t = 0: Result = A (100% first matrix)
  • t = 0.25: Result = 75% A + 25% B
  • t = 0.5: Result = 50% A + 50% B (midpoint)
  • t = 0.75: Result = 25% A + 75% B
  • t = 1: Result = B (100% second matrix)
Visual Representation

Interpolation Timeline:

A (t=0) → t=0.25 → t=0.5 (midpoint) → t=0.75 → B (t=1)
Practical Applications

Animation & Graphics:

  • Smooth animations between keyframes
  • Camera transitions (view matrices)
  • Morphing between transformations
  • Fade effects in 3D scenes

Engineering & Simulation:

  • Motion planning for robots
  • State transitions in control systems
  • Blending transformation matrices
  • Time-based simulations
Important Notes
  • Element-wise operation: Matrix interpolation is performed element by element, not as a matrix operation.
  • Range [0,1]: While values outside 0-1 are allowed (extrapolation), the most common use is within this range for smooth transitions.
  • Not for rotations: For rotation matrices, use SLERP (Spherical Linear Interpolation) or quaternion interpolation instead of LERP to avoid distortion.
  • Linear path: LERP follows a straight line in parameter space, which may not always be the desired behavior for all types of matrices.
Advanced Concepts

Extrapolation (t < 0 or t > 1):

  • t < 0: Extends backward beyond A
  • t > 1: Extends forward beyond B
  • Useful for prediction or overshooting effects

Alternative Formulations:

  • Mix formula: A + t(B - A)
  • Weighted average: (1-t)A + tB
  • Both are mathematically equivalent
  • Choice depends on computational efficiency
Performance Considerations

Matrix interpolation for 4×4 matrices requires 16 scalar interpolations (one per element). This is computationally efficient and can be easily parallelized on modern GPUs. For real-time applications like game engines or 3D graphics, LERP is one of the fastest methods for smooth transitions. However, for rotation matrices, consider using quaternion SLERP which better preserves rotational properties despite being slightly more expensive computationally.