Calculate Vector Interpolation

Calculator and formulas for linear interpolation between two vectors (LERP)

Vector Interpolation Calculator

Linear Vector Interpolation (LERP)

Calculates an intermediate vector between two vectors through linear interpolation: lerp(v₁, v₂, t) = v₁ + t(v₂ - v₁)

Select Vector Dimension
Start Vector (v₁)
Target Vector (v₂)
Weighting between 0 and 1 (0 = v₁, 1 = v₂)
t = 0.2 (closer to v₁)
Interpolation Result
X:
Y:
Z:
W:
Interpolated Vector:
Calculation: lerp(v₁, v₂, t) = v₁ + t(v₂ - v₁)

Interpolation Info

LERP Properties

Parameter t: Weighting between vectors

t = 0 → v₁ t = 1 → v₂ Linear

Position: t ∈ [0,1] between vectors
Extrapolation: t < 0 or t > 1

t-Value Meaning
t = 0.0: Start vector v₁
t = 0.5: Midpoint between v₁ and v₂
t = 1.0: Target vector v₂

Formulas for Vector Interpolation

LERP Basic Formula
\[\text{lerp}(\vec{v_1}, \vec{v_2}, t) = \vec{v_1} + t(\vec{v_2} - \vec{v_1})\]

Linear interpolation between two vectors

Alternative Form
\[\text{lerp}(\vec{v_1}, \vec{v_2}, t) = (1-t)\vec{v_1} + t\vec{v_2}\]

Weighted sum of vectors

Component-wise
\[\text{lerp}_i = v_{1i} + t(v_{2i} - v_{1i})\]

Each component interpolated separately

3D Example
\[\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x_1 + t(x_2-x_1) \\ y_1 + t(y_2-y_1) \\ z_1 + t(z_2-z_1) \end{bmatrix}\]

3D vector interpolation

Calculation Examples for Vector Interpolation

Example 1: Midpoint
v₁ = [2, 4], v₂ = [6, 8], t = 0.5
\[\begin{aligned} \text{lerp} &= [2, 4] + 0.5([6, 8] - [2, 4]) \\ &= [2, 4] + 0.5[4, 4] \\ &= [2, 4] + [2, 2] = [4, 6] \end{aligned}\]

Result: [4, 6] (Midpoint)

Example 2: 20% Interpolation
v₁ = [2, 4, 1], v₂ = [3, 5, 2], t = 0.2
\[\begin{aligned} \text{lerp} &= [2, 4, 1] + 0.2([3, 5, 2] - [2, 4, 1]) \\ &= [2, 4, 1] + 0.2[1, 1, 1] \\ &= [2.2, 4.2, 1.2] \end{aligned}\]

Result: [2.2, 4.2, 1.2] (closer to v₁)

Geometric Interpretation
t = 0: Start Point
At vector v₁
t = 0.5: Middle
Between v₁ and v₂
t = 1: End Point
At vector v₂

Interpolation produces a point on the line between vectors

t-Parameter Understanding
t ∈ [0, 1]

• Interpolation between vectors

• Result lies on connection segment

• Safe and predictable

t < 0

• Extrapolation beyond v₁

• Extends line backward

• Result outside segment

t > 1

• Extrapolation beyond v₂

• Extends line forward

• Result outside segment

Applications of Vector Interpolation

Vector interpolation (LERP) is a fundamental technique in many fields:

Computer Graphics & Animation
  • Object movements and path animations
  • Camera smoothing and transitions
  • Color gradients and shader interpolation
  • Keyframe animation between poses
Robotics & Control
  • Path planning and trajectories
  • Smooth motion control
  • Robot arm positioning
  • Velocity profiles
Data Processing
  • Time series interpolation
  • Estimate missing data points
  • Signal smoothing and filtering
  • Numerical approximation
Engineering
  • CAD systems and curve design
  • Finite-element methods
  • Flow simulations
  • Structural transitions

Vector Interpolation: Linear Transitions in Vector Space

Linear vector interpolation (LERP) is a fundamental method for computing intermediate points on a straight line between two vectors. This elegant technique enables smooth transitions and continuous movements in applications ranging from computer graphics through robotics to numerical simulation. The parameter t precisely controls the position of the resulting vector along the connecting line.

Summary

Vector interpolation combines mathematical elegance with practical versatility. The simple LERP formula - weighted combination of two vectors - enables precise control over transitions and movements in any dimension. From 2D animation through 3D robotics to high-dimensional data analysis, LERP remains an indispensable tool. The method demonstrates how fundamental mathematical concepts form the foundation for advanced applications in engineering and science.




Vector Functions

Addition • Subtraction • Multiplication • Scalar Multiplication • Division • Scalar Division • Dot Product • Cross Product • Interpolation • Distance • Distance Squaret • Normalization • Reflection • Magnitude • Squared-Magnitude • Triple-Product