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What is the linear interpolation formula for three variables?

Published in Mathematics 3 mins read

Linear interpolation in three variables is a method to estimate the value of a function at a point within a three-dimensional space, given its values at known points. The formula for linear interpolation in three variables is derived from the equation of a plane passing through three non-collinear points.

Here's how it works:

Linear Interpolation Formula

Let's say you have three points in 3D space, denoted as:

  • P1: (x1, y1, z1)
  • P2: (x2, y2, z2)
  • P3: (x3, y3, z3)

And you want to find the interpolated value at a point P: (x, y, z).

The linear interpolation formula is:

f(x, y, z) = f(x1, y1, z1) * ((y2 - y) * (z3 - z) - (z2 - z) * (y3 - y)) / ((y2 - y1) * (z3 - z1) - (z2 - z1) * (y3 - y1)) + 
f(x2, y2, z2) * ((y3 - y) * (z1 - z) - (z3 - z) * (y1 - y)) / ((y3 - y2) * (z1 - z2) - (z3 - z2) * (y1 - y2)) +
f(x3, y3, z3) * ((y1 - y) * (z2 - z) - (z1 - z) * (y2 - y)) / ((y1 - y3) * (z2 - z3) - (z1 - z3) * (y2 - y3))

Where:

  • f(x, y, z) is the interpolated value at point P.
  • f(x1, y1, z1), f(x2, y2, z2), f(x3, y3, z3) are the known values of the function at points P1, P2, and P3, respectively.

Example

Let's say you have the following data points:

  • P1: (1, 2, 3) with f(1, 2, 3) = 5
  • P2: (4, 5, 6) with f(4, 5, 6) = 10
  • P3: (7, 8, 9) with f(7, 8, 9) = 15

You want to estimate the value of the function at point P: (3, 4, 5).

Plugging these values into the formula, we get:

f(3, 4, 5) = 5 * ((5 - 4) * (9 - 5) - (6 - 5) * (8 - 4)) / ((5 - 2) * (9 - 3) - (6 - 3) * (8 - 2)) + 
10 * ((8 - 4) * (3 - 5) - (9 - 5) * (2 - 4)) / ((8 - 5) * (3 - 6) - (9 - 6) * (2 - 5)) +
15 * ((2 - 4) * (6 - 5) - (3 - 5) * (5 - 4)) / ((2 - 8) * (6 - 9) - (3 - 9) * (5 - 8))

Solving this equation, we get f(3, 4, 5) = 8.

Practical Insights

Linear interpolation in three variables is a simple and efficient method for estimating values within a 3D space. It is commonly used in:

  • Data analysis: For interpolating data points in three-dimensional datasets.
  • Computer graphics: For generating smooth surfaces and textures.
  • Engineering: For approximating physical quantities like temperature, pressure, or stress.

Limitations

  • Accuracy: The accuracy of the interpolation depends on the distribution of the known data points. If the points are widely spaced, the interpolation may not be accurate.
  • Assumptions: Linear interpolation assumes that the function being interpolated is linear between the known points. This may not be true for all functions.

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