What is the difference between interpolation and approximation in the bivariate formulation?

Conditions

• Given m points $(x_i, y_i, z_i)$ with $i=1,m$.
• Find a function $F(x,y)$.
• Interpolation requires exact matching at the sample points.
• Approximation allows an error term $e(x_i, y_i)$.

Reasoning, step by step

1. Define the bivariate problem with m given points $(x_i, y_i, z_i)$.
2. Seek a function $F(x,y)$ to represent the data.
3. For interpolation, enforce the exact condition $z_i = F(x_i, y_i)$ at each sample point.
4. For approximation, allow the condition $z_i = F(x_i, y_i) + e(x_i, y_i)$, where $e(x_i, y_i)$ is a deviation term.
5. Use approximation when dealing with noisy data to achieve smoothing.

Example

The slide explicitly labels $z_i = F(x_i, y_i)$ as interpolation and $z_i = F(x_i, y_i) + e(x_i, y_i)$ as approximation, with the lecturer noting that approximation is needed if the data are noisy and we are trying to smooth it out.

Common misconceptions

• Using “interpolation” loosely for any fitted surface through data, ignoring the strict exact-match requirement.
• Believing that approximation forces the function to pass exactly through every data point.

Watch the explanation

Explore next

• Why does the lecturer introduce approximation instead of always using exact interpolation?
• Why is interpolation said to have no unique solution?
• What must be added to obtain a unique interpolation function?