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In the bivariate formulation, interpolation requires the fitted function to match the measured values exactly at every given data point, expressed as $z_i = F(x_i, y_i)$. Approximation relaxes this exact fitting by allowing a deviation or error term $e(x_i, y_i)$, expressed as $z_i = F(x_i, y_i) + e(x_i, y_i)$.

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 $e(x_i, y_i)$.

In the bivariate formulation, interpolation requires the fitted function to match the measured values exactly at every given data point, expressed as $z_i = F(x_i, y_i)$. Approximation relaxes this exact fitting by allowing a deviation or error term $e(x_i, y_i)$, expressed as $z_i = F(x_i, y_i) + e(x_i, y_i)$.

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 $e(x_i, y_i)$.