Surface Equation
Two-variable mathematical functions model continuous multidimensional responses by combining polynomial terms of both input variables into a single surface equation. Calculating a surface fitting polynomial maps raw sensor output and temperature readings simultaneously to calculate corrected physical values. Fitting calibration data across a two-dimensional grid compensates for non-linear sensor behavior and temperature-dependent span shifts in one unified mathematical model.
High-precision pressure transducers use surface fitting algorithms inside microcontrollers for multi-variable correction.
Matrix Determination
Bivariate equations express output variables as sums of joint power terms multiplied by solved coefficient constants. Solved coefficients determine the shape of the fitted surface over the calibration domain. Linear algebra solvers compute coefficient vectors using multi-variable least squares matrix inversion.
Cross-Term Fitting
Cross-power terms capture temperature-dependent non-linearity shifts that independent one-dimensional polynomials miss. Second-order and third-order cross terms model complex sensor drift shapes across extreme operating temperatures. Including high-order cross terms improves surface fit accuracy across transition regions.
Extrapolation Limit
Model accuracy breaks down rapidly when evaluating inputs outside the calibrated temperature and pressure boundary grid. High-order polynomial surfaces exhibit severe Runge phenomenon oscillations near domain boundaries when over-parameterized. A surface fitting polynomial requires evenly distributed calibration matrix points to prevent surface warping between test coordinates.
