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Mathematics plays an important role in the Science of Metrology and Computer related Design and FEA, FEM. Mathematical models are needed to understand how to design effective measurement systems, and to analyze the results they produce. Mathematical techniques are used to develop and analyze idealized models of physical phenomena to be measured, and mathematical algorithms are necessary to find optimal system parameters. Finally, mathematical and statistical techniques are needed to transform the resulting data into useful information. In physical metrology it is often necessary to fit a mathematical model to experimental results in order to recover the quantities being measured. In some cases the desired variables can be measured more or less directly, but the measuring instruments distort the measured function so much that mathematical modeling is required to recover it. In other cases the desired quantities cannot be measured directly and must be inferred by applying a model to the measured variables.
Curve fitting is an important aspect of Scanning based Metrology or Reverse engineering. Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing in which a "smooth" function is constructed that approximately fits the data. A related topic is regression analysis, which focuses more on questions of statistical inference such as how much uncertainty is present in a curve that is fit to data observed with random errors. Fitted curves can be used as an aid for data visualization, to infer values of a function where no data are available, and to summarize the relationships among two or more variables. Extrapolation refers to the use of a fitted curve beyond the range of the observed data, and is subject to a degree of uncertainty since it may reflect the method used to construct the curve as much as it reflects the observed data.
Least Square Best Fit, Newton Rapson Method, Rangakutta Method etc., are many modules we study in engineering. At that point of time we never know where it is going to be utilized. Above methods are heavily used in 3D Inspection while creating mathematical features.
Statistics plays extremely important role in Mechanical Engineer's life. Accuracy, repeatability, R&R(repeatability and reproducibility) are all off shoots of statistical analysis.