In our research of iridium oxide film microelectrodes for neural stimulation, the surface area of the microelectrode is an important parameter that needs to be calculated quite accurately. Each electrode is conical shaped, which is etched from a piece of iridium metal wire. And its dimensions are on the order of 10-micrometer. Using a digital microscope (Hirox) system with the image stacking software (Helicon Focus) (we call our system "poor-man's confocal microscope"), we can get clear images of the electrodes, from which we'd like to calculate their surface area.
This notebook documents my approaches to solving such a trivial problem at first sight, which gradually turns out to be involving more mathematics than formulae from high school textbook. This notebook consists of four sections.
Section I describes the ideal conical shaped electrode and the formula for calculating its surface area, which, I guess, can be found in grade school math textbook.
Section II deals with the real electrode shape which to some extent deviates from the ideal. To achieve more accuracy, I decided to pick data points directly from the electrode image and try to fit an explicit polynomial function a to it. Thus the surface area of the electrode can be calculated accordingly. It involves no more math than simple high school analytic geometry, whereas the data fitting is more or less learnt in college through numerical method or physics experiment class.
Section III becomes more ambitious as to reduce the "human factor" in this calculation. A second-degree implicit polynomial a was used to fit the 2D curve of the electrode. I came to know about implicit polynomial curve fitting after reading some of the research papers in the pattern recognition and machine vision area. Though the math involved doesn't go beyond some linear algebra or matrix theory, it's no longer directly from the textbook.
Section IV experiments with higher degree implicit polynomial curve fitting algorithms.
Put aside the argument whether it's worth all the effort to make a simple problem so complex, by throwing math tools heuristically onto it, this notebook at least showed that Mathematica can accommodate such mathematical experiments at all levels, especially in turns of making otherwise expensive algorithms easy to implement.