Differential equation to describe beam width

[gobberpooper]

I have two independent variables (rotation in degrees, and distance from best focus distance in mm) and one dependent variable (1/e^2 width squared in mm^2). How would I go about creating either a multivariable function or a differential equation to describe the data? I've attached the raw data and resulting family of curves for reference.

The data was acquired by attaching a camera and a laser to a rail system with beam splitters and filters to protect the lens, and captured using FireWire BeamPro. Once the focus distance was found (~770mm), I captured data at 5 distances further away from and closer to the lens at 10mm intervals. I then rotated the camera by approximately 5 degrees and repeated. This was done from 0 to 20 degrees. The rotation didn't stay perfectly constant throughout each set, so the two points below each column of rotation is the average angle and the standard deviation. The other random row doesn't matter.

Basically, I have a family of curves corresponding to multiple data sets, and I need a way to describe them. I've forgotten how to do so from raw data and I can't find a good reference. Can somebody point me in the right direction?

 

[Valerie Park]

To create a multivariable function or a differential equation to describe the data, you can use a variety of mathematical techniques, such as linear regression, nonlinear regression, or even some more sophisticated methods. Linear regression is a simple and commonly used technique for fitting a line to a set of data points. It involves minimizing the sum of squared errors between the observed data and the fitted line, and can be used for both linear and nonlinear data sets. Nonlinear regression is a slightly more complex technique that can be used for fitting more complicated curves to data sets. It involves minimizing the sum of squared errors between the observed data and a parameterized curve. Finally, more sophisticated methods such as genetic algorithms, support vector machines, and artificial neural networks can also be used to fit complex curves to data sets.