0018// Calculate Flatness by Least Square Method

This week, we will discuss how to calculate flatness if points' coordinates on surface are given to you.

First assume that we have a ideal plane which can form a best fit plane between the the points you measured and its equation as shown below:

This ideal plane also known as least square surface.

Then, distance of one the point you measured and this ideal plane is

So, if you measured N points, you will have N distances

Tolerance zone of surface flatness defined by two parallel planes with distance which equal to tolerance size.

Therefore, this least square surface must give square sum of all distance is minimum.

(Remember: Flatness don't have datum, least square surface don't have any orientation contraints)

At the minimum points, the partial derivatives must sastisfy equation below:

After partial differentiation and follow by equate to zero, we can arrange a, b and c as below:

So, the "best" least square surface is

The flatness can get by distance between peak and valley:

However, that is hard to judge which points are peak and valley. My recommendation is use Microsoft Excel's built-in Solver to get the "smallest possible" flatness (this method also known as brute-force search or exhaustive search).

All in all, if you able to get the coordinates of surface and have an Excel in your laptop, you can calculate flatness without any complex algorithm.

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