Reversed second moments of area values relative to anatomical planes



Hi, I am running cross-sectional analysis on femora with “Analyse - orientation” set on to get information on A-P and M-L second moments of area. However when the results are returned, I realized that the values for A-P and M-L orientation are reversed. My M-L dimensions are clearly larger but the value is smaller. I checked that against Moment macro and it returns results as expected.

Can this be fixed?

thanks much.


This is the expected behaviour. Second moment of area is measured around the axis, not in the direction of the axis.

The second moment of area I around the long axis is smaller than I around the short axis, because the pixels are closer to the long axis than to the short axis, and I is calculated as the sum of the pixel ‘areas’ times each pixel’s distance from the axis squared.

So, if your femur is wider in the ML direction than in the AP direction, you can expect that IML is smaller than IAP.


Dear Michael,

thanks much for the explanation. I see the logic, the only problem is then that very often Ix and Iy are used to refer to rigidity along (or in the direction of) AP and ML axes.


That is confusing (and potentially wrong).

Could it be that Ix and Iy relate to moments around the x and y axes of the image, as they pass through the section’s centroid, with the specimen in some kind of standard orientation? It is quite a lot easier to calculate moments around known axes (especially the image coordinate system) than around principal axes, which I suspect is why those numbers are reported.

As far as the relation between rigidity and second moment goes, there is an assumption that the neutral plane of bending passes through a principal axis, which it usually doesn’t. Still, I is useful as an estimate of bending stiffness, especially if you are using it to compare cross-sectional geometry among specimens.


Yes, that is confusing. As per your question, well, yes and no. Most of the times Ix refers to AP direction but that plane if often placed vertically while Iy refers to ML direction which runs horizontal, so basically the axes are reversed from the standard understanding of x and y axes. I am referring to the standard practice of CSG in anthropology. The standardized position in a lot of papers is posterior facing down, anterior up, and medial to lateral sideways.

I am aware of the neutral plane of bending and principal axis though. Thanks


Right - super confusing use of notation. The rotation around the x (horizontal) image axis relates to the pixel distances measured in an AP direction away from the axis, so Ix = I “in the AP direction” but not IAP.


Exactly, I could not have described it better. Imagine then, that you have comparative data gathered from various published sources and you are not sure how this notation has been done in each of them since very often, it is taken that everybody knows what Ix and Iy denote, hence the description is not included in the paper.