Possibly reversed results in 2nd Moment of Area

Hi all,

I’m looking into the 2nd moment of area of cross-sections of rodent incisors using .tif files. The shapes of these cross-sections are effectively ellipsoid and much taller than they are wide and the Imax and Imin results reflected this. Though the principal axes set by BoneJ were close in orientation to what I was interested in (the major axis was very very close to being perfectly cranial-caudal in orientation) I also used the orientation tool to set the orientation at 0.0 degrees as that is the direction the load will be coming from during a bite.

When I compared the results from the Imax and Imin with the ICrCa and IML I was surprised to find the results of the ICrCa and IML appeared swapped from what I would expect from what I’d seen from the Imax and Imin.

I.e.
Imin: ~35,000
Imax: ~120,000
ICrCa: ~35,000
IML: ~120,000

Bearing in mind the Cranial-Caudal axis was set to 0.0 I had expected it to show similar results to those from the major axis reported in the Imax. Is there something I’m missing that’s causing this? I expected some differences, but it’s almost as if I’d input the orientation to 90 degrees of what I actually selected.

Sorry for the essay!

Best wishes,

Phil

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The main point of confusion is about the relationship between major and minor principal axes and which of the min/max moments of area is around each one. Is the moment the one that happens around that axis, or the one that is calculated in the direction of that axis? How did Imin and Imax get their names?

EDIT (Oct 2020): this page has a great explanation and BoneJ has been “put back” into a state that is consistent with its explanation. Briefly, major ≃ max and minor ≃ min. To make the nomenclature convention more consistent, the subscript Iwhatever relates to the whatever axis around which the moment is calculated, for all cases, where whatever is x, maximum (long) principal axis, craniocaudal direction, or, well, whatever.

The effect is a bit counter-intuitive, which is why it’s confusing: the moment is calculated as the sum of the distances away from (and not in the direction of) the axis squared times a small area. So Ix is calculated by adding up squares of distances in y times little areas.

Meanwhile Imin and Imax are calculated around shorter (minor, minimum) and longer (major, maximum) principal axes respectively, so that Imin > Imax

In general, you can expect a small second moment of area to relate to your longer axis subscript, and a complementary large second moment of area relating to your shorter axis subscript. This is intuitive once you realise that the second moment of area is calculated by adding up distances away from the axis squared: the pixels of the cross section are on average closer to the long axis than they are to the short axis.

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Ah, I see! So it’s effectively a matter of naming conventions that had me confused.That makes much more sense, thank you!

I just wanted to let you know that I think you made a typo in the bonej documentation in relation to Imin and Imax. You said in this post that Imin is calculated around the major principal axis, but in the documentation of slice geometry, you have written the following,
Imin: Second moment of area around minor axis
Imax: Second moment of area around major axis

You may want to change this on the website to avoid further confusion.

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Done

EDIT (Oct 2020). Need to put this back to how it was before, which was correct the first time.

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