Cutting Sloping Root Fiber

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When you think about the fiber low on a tree, you realize it must be curving -- slanting out toward the roots. What will happen when you cut it? Will you cut through all the fibers because you are cutting your backcut higher than the notch?

Let's let the pencils represent the grain of the tree down by the roots. Sure, it does grow at an angle (though the pencils have exaggerated the effect). Note the angle of the black pencil.

Here we've laid out the cuts to fell the tree in red, with the black pencil within the hinge. Notice that the black pencil is not cut. The pencil on the extreme right will be cut through -- but where is it? It's coming out of the center of the tree toward the direction of fall. Yes, it will be cut -- but since it is in the center of the hinge, its strength contributed nothing to the strength of the hinge's ability to prevent the tree from going sideways. (Remember that the strength of the fibers in the hinge are proportional to the distance of the fiber from the center cubed -- since the distance of the fibers near the center of the tree is zero or very small, they have practically no effect on the strength of the hinge.) And, of course, the step has a profound effect on preventing the tree from coming back over the stump -- even if all the hinge were cut. So having the backcut leave a step does not weaken the hinge appreciably. (And, yes, making a deeper undercut does help.) This result, surprising to some, can be further studied by investigating the mathematical topic, "descriptive geometry".