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Shouldn't this be the magnitude of the cross product?

Disregard, realized I completely missed the point of this slide... whoops

Though I'm still having trouble understanding the meaning of the derivative.

The change in area of the triangle with respect to vertex p is a function of the normal and edge e. Which normal are we talking about here? The normal of the triangle? Edge? Either way, I'm not seeing how that tells us anything


@adam The vector N is the unit normal of the triangle, and is just used here for convenience to express the vector orthogonal to the base of the triangle, via the cross product N x e with the vector parallel to the base---and whose magnitude is equal to the base length. It's pretty easy to see that this direction, N x e, is the direction of motion for p that will increase the area quickest, since area is 1/2 base times height. Likewise, we know that the change in area for a unit change in height is 1/2 times the base length |e|, hence the factor 1/2. Glad to clarify if there's still confusion!