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
keenan
@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!
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!