2011-03-20, 07:00 AM
(This post was last modified: 2011-03-21, 04:21 AM by 2147483647.)
The first "formula" should be very intuitive. Any vector can be broken down into a number of "components" that can be summed together. In this case, the vector is split up into two components: one tangential to a different function and the other normal to the function.
a = aT + aN
This makes sense, since dotting a and T gives the projection of a onto T, and likewise for a and N. The second formula isn't really a formula. It's a trick to find the N vector, which is usually really annoying to find. The original formula for the N vector is N = T'/|T'|. As you can tell from the formula, it takes two steps to solve for it from a. That's why the second formula is used: solving N from a directly is much faster.
I'm not really sure what's meant by your third formula. Maybe your book provides a proof or example?
Edit: Very elegant, Russt.
a = aT + aN
This makes sense, since dotting a and T gives the projection of a onto T, and likewise for a and N. The second formula isn't really a formula. It's a trick to find the N vector, which is usually really annoying to find. The original formula for the N vector is N = T'/|T'|. As you can tell from the formula, it takes two steps to solve for it from a. That's why the second formula is used: solving N from a directly is much faster.
I'm not really sure what's meant by your third formula. Maybe your book provides a proof or example?
Edit: Very elegant, Russt.