Could someone explain what M and M_1, M_2, M_3 all are meant to represent?

Gabe-Mitnick

I think the bold letters $a, b, c, d$ to the right are not related to the italic $a, b, c, d$ elements of the matrix $\mathbf{M}$. The bold equations are saying this: Say you have a point $a$ and transform it by some matrix $\mathbf{M_1}$ to get a point $b$, and then transform b by some matrix $\mathbf{M_2}$ to get a point $c$, etc. Then you can compose those three transformations into one matrix $\mathbf{M = M_1 M_2 M_3}$, whose effect is the same as applying the three transformations in order. So you could transform point a directly to point $d$ by multiplying by that combined matrix $\mathbf{M}$. Overall, it means that composing transformations is equivalent to matrix multiplication, and both of them are associative (but not commutative).

Staffjamesfobrien

Yes, the bold variables are distinct from the scalars. I can see how that might be confusing!

Could someone explain what M and M_1, M_2, M_3 all are meant to represent?

I think the bold letters $a, b, c, d$ to the right are not related to the italic $a, b, c, d$ elements of the matrix $\mathbf{M}$. The bold equations are saying this: Say you have a point $a$ and transform it by some matrix $\mathbf{M_1}$ to get a point $b$, and then transform b by some matrix $\mathbf{M_2}$ to get a point $c$, etc. Then you can compose those three transformations into one matrix $\mathbf{M = M_1 M_2 M_3}$, whose effect is the same as applying the three transformations in order. So you could transform point a directly to point $d$ by multiplying by that combined matrix $\mathbf{M}$. Overall, it means that composing transformations is equivalent to matrix multiplication, and both of them are associative (but not commutative).

Yes, the bold variables are distinct from the scalars. I can see how that might be confusing!