question on rotational transform matrix, I

cncnewbee

Hi,
I'm reading a book called Robotics, written by Tadej Bajd on my own to learn about robotics and have no one else to put my questions other than to the forums.

Here the writer on 11th page writes:

"By considering the similarity of triangles in Figure 2.3, it is not difficult to derive
the following two equations
x...."

where I can't get which triangles to look for similarities as there could be various (I'm no expert in math) and also, can't get how the formula is derived. Please explain

screen shot of book:
i46.tinypic.com/30i9nih.png

Thank you in advance

Quantumjump

look at the attachment, I have added new points and lines to your picture to make the proof clear.

We have:


x= AB+BC
but BC = x' cos (beta)
and AB = TM sin beta + Mx' sin beta
= (TM+Mx') sin beta
= Tx' sin(beta)
= z' sin (beta) (because Tx' parallel to z')

hence x= x' cos beta + z' sin beta

Similarly,

z = CK -zK
but CK= z' cos(beta)
and zK= z'D= Tz'sin(beta)= x' sin(beta)

hence

z= z'cos beta - (x'sin beta)

Attached Thumbnails

 

cncnewbee

Thanks you very much! Now I understand it clearly!