In a rigid body system, two coordinate systems can be related using a single 4 x 4 transformation matrix to describe any rotations and/or translations necessary to transform coordinates from the related coordinate system (frame of reference) to the primary system. The equation performing the transform from a point (X’,Y’,Z’) in the related coordinate system to a point (X,Y,Z) in the current coordinate system can be shown as follows, where for homogeneous transforms M
_{
41
}
= M
_{
42
}
= M
_{
43
}
= 0 and M
_{
44
}
= 1:

X M
_{
11
}
M
_{
12
}
M
_{
13
}
M
_{
14
}
X’

Y = M
_{
21
}
M
_{
22
}
M
_{
23
}
M
_{
24
}
x Y’

Z M
_{
31
}
M
_{
32
}
M
_{
33
}
M
_{
34
}
Z’

1 M
_{
41
}
M
_{
42
}
M
_{
43
}
M
_{
44
}
1