Given three points placed on the pelvis one on each anterior

Given three points placed on the pelvis one on each anterior superior iliac spine (ASIS) and one on the sacrum create a coordinate system centered at the midpoint of the ASIS vector. The right ASIS is located at (25, 8, 35) the left ASIS is located at (-18, 7, 34) and the sacrum is located at (2, 16, 30).

Solution

The centroid of the three points the right ASIS (25, 8, 35), the left ASIS (-18, 7, 34) and the sacrum (2, 16, 30) is given by

\\frac{1}{3} [(25, 8, 35)+(-18, 7, 34)+ (2, 16, 30)

=\\frac{1}{3} (25-18+2, 8+7+16, 35+34+30)

==\\frac{1}{3} (9, 31, 99) =(3, 31/3, 33)

So the new coordinate system has origin at =(3, 31/3, 33)

So if the original coordinate system (x,y,z) has origin at (0,0, 0)

the new coordinate system (x’,y’,z’) has origin at =(3, 31/3, 33)

Thus

(x’,y’,z’) =(x-3, y-31/3, z-33)

Then in the new coordinate system

the right ASIS =(25-3, 8-31/3, 35-33)=(22,-7/3, 2)

the left ASIS (-18-3, 7-31/3, 34-33)=(-21, -10/3, 1)

and the sacrum (2-3, 16-31/3, 30-33)=(-1, 17/3, -3)