1. Chapter 3 Transformation of Coordinates
36. Translation.
Let the coordinates of a point P with respect to a set of rectangular axes OX,OY,OZ be (x,y,z) and with respect to a set of axes O’X’,O’Y’,O’Z’, parallel respectively to the first set, be (x’y’z’). If O’=(h,k,l) referred to the axes OX,OY,OZ, we have

2. Z’ Z
O’=(h,k,l) X’ O X Y’ Y

3. Rotation.
Let the coordinates of a point P, referred to O-XYZ, be (x,y,z), and referred to O’X’Y’Z’, be (x’,y’,z’). Let the direction cosines of O’X’, O’Y’, O’Z’,referred to O-XYZ, be respectively,

4. Namely,coordinates x’,y’,z’ and x,y,z satisfy

5. Since the related matrix A is an orthogonal matrix, the inverse is its transpose matrix

6. Therefore,

7. Rotation and reflection of axes
Rotation about an axis. For example z axis, then z’=z and thus

8. It is called a rotation or reflection if |G|=1 or |G|= -1. Where

9. For two coordinate systems, OXYZ and O’X’Y’Z’, consider the intersection line of XOY and X’OY’, NN’, we make the following steps:
O-XYZ to (OXON)
to (OZOZ’)
to (ONOX’)
Notice that steps 1,2 and 3 are rotations, and

10. For O-XYZ to , it is simply a rotation with Z axis, from plane analytic geometry,

11. For to , it is simply a rotation with N axis,

12. For to , it is simply a rotation with OZ’ axis,

13. If O-X’Y’Z’ cannot be obtained from O-XYZ by rotation, the sign of y’ should be changed. These formulas are known as Euler’s formulas

14. Notice: Degree of an equation unchanged by transformation of coordinates
Since they are all linear transformations
Exercises: P43, No. 2,3, 6