# Coordinate Rotation in a Cartesian 3-Space

• ForMyThunder
In summary, the conversation discusses the process of deriving a set of equations for a new Cartesian coordinate system after a rotation of an original coordinate system. The steps involved transforming the coordinates into spherical coordinates, rotating them by angles p0 and q0, and simplifying and substituting the original values. The conversation also mentions the need for a coordinate-independent set of equations and asks for any simpler equations for general rotations in three dimensions. It is suggested that spherical coordinates may only be useful for rotations around the z-axis.

#### ForMyThunder

I have been trying to derive a set of equations for a new Cartesian coordinate system after a rotation of an original coordinate system. This is what I did:

1) I transformed the Cartesian coordinates (x,y,z) into spherical coordinates (r,p,q):
x= r cos(q) cos(p)
y= r cos(q) sin(p)
z= r sin (q)

2) The coordinates are to be rotated by angles of p0 and q0 so that:

p'= p-p0
q'= q-q0
r'= r

3) Substitution:

x'= r cos(q-q0) cos (p-p0)
y'= r cos(q-q0) sin (p-p0)
z'= r sin(q-q0)

4) Simplifying and substituting the original values of x, y, and z:

x'= (r cos(q) cos(q0) + r sin(q) sin(q0)) (cos(p) cos(p0) + sin(p) sin(p0))
= r cos(q) cos(q0) cos(p) cos(p0) + r cos(q) cos(q0) sin(p) sin(p0) + r sin(q) sin(q0)cos(p) cos(p0) + r sin(q) sin(q0) sin(p) sin(p0)
= x cos(q0) cos(p0) + y cos(q0) sin(p0) + z sin(p0)cos(p) cos(p0) + z sin(q0) sin(p)
sin(p0)

y'=(r cos(q) cos(q0) + r sin(q) sin(q0)) (sin(p) cos(p0) - cos(p) sin(p0))
=r cos(q) cos(q0) sin(p) cos(p0) - r cos(q) cos(q0) cos(p) sin(p0) + r sin(q) sin(q0)
sin(p) cos(p0) - r sin(q) sin(q0) cos(p) sin(p0)
=y cos(q0) cos(p0) - x cos(q0) sin(p0) + z sin(q0) sin(p) cos(p0) - z sin(q0) cos(p)
sin(p0)

z'= r sin(q) cos(q0) - r cos(q) sin(q0)
= z cos(q0) - r cos(q) sin(q0)

This is as far as I got, but in the equations for x and y, there are still some sin(p)'s and
cos(p)'s left in there which cannot be evaluated without the original coordinates and I want to find a coordinate-independent set of equations so that the same equations can be used for every point in the original coordinate system.

My question is: Is there any way to get rid of these sine's and cosine's? Or do you see anything that I could have done wrong or different?

If you know any simpler equations, please send them to me.

Thanks.

The spherical coordinates are useful only for the rotation around z-axis. With your notation, that means translation of variable p. For more general rotations, the spherical coordinates are making things only more difficult.

Here's something about rotations in three dimensions: Elements of SO(3)?