问题描述:

I'm trying to make sense of all the ways I can use the Rotation matrix (sorry for unreadability):

R(x) = [ cos(x) -sin(x) ; sin(x) cos(x) ]

There are two types of coordinate systems (CS) that you can have:

CS1 --> positive x-axis going right and positive y-axis going up which is apparently the standard orientation people use.

CS2 --> positive x-axis going right and positive y-axis going down.

What I'm confused with is the sign of angle x when plugged into R(x). Depending on the sign of the numerical, the angle either makes a counter-clockwise or clockwise rotation (on an arbitrary vector) but for which CS orientation and how would the other orientation behave?

Also, if you could explain a bit more than what wikipedia has on alibi and alias transformations, that would be great. Which would be best to think of rotations in? If I think in the alias approach, would I have to use the opposite sign for angle x.

This is what I'm referring to:

http://en.wikipedia.org/wiki/Rotation_matrix#Ambiguities

Thanks!

Would this be right...

Making use of R(x).

The vector undergoes a counter clockwise rotation of 'x' in CS1 (alibi). This relates to a clockwise rotation of CS1 of an angle 'x' (with the vector now having new coordinates). If x is negative then counter-clockwise becomes clockwise (alibi) and clockwise becomes counter-clockwise (alias)

If we were to use CS2...the above can be taken but with again swapping counter-clockwise with clockwise and vice versa.

In standard cartesian coordinates (what you've called "CS1") that matrix will cause a *counter clockwise* rotation to the *vector* (i.e. "alibi")

In the inverted Y axis case it'll be clockwise.