1. Relationship between the third Euler angle and the roll angle
第3オイラー角とロール角の関係
Relationship between the third Euler angle and the roll angle
North Pole
Initial attitude:ZYZ Euler angles (0,0,0)
Satellite Z-axis points North pole
Satellite X-axis points Spring equinox RA
DEC
Spring Equinox Z Z X Y X
No rotation around Z
90 deg around Y
ZYZ Euler angles (0,90,0)
Satellite Z-axis points Spring equinox
Satellite X-axis points South pole

2. Relationship between the third Euler angle and the roll angle
X-ray telescope is like a pin-hole camera, such that the “mirror image” is projected on the detector plane.
In order to reproduce the correct sky image, it is necessary to see the image from behind, or “flip” the image.
Usually, we define the detector X- and Y-axis (DETX, DETY), such that
DETX=SATX　
DETY=-SATY.
第3オイラー角とロール角の関係
Relationship between the third Euler angle and the roll angle
North Pole RA
DEC
Spring Equinox RA
DEC
Detector plane
DETX
DETY
X-ray telesope Z Y
DEC RA
Relationship of sky coordinate and detector coordinate
Euler angle :(0,90,0)
North
DETY
DETX X
East
ZYZ Euler angles (0,90,0)
Satellite Z-axis points Spring equinox
Satellite X-axis points South pole

3. Relationship between the third Euler angle and the roll angle
第3オイラー角とロール角の関係
Relationship between the third Euler angle and the roll angle
North Pole
ZYZ Euler angles (0,90,0)
Satellite Z-axis points Spring equinox
Satellite X-axis points South pole RA
DEC
Spring Equinox RA
DEC
Rotate y around Z axis
ZYZ Euler angles (0,90,y)
Detector Plane
DETY
DETY y Z Y
North
DETX
DETY
DETX
DETX
DETY
DETX
DEC y X
Normally, “roll-angle” is defined as the angle from North to DETY axis measured in counter-clockwise
Roll-angle
East RA
Euler angle:(0,90,0)
Euler angle:(0,90, y)
Now, we see that
Roll-angle=90-y