Three types of coordinate system are used frequently in SHAPE: the observer Cartesian system; the natural crystal system; and a clinographic system.
The observer system is fixed with respect to the display screen or paper of dot-matrix or pen-plotter copy. The origin is at the center of the screen or paper, and the x axis points directly toward the observer. The y axis points to the observer's right and the z axis vertically, both in the plane of the screen or paper.
The parameters of crystal faces are given in terms of the natural crystal axis system, although in the QSHAPE or Cartesian mode this is itself a Cartesian system. Once all the faces have been generated by symmetry, their coefficients are converted to Cartesian coefficients. Except for crystals using rhombohedral axes, the natural c axis is originally placed parallel to the observer z axis, and the natural b axis is aligned in the y-z plane (or in crystallographic terminology the a * axis is parallel to x ). If rhombohedral crystal axes are chosen, the three-fold symmetry axis is parallel to z , and the the vertical plane containing the first rhombohedral axis is aligned in the x-z plane.
This conversion defines the "original'' orientation of the crystal. After the location of corners etc., and before the initial display, the orientation may optionally be modified by the rotations in the Initial Orientation dialog in the Input2 menu, and by clinographic rotation, defined in the next paragraph.
Clinographic Viewing . One of the standard methods of drawing crystals is clinographic projection, in which the crystal is in effect viewed from above and slightly to the right of the a axis, or to be more precise, in towards the center along the vector [621]. If the option for clinographic viewing is chosen, after the faces, corners etc. are found, the crystal and its associated Cartesian system are rotated -18.4 degrees on the z axis, then 9.5 degrees on the y axis. This is not true clinographic projection, but the results are almost the same.
After the drawing appears, additional rotations may be made either on (1) specified vectors in the natural crystal system; (2) the observer Cartesian axes; or (3) if clinographic viewing is on, the rotated Cartesian axes, or clinographic system. The clinographic system does not rotate with the crystal; it is always related to the observer system by the two rotations given above.
If clinographic viewing is not on, the clinographic and observer systems have the same orientation, and if the natural crystal system is Cartesian or cubic all three systems have the same orientation.
Stereonet Viewing. When this display mode is selected (Modes menu) the crystal is viewed directly down the c axis, if in standard or original orientation. The a* axis is downward. Rotations made in standard mode are carried into this mode, but the clinographic rotations are not used.
Summary of axial systems:
1) Observer Cartesian system . Reference system, fixed with respect to the display screen or paper.
2) Natural or crystal system . The axes are the unit-cell edges. Axes may have unequal lengths and may be at angles of other than 90 degrees.
3) Clinographic system. Only applies when clinographic projection or viewing is on. Axes are those of the observer Cartesian system, but rotated in a certain manner to avoid a straight-on view of the crystal.
If Perspective viewing is not selected, the drawing is a direct projection in the x axis direction (observer system). If perspective viewing is selected, the corners are projected onto the y-z plane from a point on the x- axis at the specified distance in the positive direction (toward the observer).
In SHAPE, a, b and c (or A, B and C) refer to the natural crystal axes. The three axes will always have these names or a1, a2 and a3 if the axes are equal by symmetry. The letters x, y and z usually refer to the observer Cartesian axes. If it is necessary to refer to the clinographic system, this is mentioned explicitly.