You can see that the quaternion differences align along the centers of the helix, thus basically defining that axis. The actual placement of a quaternion vector in space in general is not defined. However, there is only a single position in space where vector representing the quaternion difference can be placed such that the projected angle (looking down the axis) from the alpha carbon of one residue to the next matches the quaternion's theta value. That turns out to be the exact position of the local helical axis. For an actual protein secondary structure helix, this is the center of the helix; for a sheet structure, this is right down the core of the sheet strand. Mathematically, Jmol's "straightness" parameter involves a (four-dimensiona, ahem) dot product of two adjacent quaternion differences. Straightness in Jmol is specifically defined as:



where dq_i and dq_i-1 are quaternions representing the local helical axis of amino acid residues i and i-1, respectively. The dot product is a four-dimensional dot product. If you were to color these models by straightness -- [go ahead and do that -- call up a console and enter color cartoons straightness] -- you would see that within the helices, where the vectors align nicely, the colors are bright red, and that for less regular regions, where the vectors don't align well, are white.

load =1a6g
quaternion difference draw;
color atoms translucent;
cartoons only;
set echo top left;
echo 1a6g;
color group;
color translucent;
write state 1a6ghelix.spt";

      

Sheets are helical in nature as well. It's just that they have larger angles -- close to 180 degress -- between residues. The commands given were:

load =1ag6
quaternion difference draw;
color atoms translucent;
cartoons only;
set echo top left;
echo 1ag6;
color group;
color translucent;
write state 1ag6sheet.spt";