## Kinematic View of Alignment

An optical surface is a three-dimensional object of finite size. To locate such an object in space, one must first find some point (usually its geometric center) and move that point's three linear coordinates. Once its center is placed, two remaining orientation angles must be fixed to specify its position. Actually, mathematicians describe three such rotation angles, but symmetry usually allows one to be ignored. For alignment, this hidden angle is a final pointless rotation around the symmetrical optical axis.

The three coordinates of the center as well as these three angles make up the six degrees of freedom necessary to know the position and orientation of a solid object precisely. The mounting points are well designed if they are sufficient to fix the object's location but not to overconstrain it. Three-legged stools are stable on the roughest floors because their three-point design is both necessary and sufficient. Stiff four-legged stools usually rock because the design has too many supports. The fourth leg cannot be precisely made to occupy the plane of the floor. This difficulty also ac counts for the reason that optics mounting cells are so often multiples of three points.

Alignment is usually separated into two independent tasks. First, the centers of all optical elements are placed on a single axis line and properly spaced. Second, the tilts of each optical element are oriented so that each becomes a circle of revolution around the axis. Thus, if one twirls the optical system around the axis, it looks the same. The placement of the center on-axis is called centering, and obtaining the correct tilt orientation is called squaring-on by some authors.1