Matrix Analysis of Gimbaled Mirror and Prism Systems*

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Abstract
This paper shows a general matrix method for analyzing systems of gimbaled mirrors, prisms, and ray projectors. The method shows how to find the gimbal angles which are necessary to ensure that a ray traversing the system obeys some requirement of collimation or orientation. Gimbal angles are considered to be the important measurement parameters, and more meaningful than the usual direction cosines. The methods apply to a large class of optical systems, including those in which gimbals are not specifically used. The gimbal concept will be found very useful in describing the orientation of images in prism systems, such as the peculiar behavior of a penta prism. A gimbal operator is defined and used throughout. The appendix contains a formal development of the mirror as a second-rank tensor together with important properties of prisms as products of mirror matrices.

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