We first outline an overall design philosophy for rigid geometric manipulations, then examine a manipulation's characteristics: nesting, scope, pivot constraints, and axis,uonstraints. We show how a mnemonic notation helps us explain how a simple matrix operation can make manipulations (both rotations and translations) nested within rotations easy to control. Finally, we mention some practical considerations to increase calculation speed and control numerical error. An appendix collects formulas useful for working with rotations.Design Philosophy A computer graphics system is a realization of a mathematical model. The model is. an abstraction of some part of the real universe, and need not faithfully reproduce every aspect of the original. The system designer is free to select and include the attributes of the actual world which he decides are relevant to the task aided by the system. Because the system is not restricted to capabilities possible with physical objects, the graphics user can be given power and assistance impossible with a mechanical realization of the application's model.Freed from the limitations of the physical structure underlying the mathematical model of the application, the system designer is obligated to judge the user's requirements carefully. The system user will generally be unaware of the existence of the interposed mathematical model, and will be viewing and manipulating an object which is apparently physically real. This impression must be encouraged, and the designer should take care not to destroy the illusion without overwhelmingly good cause. Real objects move the direction they are pushed. A computer-driven rotation system can be programmed to provide the user with facilities of arbitrary complexity. It is the designer's responsibility to conceal this complexity from the user, while transmitting as much useful power as possible.Birmingham & Taylor see the designer as engineer and psychologist [1954]:It is through the controls, the levers, knobs, handwheels, and switches that human response takes its effect upon the mechanism. On the other hand, it is through the displays, the dials, light panels, cathode ray tubes, horns, buzzers, and cross-pointer indicators that the operator is presented with information concerning the activities of the mechanism....In general, it is the task of the engineering psychologist to assist the engineer in designing the displays, controls, and intervening mechanism so that the output of the man-machine system is optimized, while the human operator requirements in regard to native ability and training am minimized.