The projection moiré technique enables quantitative measurements of the shape of an opaque curved object surface in space. In contrast to the actual literature, where the interpretation of moiré fringes is mostly limited to particular geometrical configurations, we present a strong mathematical tool that makes it possible to accurately analyze more general and complicated geometrical cases. We therefore introduce a mathematical model based on the calculation with projectors and use tensor calculus to deduce the general equations of projection moiré. Emphasis is put on relative moiré, which is used in most experiments, and on difference moiré, which is generally used to calibrate optical systems. The concept of a sensitivity vector, which comes essentially from holographic interferometry, is also introduced. Using a computer-based image processing system, a numerical experimental verification of the obtained theoretical tensor equations is performed. Simultaneously, we describe how to calibrate an optical setup and gain evidence of a few nonlinear effects. This shows which parameters of the setup are of importance and should be carefully controlled. Moreover, we treat some well-known special cases to demonstrate the agreement with the existing literature.