The invariator principle is a measure decomposition that was rediscovered in local stereology in 2005 and has since been used widely in the stereological literature. We give an exposition of invariator related results where existing formulae are generalized and new ones proposed. In particular, we look at rotational Crofton-type formulae that are obtained by combining the invariator principle and classical Crofton formulae. This results in geometrical quantities represented as averages over weighted Crofton-type integrals in linear sections. We refer to these weighted integrals as measurement functions and derive several, more explicit representations of these functions. In particular, we use Morse theory to write the measurement functions in terms of critical values of the sectioned object. This is very useful for surface area estimation.

Keywords: Local stereology; invariator principle; rotational Crofton-type formulae; Morse theory; Hadwiger's index; surface area estimation