  Symmetry detection and discrimination are of fundamental meaning in science,
technology, and engineering. This paper introduces reflection invariants and
defines the directional moment to detect symmetry for shape analysis and object
recognition. And it demonstrates that detection of reflection symmetry can be
done in a simple way by solving a trigonometric system derived from the
directional moment, and discrimination of reflection symmetry can be achieved
by application of the reflection invariants in 2D and 3D. Rotation symmetry can
also be determined based on that.The experiments in 2D and 3D, including the
regular triangle, the square, and the five Platonic objects, show that all the
reflection lines or planes can be deterministically found using directional
moments up to order six. This result can be used to simplify the efforts of
symmetry detection in research areas, such as protein structure, model
retrieval, inverse engineering, and machine vision etc.
