Basic structure and principles of roof prisms
Roof surface: one of the reflective surfaces of a prism is designed as a “roof” (similar to the angle of two sloping surfaces of a roof, usually 90°), and these two sloping surfaces flip the direction of the light through secondary reflections, thus correcting for left-right inversion of the image.
Optical path adjustment: after many times of total reflection inside the prism, the light ray outgoing direction is parallel to the incoming direction, but the optical path is collapsed, which shortens the length of the instrument.
Phase correction: high-end roof prisms are coated with a phase coating to solve the problem of optical range difference caused by beam splitting on the roof surface, and to improve imaging contrast and resolution.

Comparison of roof prism and paul prism
Characteristic	Roof Prism	Paul Prism
Volume	Compact, straight design	Large volume and wide objective lens spacing
Imaging Quality	Dependent on coating technology	Naturally high contrast
Cost	Relatively high	Relatively low
Typical Application	High-end portable binoculars	Conventional telescopes (for navigation)