What conceptual model is used to project the locations of all stars for observing daily paths?

The Celestial Sphere, an imaginary, vast sphere surrounding the Earth onto which all stars are projected.

To accurately describe and understand the paths stars appear to follow, astronomers utilize the concept of the Celestial Sphere. This is an imaginary, massive sphere envisioned to surround the Earth entirely. All observed stars are treated as if they are fixed points projected onto the inner surface of this sphere. When we watch the stars move across the sky during the night, we are observing their apparent movement along specific circles traced on this conceptual sphere. This model allows for the calculation and visualization of key coordinates like declination and the location of the Celestial Poles relative to any terrestrial observer.