This paper is related to a detailed study of the ancient astronomical instrument that dates back to the 14th century. It is a centerpiece of the ancient observatory built in Samarkand, Uzbekistan. The famous ancient astronomer and mathematician, Ulugbek, designed and built it. This observatory contained one of the most precise medieval instruments in the world. It used to be housed in a 3-story building with a basement. Unfortunately, the building was destroyed; currently, only the underground part is preserved. Ulugbek’s studies were well known throughout the rest of the world, and as such, his name was mentioned by many famous medieval scientists. A terrestrial laser scanner was used to document the monument’s geometry accurately. As a result, a 3D digital twin of the monument was created in the form of a point cloud. The latter was studied from a few points of view. First, the radius of the main circular surface was estimated based on the least-squares fitting procedure. Second, the location of the circle center was estimated, including its elevation above the grade level. Third, deviations of the main instrument from the cylindrical surface were calculated. The color maps showing the deviations were generated. All this information is used to evaluate the quality of ancient construction, and the ancient science used to design and build it. This information will be used to monitor the heritage monument and assess its deterioration over time.