Which methods are used to represent the curved surface of the earth on a flat surface?

The correct choice is projections because they are specific mathematical transformations that allow the representation of the three-dimensional curved surface of the Earth on a two-dimensional flat surface, such as a map. Different types of projections are designed to minimize certain types of distortions that can occur when depicting the Earth's surface, which may include distortions in area, shape, distance, and direction. For instance, the Mercator projection preserves angles but distorts area, while the equal-area projections maintain the relative sizes of regions but may distort shapes.

The other options do not accurately describe the specific techniques used for this task. Illuminations typically refer to the lighting conditions in a visual display or graphic representation rather than mapping. Representations is too broad and vague in this context, as it does not convey the specific method used for translating geographic information onto a flat surface. Models, while they can represent physical processes or phenomena (such as a 3D model of landform), tend to depict objects or systems rather than serving as a systematic approach to represent geographical spaces specifically in map-making. Thus, projections are the definitive choice when discussing how to convert the curved surface of the Earth to a flat representation.