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I know that a geographic coordinate systems are inherently three dimensional, while they are usually represented by two coordinates (lat and long) because the height is about the same across the surface.

These 3D coordinates are eventually projected to a 2D screen when we use a GIS software. My question is doesn't that mean they are practically projected? Isn't the only way a coordinate system is geographic, and not projected is that the map information is on a 3D sphere (like on a physical globe or Google Earth) and not on a 2D surface?

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It is possible to work only with geographic (lat-long) coordinates, do all the calculations on a sphere and never project them. However, you are correct: Whenever we see a curved surface (a portion of the Earth) on a flat screen or paper there has been a map projection (or just "projection"). Note also that, sometimes, when we see only a small region, we may not know or use geographical (curved) coordinates but use 2D coordinates in a local plane coordinate system directly.

There are very many questions on map projections. Maybe start with If Geographic Coordinates are unprojected coordinates, how can GIS softwares display such unprojected data on a plane?

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