I didn't even need NASA, one flight on a passenger jet clearly showed me the curvature of the Earth.
People have known the world was round for thousands of years, they figured it out in Greece using sticks and shadows and they worked out the diameter at the time they discovered the nature of the world. Everyone knew, everyone studied it. Columbus studied it but got his math wrong and decided the world was ridiculously smaller than it is, and had a hard time convincing anyone he was correct until he met with Isabella and King Ferdinand II fundamentalist Catholic Christians that agreed with him. It's always the fundamentalist Christians getting it all wrong every time, these two also started the first Spanish Inquisition, so yea, fundamentalist.
Columbus never knew he didn't make it to India, so he was very much like an alcoholic, he didn't know where he was going, when he got there he didn't know where he was, when he got back he didn't know where he had been.
He was fortunate the New world was there, no one would have survived a voyage across an ocean that was bigger than the Atlantic and the Pacific and the Americas.